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# Quantum vs classic in NP-hard problems

Is there any quantum algorithm (algorithm for quantum computers) for any NP-hard problem that has better runtime than the best known classic algorithm's runtime?

It is conjectured that the complexity of SAT on $$n$$ variables is $$\tilde\Omega(2^n)$$ (a version of this is SETH, the strong exponential time hypothesis). In contrast, Grover's algorithm solves it in $$\tilde O(2^{n/2})$$.
On the other hand, it is conjectured that quantum computers cannot solve NP-hard problems in polynomial time, that is $$\mathsf{NP} \not\subseteq \mathsf{BQP}$$.


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Time and Distance : Aptitude Question & Answer

Directions of questions: In the following questions, Choose the correct answer from the given options and click the number of your choice.

 Quiz Name Time and Distance Quiz Category Aptitude Practice Question Type Multiple Choice Questions No. of Questions 30

format: plain


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Last edited by Gozshura
Thursday, July 16, 2020 | History

4 edition of Linear Triatomic Molecules found in the catalog.

# Linear Triatomic Molecules

## by Guy Guelachvili

• 12 Want to read
• 21 Currently reading

Published by Springer .
Written in English

Subjects:
• Electricity, magnetism & electromagnetism,
• Mathematics and Science,
• Science,
• Science/Mathematics,
• Nuclear Physics,
• Physics,
• Science / Physics,
• band centers,
• energy levels,
• intensities,
• magnetic moments,
• molecular constants,
• spectroscopic constants,
• wavenumbers,
• Magnetism

• The Physical Object
FormatHardcover
Number of Pages168
ID Numbers
Open LibraryOL12775140M
ISBN 103540433724
ISBN 109783540433729

A triatomic molecule can have a linear configuration, as does CO 2 (Fig. Pa). or it can be nonlinear, like H 2 O (Fig. Pb). Suppose the temperature of a gas of triatomic molecules is sufficiently low that vibrational motion is negligible. 2. Molecular Parameters and Energy Level Formulation The treatment of the rotational energy levels and transition frequencies for polyatomic molecules is more complicated than the formulations applied to diatomic molecules [].However, since triatomic molecules are the simplest class of polyatomic species, they demonstrate the simplest spectra and interactions .

The download non-linear triatomic molecules: part 1β of New York - which at the action let about of Manhattan - had not well annual economies, and the enterprise stopped swallowed as a heritage to RFBoy in Manhattan while dealing myth in Brooklyn.   For a linear triatomic molecule the problem is the same as for a diatomic molecule. The energy levels are. The degeneracy is. For a linear molecule the moment of inertia is. where d j is the distance of the jth mass from the center of mass. The center of mass is given by. where x j, y j, and z j are the Cartesian coordinates in an arbitrary.

Read Jess Brewers response. Linear molecules and non-linear molecules have different degrees of freedom. Linear molecules have 5 degrees of freedom, non linear, six. For linear molecules, rotation about the molecular axis does not produce any chan.   It depends if the triatomic gas in question is linear or not. Let’s take a look at two different triatomic gases For example, in the case of water vapour, H2O, a triatomic gas, is not linear because of the two lone pairs on the Oxygen. This asymm.

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### Linear Triatomic Molecules by Guy Guelachvili Download PDF EPUB FB2

Buy Linear Triatomic Molecules: CHSi (HCSi), ClHNe (NeHCl), Cl2H- (ClHCl-), FHO (FHO), FHO+(FHO+), F2H- (FHF-), FN2+ (FNN+), HN2+ (HNN+), HNSi (HNSi), Science and.

Buy Linear Triatomic Molecules: BC1H+ (HBC1+) COSe (OCSe) (Landolt-Börnstein: Numerical Data and Functional Relationships in Science and Technology - New Series) (Vol 20) on FREE SHIPPING on qualified orders.

Linear Triatomic Molecules: C2H (CCH) (Landolt-Börnstein: Numerical Data and Functional Relationships in Science and Technology - New Series (20B6)) (part 6) st Edition by G.

Guelachvili (Author), K.N. Rao (Author). Linear Triatomic Molecules: COO+ (OCO+), CFeO (FeCO) CNO- (NCO-) (Landolt-Börnstein: Numerical Data and Functional Relationships in Science and Technology. Linear Triatomic Molecules Book Subtitle CHSi (HCSi), ClHNe (NeHCl), Cl2H- (ClHCl-), FHO (FHO),FHO+(FHO+), F2H- (FHF-), FN2+ (FNN+), HN2+ (HNN+), HNSi (HNSi), HOSi+ (HOSi+), N2S (NNS), NOP (PNO), NOSi (NSiO), NOSI (SiNO), NOSI (SiON)Brand: Springer-Verlag Berlin Heidelberg.

Volume II/20 "Molecular Contants mostly from Infrared Spectroscopy" is a recent Landolt-Börnstein publication series. It is made up of several subvolumes (A, B, C, D) with comprehensive compilation of critically evaluated molecular constants of diatomic (A), linear triatomic (B), other triatomic (C) and other polyatomic (D) : Springer-Verlag Berlin Heidelberg.

Volume II/20 provides critically evaluated data on free molecules, obtained from infrared spectroscopy and related experimental and theoretical investigations. The volume is divided into four subvolumes, A: Diatomic Molecules, B: Linear Triatomic Molecules, C: Nonlinear Triatomic Molecules, D.

Volume Linear Triatomic Molecules book provides critically evaluated data on free molecules, obtained from infrared spectroscopy and related experimental and theoretical investigations.

The volume is divided into four subvolumes, A: Diatomic Molecules, B: Linear Triatomic Molecules, C: Nonlinear Triatomic Molecules, D:Brand: Springer-Verlag Berlin Heidelberg. Linear Triatomic Molecule. In Chapter 2, Sectionwe discussed a rigid triatomic molecule. Now we are going to discuss three masses held together by springs, of force constants $$k_1$$ and $$k_2$$.

We are going to allow it to vibrate, but not to rotate. Suppose the molecule is the linear molecule OCS, and the three masses 12 and 32 respectively, and, from infrared spectroscopy, it is determined that the moment of inertia is (For this hypothetical illustrative example, I am not concerning myself with units).

In that case, equation $$\ref{eq}$$ becomes. Volume II/20 "Molecular Constants mostly from Infrared Spectroscopy" is a recent Landolt-Börnstein publication series. It is made up of several subvolumes (A, B, C, D) with comprehensive compilation of critically evaluated molecular constants of diatomic (A), linear triatomic (B), other triatomic (C) and other polyatomic (D) molecules.

Abstract. Triatomic molecules are most frequently bent; less common are the linear and (the quite rare) triangular arrangements. Stable molecular geometries with precise bond angles and bond lengths can be predicted by determining quantum mechanically the molecular energy as a function of atomic positions and then finding the minima in this by: 2.

Volume II/20 "Molecular Contants mostly from Infrared Spectroscopy" is a recent Landolt-Börnstein publication series. It is made up of several subvolumes (A, B, C, D) with comprehensive compilation of critically evaluated molecular constants of diatomic (A), linear triatomic (B), other triatomic (C) and other polyatomic (D) : \$   Construct MO diagrams for simple linear triatomic molecules and/or compounds.

Book: General Chemistry Supplement (Eames) Using the electron domain model we predict that it will be linear. Be has 2s and 2p orbitals, and it is in the middle. H has 1s orbitals; there are 2 H atoms on the outside. Guy Guelachvili is the author of Linear Triatomic Molecules ( avg rating, 1 rating, 0 reviews, published ), 16o12c16o ( avg rating, 0 ratings, 5/5(1).

Linear Edit. Linear triatomic molecules owe their geometry to their sp or sp3d hybridised central atoms. Well known linear triatomic molecules include carbon dioxide (CO 2) and hydrogen cyanide (HCN). Xenon difluoride (XeF 2) is one of the rare examples of a linear triatomic molecule possessing non-bonded pairs of electrons on the central atom.

About the book. Description. Molecular Spectroscopy: Modern Research, Volume III is a collection of papers presented at the 40th Annual Molecular Spectroscopy Symposium, held at the Ohio State University. The contributors of this seven-chapter text cover the significant advances in molecular spectroscopic research and their application in chemistry.

If the triatomic molecule is linear how many rotational degrees of freedom are there. How about if it is not linear - like water.

It is commonly stated in senior secondary and introductory tertiary texts that monatomic gasses have 3dof per molecule, diatomic have 5, and triatomic+ have 7.

Book for study and reference.: 1. as, “ Molecular structure and spectroscopy”, second edition, practice – Hall of Linear Molecules: In this type, all the atoms of the molecules are arranged in a straight line.

For example, The moment of inertia for the end to end rotation of the linear triatomic molecule will 1 B. We discuss here linear triatomic molecules A–B–C, but the arguments given will hold also for linear molecules with more than three nuclei.

We describe the bending motion in terms of the instantaneous bond angle supplement ρ ¯ (= π - ∠ (A–B–C)), which measures the bond-angle deviation from linearity during the bending by: 9. Linear molecules.

As we stated above, a simple triatomic molecule of the type $$AX_2$$ has its two bonding orbitals ° apart, producing a molecule that we describe as having linear geometry. Examples of triatomic molecules for which VSEPR theory predicts a linear shape are BeCl 2 (which, you will notice, doesn't possess enough electrons to conform to the .Polyatomic Molecules: Results of Ab Initio Calculations describes the symmetry of polyatomic molecules in ground states.

This book contains 12 chapters that also cover the excited and ionized states of these molecules. The opening chapter describes the nature of the various ab initio computational methods.

The subsequent four chapters deal with. Purchase Basic Molecular Spectroscopy - 1st Edition. Print Book & E-Book. ISBNLinear Polyatomic Molecules Polyatomic Molecules Conclusions References Further Reading Problems Linear Triatomic Normal Modes 5 Raman and Electronic Spectra Essential TheoryBook Edition: 1.


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# Heat conduction

$\mathfrak{D}$

Conduction is heat transfer across a stationary medium, either solid or fluid. For an electrically nonconducting solid, conduction is attributed to atomic activity in the form of lattice vibration, while the mechanism of conduction in an electrically-conducting solid is a combination of lattice vibration and translational motion of electrons. Heat conduction in a liquid or gas is due to the random motion and interaction of the molecules. For most engineering problems, it is impractical and unnecessary to track the motion of individual molecules and electrons, which may instead be described using the macroscopic averaged temperature.

One-dimensional heat conduction, plane and radial walls, extended surface, bioheat equation, two-dimensional heat conduction, and conduction from buried object.
Lumped analysis, finite slabs, cylinders, spheres, semi-infinite body, and multidimensional conduction.
Discretization of computational domain and governing equations, one-dimensional steady and unsteady state conduction, multi-dimensional unsteady-state conduction, and solution of algebraic equations
Classifications, boundary conditions at interface,exact solutions, integral approximate solution, numerical solution, binary system, and melting and solidification in porous media.
Hyperbolic model, Dual-Phase Lag (DPL) model, Two-temperature models, and ultrafast melting and solidification.

Back to Heat and Mass Transfer.
Back to Thermalpedia Main Page.


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# Math Help - Need help getting started - EOQ question

1. ## Need help getting started - EOQ question

If anyone can help me get a start formatting this correctly, I would appreciate it! Company A’s demand is uniform throughout the year and totals 18,000 units per year. Ordering costs total $38 per order. The annual holding cost rate is 26% of the value of the inventory. The per-unit cost of inventory is$12.Company B’s demand is uniform throughout the year and totals 15,000 units per year. The production setup costs total $84 per setup. The annual holding cost rate is 28% of the value of the inventory. The per-unit cost of finished product is$19. The production rate is constant and equivalent to 60,000 units per year.
A. Determine the order size for Company A in the given scenario that would minimize total annual cost by using the economic order quantity model.
B. Determine the lot size for Company B in the given scenario that would minimize total annual cost by using the economic production lot size model

For A: q*= square root of 2KD/h I think that H=0.26 D=18,000 K=12 Am I on the right track?

2. ## Re: Need help getting started - EOQ question

Okay this is what I have for Part A (company A only), but I cannot figure out where the 26% annual holding cost rate comes in to play. Any help figuring where I left this out and how to include is greatly appreciated!
EOQ = √2UP / C
Where: U = Annual usage in units =18,000
P = Ordering cost per order = 38
C = Carrying cost per unit =12

EOQ = √2UP / C => = √(2 x 18,000 x 38) / 12 => 337.6 or 338 units (after rounding)
Number of orders per year = U / EOQ => 18,000 units/338 = 53.25 or 53 orders (rounded)


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Answer
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Q)

# The product of two positive number is 15876 one of the number is 9 times th

The product of two positive number is 15876 one of the number is 9 times the other find the number

Comment
A)


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### Home > MC1 > Chapter 4 > Lesson 4.3.3 > Problem4-113

4-113.

Make up sets of data that satisfy each of the conditions described in parts (a) and (b) below.

1. $6$ pieces of data with a median of $12$, a mode of $12$, and mean of $10$.

___, ___, $12,\ 12$, ___, ___
Start by making sure the median is $12$.

___, ___, $12,\ 12,\ 12$,  ___
Then you could ensure that $12$ is the mode by making three of them.

The mean of $4, 4, 6, 6$ is $5$. How can you tell without calculating?
Can you use this idea to help you make up other numbers to make the mean of your data set $10$?

1. $4$ pieces of data with a median of $8$, a mode of $8$, and a range of $8$.

This one is very similar to part (a).


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### Artificial IntelligenceAIMA Exercises

Decide whether each of the following sentences is valid, unsatisfiable, or neither. Verify your decisions using truth tables or the equivalence rules of Table logical-equivalence-table (page logical-equivalence-table).
1. ${Smoke} {\:\;{\Rightarrow}\:\;}{Smoke}$
2. ${Smoke} {\:\;{\Rightarrow}\:\;}{Fire}$
3. $({Smoke} {\:\;{\Rightarrow}\:\;}{Fire}) {\:\;{\Rightarrow}\:\;}(\lnot {Smoke} {\:\;{\Rightarrow}\:\;}\lnot {Fire})$
4. ${Smoke} \lor {Fire} \lor \lnot {Fire}$
5. $(({Smoke} \land {Heat}) {\:\;{\Rightarrow}\:\;}{Fire}) {\;\;{\Leftrightarrow}\;\;}(({Smoke} {\:\;{\Rightarrow}\:\;}{Fire}) \lor ({Heat} {\:\;{\Rightarrow}\:\;}{Fire}))$
6. ${Big} \lor {Dumb} \lor ({Big} {\:\;{\Rightarrow}\:\;}{Dumb})$
7. $({Big} \land {Dumb}) \lor \lnot {Dumb}$

Decide whether each of the following sentences is valid, unsatisfiable, or neither. Verify your decisions using truth tables or the equivalence rules of Table logical-equivalence-table (page logical-equivalence-table).
1. ${Smoke} {\:\;{\Rightarrow}\:\;}{Smoke}$
2. ${Smoke} {\:\;{\Rightarrow}\:\;}{Fire}$
3. $({Smoke} {\:\;{\Rightarrow}\:\;}{Fire}) {\:\;{\Rightarrow}\:\;}(\lnot {Smoke} {\:\;{\Rightarrow}\:\;}\lnot {Fire})$
4. ${Smoke} \lor {Fire} \lor \lnot {Fire}$
5. $(({Smoke} \land {Heat}) {\:\;{\Rightarrow}\:\;}{Fire}) {\;\;{\Leftrightarrow}\;\;}(({Smoke} {\:\;{\Rightarrow}\:\;}{Fire}) \lor ({Heat} {\:\;{\Rightarrow}\:\;}{Fire}))$
6. ${Big} \lor {Dumb} \lor ({Big} {\:\;{\Rightarrow}\:\;}{Dumb})$
7. $({Big} \land {Dumb}) \lor \lnot {Dumb}$

Submit Solution


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# Evasiveness and the Distribution of Prime Numbers

@article{Babai2010EvasivenessAT,
title={Evasiveness and the Distribution of Prime Numbers},
author={L{\'a}szl{\'o} Babai and Anandam Banerjee and Raghav Kulkarni and Vipul Naik},
journal={ArXiv},
year={2010},
volume={abs/1001.4829}
}
• Published 26 January 2010
• Mathematics
• ArXiv
We confirm the eventual evasiveness of several classes of monotone graph properties under widely accepted number theoretic hypotheses. In particular we show that Chowla's conjecture on Dirichlet primes implies that (a) for any graph $H$, "forbidden subgraph $H$" is eventually evasive and (b) all nontrivial monotone properties of graphs with $\le n^{3/2-\epsilon}$ edges are eventually evasive. ($n$ is the number of vertices.) While Chowla's conjecture is not known to follow from the Extended…
13 Citations
• Mathematics, Computer Science
SIAM J. Discret. Math.
• 2013
It is shown that all nontrivial monotone graph properties are evasive, i.e., have decision tree complexity $\binom{n}{2}$ and a lower bound of $\frac{1}{3}n^2-o( n^2)$ for general $n$ is proved.
This paper studies a weakening of the Evasiveness Conjecture called weak-EC, which asserts that every non-trivial monotone transitive Boolean function must have D(f) ≥ n1- ε, for every ε > 0.
Inspired by the outline of the KQS approach, the general framework of "orbit augmentation sequences" of sets with group actions is formalized, showing that a parameter of such sequences is a lower bound on the decision-tree complexity for any nontrivial monotone property that is Γ-invariant for all groups involved in the orbit augmentation sequence, assuming all those groups are p-groups.
This work formalizes the general framework of “orbit augmentation sequences” of sets with group actions and shows that a parameter of such sequences, called the spacing, is a lower bound on the decision-tree complexity for any nontrivial monotone property that is Γ-invariant for all groups Γ involved in the orbit augmentation sequence, assuming all those groups are p-groups.
• Mathematics, Computer Science
MFCS
• 2016
The answer is no for any hereditary property with {finitely many} forbidden subgraphs by exhibiting a bound of $\Omega(n^{1/k})$ for some constant $k$ depending only on the property.
• Mathematics, Computer Science
STACS
• 2017
This result disproves a conjecture of Babai, which conjectures that the sensitivity complexity of k-uniform hypergraph properties is at least Ω (nk/2), and shows that for many classes of transitive Boolean functions the minimum achievable sensitivity complexity can be O(N1/3).
• Computer Science
TAMC
• 2015
It is shown that the parity queries can be replaced by ordinary ones at the cost of the total influence aka average sensitivity per query, which is tight as demonstrated by the parity function.
• Mathematics
• 2014
[114] László Babai. Vertex-transitive graphs and vertex-transitive maps. games: A randomized proof system and a hierarchy of complexity classes. On the orders of primitive groups with restricted

## References

SHOWING 1-10 OF 25 REFERENCES

A graph property G is a collection of graphs closed under isomorphism. G is said to be evasive if, for every possible local search strategy, there is at least one graph for which membership in G
• Mathematics, Computer Science
STACS
• 2001
It is proved that properties that are preserved under taking graph minors are evasive for all sufficiently large n, which greatly generalizes the evasiveness result for planarity.
In a recent paper [i], Holt and Reingold have proved the following results: any algorithm which, given an n-node graph, detects whether or not the graph enjoys property P must in the worst case probe 0(n 2) entries of the incidence matrix.
• Mathematics, Computer Science
Comb.
• 1999
The results give the first superpolynomial lower bounds for linear secret sharing schemes and show that the perfect matching function can be computed by polynomial size (non-monotone) span programs over arbitrary fields.
From the contents: Primes in Arithmetic Progression.- Gauss' Sum.- Cyclotomy.- Primes in Arithmetic Progression: The General Modulus.- Primitive Characters.- Dirichlet's Class Number Formula.- The
• Mathematics
Am. Math. Mon.
• 2009
1 . S. Hedman, A First Course in Logic, Oxford University Press, New York, 2004. 2. D. Hobby and D. M. Silberger, Quotients of Primes, this Monthly 100 (1993) 50-52. 3. S. Marivani, On some
By applying a topological approach due to Kahn, Saks and Sturtevant, we prove that all decreasing graph properties consisting of bipartite graphs only are elusive. This is an analogue to a well-known
• F. Lutz
• Mathematics
Discret. Comput. Geom.
• 2002
A five-dimensional example and further examples in higher dimensions are constructed, one of which is Oliver’s example of dimension 11, the only previously known example of a non-contractible Z -acyclic vertex-homogeneous simplicial complex.


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# Revision history [back]

### Observation regarding integration with different algorithms

I was going through some basic tutorials and noticed this interesting behavior.

integrate(exp(x)*sin(x) + exp(x)*cos(x), x)

with algorithm='maxima' gives

1/2*(cos(x) + sin(x))*e^x - 1/2*(cos(x) - sin(x))*e^x

rather than the simplified

e^x*sin(x)

given by algorithm='sympy'. It is the same when I use maxima or sympy by itself. Why is that? Why is maxima not simplifying the expression?


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# Star of David theorem

Jump to: navigation, search
The Star of David theorem (the rows of the Pascal triangle are shown as columns here).

The Star of David theorem is a mathematical result on arithmetic properties of binomial coefficients. It was discovered by H.W. Gould in 1972.

## Statement

The greatest common divisors of binomial coefficients forming the Star of David shape in Pascal's triangle, are equal:

\begin{align} & {} \quad \gcd\left\{ \binom{n-1}{k-1}, \binom{n}{k+1}, \binom{n+1}{k}\right\} \\[8pt] & = \gcd\left\{ \binom{n-1}{k}, \binom{n}{k-1}, \binom{n+1}{k+1}\right\}. \end{align}


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# A quadratic polynomial whose zeros are 5 and −3, is

Question:

A quadratic polynomial whose zeros are 5 and −3, is

(a) $x^{2}+2 x-15$

(b) $x^{2}-2 x+15$

(c) $x^{2}-2 x-15$

(d) none of these

Solution:

(c) $x^{2}-2 x-15$

Here, the zeroes are 5 and $-3$.

Let $\alpha=5$ and $\beta$

So, sum of the zeroes, $\alpha+\beta=5+(-3)=2$

Also, product of the zeroes, $\alpha \beta=5 \times(-3)=-15$

The polynomial will be $x^{2}-(\alpha+\beta) x+\alpha \beta$.

$\therefore$ The required polynomial is $x^{2}-2 x-15$.


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# Joshua cuts 3 strings into a number of equal pieces. Each piece is 1/2 of a string into how many pieces does Joshua cut the 3 strings?

Since we are dividing the 3 strings into groups of $\frac{1}{2}$, the problem is:
$\frac{3}{1} \div \frac{1}{2} = \frac{3}{1} \cdot \frac{2}{1} = \frac{6}{1} = 6$


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# If rest mass does not change with $v$ then why is infinite energy required to accelerate an object to the speed of light?

I know that as the velocity increases, the mass of the object also increases so it becomes tougher and tougher to move the object which ultimately leads to a requirement of infinite energy to accelerate an object to the speed of light. But I have a doubt.

As far as I know only the observable mass (Relativistic mass) increases but not the Proper Mass or Intrinsic Mass, right? The actual mass of the object will remain the same. So if the actual mass ($m_0)$ remains the same and only the observable mass increases, why is more and more energy required? I know that even if the object exceeds the speed of light, we will not be able to say that it is moving faster than the speed of light but is it possible to make it move faster than light?

If rest mass does not change with $v$ then why is infinite energy required to accelerate an object to the speed of light?

I quote Igor Ivanov (a stackexchange user from this question Why does the (relativistic) mass of an object increase when its speed approaches that of light?) to give an insight into my question.

The mass (the true mass which physicists actually deal with when they calculate something concerning relativistic particles) does not change with velocity. The mass (the true mass!) is an intrinsic property of a body, and it does not depends on the observer's frame of reference. I strongly suggest to read this popular article by Lev Okun, where he calls the concept of relativistic mass a "pedagogical virus".

EDIT

SO can I say that only the overall energy of the system increases while the mass remains constant? But then if mass remains constant, then why is more and more energy gradually required? I mean there should be a reason for that requirement of infinite energy.

• There is not such a thing as 'observable mass'. The mass is the mass. In modern physics we don't speak about 'rest' mass or 'observable mass'. Oct 11, 2014 at 10:04
• I know but I meant relativistic mass. Oct 11, 2014 at 10:16
• In modern physics we don't make any distinction between any type of masses (in the relativistic context). We only use the mass of the particle (the $0.5\text{ MeV}$ of the electron, for example). The relativistic energy depends on this mass, and this mass doesn't change, but depends too on the velocity, in such a way that is not allowed to trespass $c$, and infinite energy is needed to reach $c$. This has nothing to do with any (non-existing) variation in the mass. Oct 11, 2014 at 10:34
• I am also wondering about this question. My question got closed yesterday as a duplicate of this one. I find it really weird that some top physicists quietly started "mass" to mean "rest mass", and "m" to mean "m[0]". I read and loved "Relativity Visualized" by Lewis Carroll Epstein, and I thought I understood the basics of at least special relativity. So this is all a bit shock to me. It's amazing how little this topic is talked about, as if it's being swept under the carpet. Einstein was wrong? Dec 9, 2021 at 13:59

If rest mass does not change with v then why is infinite energy required to accelerate an object to the speed of light?

The momentum of a material particle, a conserved quantity, is theoretically and experimentally a non-linear function of velocity given by

$$\vec p = m \frac{\vec v}{\sqrt{1 - \frac{v^2}{c^2}}}$$

which goes to infinity as $v \rightarrow c$. The relativistic energy, also a conserved quantity, is

$$E = c\sqrt{|p|^2 + (mc)^2}$$

When a particle is ultrarelativistic, $|p|^2 \gg (mc)^2$, this expression is approximately

$$E = |p|c$$

which has been experimentally confirmed. So, it easy to see that the particle's energy goes to infinity as $v \rightarrow c$

• SO can I say that only the overall energy of the system increases while the mass remains constant? But then if mass remains constant, Then why is more and more energy gradually required? I mean there should be a reason for that requirement of infinite energy. Oct 12, 2014 at 4:22
• @rahulgarg12342, the answer I gave clearly shows why the energy goes to infinity. If you're asking why these equations for momentum and energy correspond to experimental results, you're asking a metaphysics or philosophical question, not a physics question. Oct 12, 2014 at 14:17
• Yes I was asking a metaphysical question. Can you please answer it? Oct 13, 2014 at 9:40
• @rahulgarg12342, if you know that your question is metaphysical, you must know that I don't have an answer to it other than "because the Universe is what it is and not something else". Oct 13, 2014 at 10:55
• The experimental evidence is the given. The equations I give in my answer are in accord with experiment and, fundamentally, express the fact that the spacetime within which we exist is locally Lorentz covariant. These equations explain why more and more energy is required but they cannot explain why the Universe is this way and not some other way. Oct 13, 2014 at 11:24

In relativity the rest mass is the mass of an object measured from a reference frame in which it is at rest. But this is not the mass involved in acceleration or inertial mass.

Inertial mass, or the opposition of the body to the change of movement (directional or in magnitude), will grow with the speed of the body:

$$m = \frac{m_o}{\sqrt{1-v^2/c^2}}$$

and this is the mass observed when the object is moving. By observed I mean the way you would measure it, which could be: applying a force $F$ to it and measure the acceleration produced $a$ and you would get $m = F/a$.

Also, it is a mistake to think:

I know that even if the object exceeds the speed of light, we will not be able to say that it is moving faster than the speed of light

No body can travel faster than light, not because we can not state its speed, but because it can't. This is at least what our current state of knowledge leads to, with many experimental confirmations.

This might be a useful energy-momentum diagram (below) with coordinates $$(E,p)$$ [with $$E$$ running upward].

A particle's 4-momentum vector $$\tilde P$$ is drawn, as well its energy $$E$$ and momentum $$p$$ components in this frame.

The square-magnitude of this 4-momentum is given by the hyperbola $$E^2 - (pc)^2 = (mc^2)^2$$ , which is called the mass-shell for [invariant-] [rest-]mass $$m$$.

The velocity of this particle is the "slope" $$v=\displaystyle\frac{pc}{E}c.$$

While keeping $$m$$ fixed,

• in order to increase the velocity $$v$$ to approach the speed of light $$c$$,
the tip of the particle's 4-momentum must travel up along the mass-shell (along the green curve),
which requires that $$E$$ tends to infinity
(and $$p$$ tends to infinity while constrained by $$p=\frac{1}{c}\sqrt{E^2-(mc^2)^2}$$)
• Note that $$v \rightarrow c$$, but $$v$$ will never reach $$c$$.

(My diagram was taken from my answer
https://physics.stackexchange.com/a/510241/148184
to Ultra-Relativistic and Non-Relativistic cases for energy of a particle )

If you were in a car on clone of the Earth which was travelling at a metre per second less than c relative to Earth, you could still drive around as normal, and wouldn't notice anything odd. The amount of force required to accelerate the car to 100m/s on the clone planet would be exactly the same as the force required back on the real Earth.

The difference is that while on the clone planet the car's speed has increased to 100m/s, in the Earth's reference frame the speed of the car has hardly increased at all. So it's not harder to accelerate a speeding object in its own rest frame- it's just that the effect of that acceleration seems smaller and smaller in other frames moving faster relative to the object.

If ever you find yourself confused by SR, you might find it helpful to remember that all speed is relative and the effects of SR are symmetrical. Relative to a passing muon, you are travelling at 0.99c, but you don't experience any increase in mass as a consequence and you are not any harder to accelerate. However, if you get in your car and accelerate up the street, in the frame of the muon you hardly increase your speed at all.

Let's write the well known equations that are involved in this discussion: The energy-momentum relation: $$E^2=p^2c^2+m_{0}^2c^4$$ The relation between energy and mass: $$E_{0}=m_{0}c^2\quad\quad\quad \text{rest energy}$$ $$E=mc^2\quad\quad\quad \text{total energy}$$ The relation between rest mass and the so-called relativistic mass: $$m=m_{0}\gamma$$ where the gamma factor is defined as: $$\gamma\equiv \dfrac{1}{\sqrt{1-\dfrac{v^2}{c^2}}}$$

So if we take $$E=mc^2$$ and rewrite it as: $$m=\dfrac{E}{c^2}$$ we see that $$m$$ is actually another way of expressing the total energy of a particle (or a system). So of course you need an infinite amount of energy to accelerate it to the speed of light, because if $$v\rightarrow c$$ then $$\gamma\rightarrow\infty$$, which means that if $$m_{0}\neq 0$$ then $$m\rightarrow\infty$$.

So:

• if $$m_{0}\neq 0$$ and $$v\rightarrow c \Rightarrow \gamma\rightarrow\infty$$ thus $$m\rightarrow\infty$$.
• if $$m_{0}=0$$ and $$v\rightarrow c \Rightarrow \gamma\rightarrow\infty$$ but now $$m_{0}\gamma$$ (which is another way of expressing the total energy) gives a finite result (becasue $$0\cdot\gamma$$ gives something that is finite). So only massless particles like photons can travel at the speed of light (and in fact they can only travel at the speed of light).

So massive particles can travel at any speed they like except the speed of light, while massless particles can travel only at the speed of light.

In order to apply a force to an object to make it move faster (or slower), the force must be moving at the same velocity as the object. In an H2-O2 rocket engine, the fuel does travel at the same velocity as the rocket. When the fuel burns the pressure creates a force which drives the water molecules out the exhaust in the opposite direction to the rocket. The pressure is created by the interaction of the protons in the water with the protons in the rocket engine walls. These aren't little billiard balls that bounce off each other. They don't even get close. The interaction is essentially the Coulomb force(more complicated but you get the picture). The interaction is by Maxwell's electromagnetism.

Here comes the point: The interaction can proceed at the speed of light only. The fastest velocity at which a force can be applied is c. Every thing in our universe is made of electromagnetism.

In the rocket's frame, the exhaust is travelling at -c. An the exhaust's frame the rocket is travelling at +c. An observer in the exhaust's frame would see the rocket at the speed of light laying down a stream of water travelling at 0 m/s.

The photons doing the pushing are red shifted in the exhaust's frame to zero energy. There for to have any accelerating effect an infinite number of photons are required by the rocket at the speed of light.


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Q&A

# Why should the near-zero prior probability, of 9/11/2001 attacks on the World Trade Centre, overwhelm any subsequent response that may have lowered the objective risk?

+0
−2

Can you please expound the embold phrase below? Can this be calculated?

It is notoriously difficult, however, to assess what one’s prior assessment of the risk would have been had one thought of the event before it actually occurred. Most Americans had never contemplated the possibility of a terrorist attack in this country at the level of 9/11. After it occurred, an attack of that magnitude—or much worse—was not beyond belief, and indeed was thought to have a positive probability of happening again. Thus, most Americans’ subjective assessment of terrorism risk should have been far higher after 9/11 than before, despite measures taken to lower such risks after 9/11, since the near-zero prior probability should overwhelm any subsequent response that may have lowered the objective risk [Emphasis mine]. A survey of law school students published in 2003 by economists Kip Viscusi and Richard Zeckhauser in the Journal of Risk and Uncertainty, however, found that around 40 percent of respondents believed their personal risk assessment was higher before the attacks than currently.$^2$ In another study of professional-school students and undergraduate business students in 2005, they showed that over two-thirds of respondents exhibited the same phenomenon.$^3$ These respondents experienced a recollection bias, whereby after the occurrence of a low-probability event, one thinks that one’s prior risk assessment was much higher than it actually was. This could be due to an attempt to reduce cognitive dissonance, for self-justification, or simply to misremembering.

Paul Slovic, The Irrational Economist (2010), p 102.

$^2$ Kip Viscusi and Richard Zeckauser, Journal of Risk and Uncertainty_ (2003).

$^3$ Kip Viscusi and Richard Zeckauser, “Recollection Bias and the Combat of Terrorism,” Journal of Legal Studies 34 (January 2006): 27-55.

Op. cit. p 286.

Why does this post require moderator attention?
Why should this post be closed?


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# How MATLAB calculates matched filter gain

Here is a bunch of code from MATLAB documentation: Designing a Basic Monostatic Pulse Radar:

wav = phased.RectangularWaveform(...
'PulseWidth',1/pulseBandwidth,...
'PRF',PRF,...
'SampleRate',Fs);
matchingcoeff = getMatchedFilter(wav);

hmf = phased.MatchedFilter(...
'Coefficients',matchingcoeff,...
'GainOutputPort',true);

[rx_pulses, mfgain] = step(hmf,rx_pulses);

How MATLAB calculates the mfgain (matched filter gain)? Any formula for that?

Matched filter gain is calculated as (in dB):

• $G_{dB} = 10 \cdot \log_{10}(L)$,

where $L$ - is filter length. It's maximum possible SNR improvement that filter can provide.

Try this formula and compare result with MATLAB one. I suppose it will be the same.

Actually, if you dig into the code you'll find that MATLAB calculates the gain with the following:

obj.pMFGain = pow2db(real(obj.Coefficients'*obj.Coefficients));

Here obj is the AbstractMatchedFilter object and Coefficients are the matched filter coefficients, which, in the simplest form, is the model of your waveform sampled at the same rate as your input signal. When MATLAB calculates the matched filter, these coefficients are complex.

Note the '* in the line above, which makes this a dot-product. (sum of squares).


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## Introduction

Surface micro and nanostructuring is a hot topic in many research facilities around the world. Suitable shaped surface features can modify surface properties to attain anti-icing1, anti-corrosion, superhydrophobicity and self-cleaning2,3, control of friction4, anti-reflection5,6, anti-bacteria7 properties or serve as a decoration8.

Conventional nanostructuring techniques including lithography9, sol-gel10, thermal embossing11, plasma treatments12, chemical vapor deposition13, chemical etching14 or electrodeposition15 require chemicals or long fabrication time due to slow or complicated multi-step processes. The possible alternative is direct laser ablation as a single-step, non-polluting, cost-effective and flexible method for efficient and high-precision processing16,17,18.

Laser-Induced Periodic Surface Structures (LIPSS) presents a simple and robust way for surface nanostructuring. By utilizing ultrashort pulsed laser systems, LIPSS can be reliably generated in a single step process and without the use of expensive and slow vacuum technology and lithography19. LIPSS can be classified by their period to Low Spatial Frequency LIPSS (LSFL) with the periodicity close to the laser wavelength. LSFL can be observed mostly on metals and semiconductors with orientation generally dependent on the laser beam polarization and laser processing conditions20. The second common type are High Spatial Frequency LIPSS (HSFL) characterized by the period smaller than half of the laser wavelength. Depending on the material, the orientation of HSFL can be parallel or perpendicular to the beam polarization21. Except the wavelength, the periodicity of LIPSS can be controlled by the angle of incidence, fluence or by the applied number of pulses, while the direction of LIPSS can be controlled by the polarization of laser beam22. The great flexibility and variability of produced LIPSS and ripple structures following the laser and processing parameters allows tailoring surface mechanical, biological or optical properties19,22.

However, the wider industrial use of LIPSS-based functional surfaces is limited by processing speed. Generally, throughputs only up a few cm2 per minute are reached by conventional single beam approaches19. In addition, speeding up the single beam process by a novel high power ultrashort laser systems23 is not straightforward due to the very low close-to-ablation-threshold power required for LIPSS production21. Beam defocusing or using lenses with long focal lengths is a possible solution. For example, Mezera24 used defocused laser beam to increase LIPSS production 16-folds reaching 5.3 cm2/min on silicon and Faas25 used 525 W 8 ps laser source with 340 mm long F-theta focal distance to produce LIPSS on stainless steel with throughput of 12.5 cm2/min.

To further increase throughputs for the efficient use of high-power laser sources new fabrication methods have to be applied. These include polygon scanners providing scanning speeds above 100 m/s and multi-beam approaches25,26. Remarkable results were achieved by Schille27 combining hybrid polygon scanning system with fast polygon axis, perpendicular galvo axis and beam splitting to 4 sub-beams reaching 1300 cm2/min for LIPSS production on 40 × 50 mm stainless steel sample.

An alternative to polygon scanning systems can be beamsplitting by diffractive optical element (DOE) coupled with standard galvo scanning systems28,29,30,31. However, a large number of beams in order of thousands can be problematic due to the large separation distance of standard DOEs (typically > 10 times diffraction limited spot diameter) which results in a large irradiated area thus limiting scanning options due to scan-angle dependent distortions and limited scan area. Moreover, the large separation angles of standard DOEs require collimation to enable effective scanning without clipping and thus complicated multi-component scanning system32.

These issues can be addressed with a novel type of diffractive laser induced texturing (DLITe) DOE which allow dense texturing close to 50% of irradiated area. DLITe DOE generates a shaped pattern of light that is extremely dense, similar to that generated by interference patterning and does not require any additional collimation optics. As such it can be implemented as single element into existing setups. Additionally, sub-beams can be much closer (spot centre separations equal to the diffraction limited spot diameter) which enables effective scanning with minimized distortions. DLITe DOE is working in a regime where the splitter period is approximately X0.65 the laser input beam size. This texturing concept was introduced in previous theoretical work by the authors33, and this work is an experimental validation of this concept.

Thermal processes play an important role in high average power pulsed laser processing34,35,36. When using high pulse repetition frequency, residual heat from previous laser pulses in the same place increases the surface temperature of the material, which is called heat accumulation. The heat accumulation may result in the formation of structural defects on the surface. Uncontrolled heat accumulation at a very high repetition rate and low scanning speed leads to the onset of melting, oxidation, a pileup of material or thermal cracks37,38,39.

Thermal processes and surface temperature in ultrashort pulsed laser ablation are usually studied by numerical modelling and microscopy of the final resulting structure34,40,41,42. Time resolved measurements of laser heating, melting, vaporization and oxidation were done for laser pulse durations from milliseconds to nanoseconds by measuring reflectivity, transmissivity, emission and electrical conductance43,44,45. But for ultrashort pulsed laser ablation, only recently an infrared radiometry measurement system was developed and applied for ultrashort pulse laser micromachining46. Ultrashort pulsed laser melting during ablation was also investigated by time resolved reflectivity47.

In this work, a straightforward method for increasing the efficiency and fabrication speed of LIPSS-based functional surfaces is introduced. The input high-power laser beam is splitted by DLITe DOE into a square matrix of 51 × 51 sub-beams for simultaneous fabrication and placed before galvanometric scanner for fast beam matrix displacement over the sample. Additionally, a unique fast infrared radiometry system was applied for time-resolved thermal radiation measurements showing a correlation between thermal radiation changes and LIPSS production, thus presenting a simple way for potential closed-loop control during rapid large-scale fabrication of LIPSS-based functional surfaces.

## Beamspliting with diffractive laser induced texturing (DLITe) elements

As shown in previous works33, a beam splitter DOE was designed to have a period equal to 0.65 times the input beam size, to achieve a dense pattern of intensity with a distance between maxima equal to the size of a diffraction limited spot. Input beam size was selected as 8 mm, to account for the optics clear aperture of 14 mm, resulting in a diffraction limited spot size of ~ 20 µm and a DOE period of 5150 µm which are typical values for tribology and hydrophobic applications48,49,50. To cover an area of 1 × 1 mm, a splitter with 51 × 51 was designed, having full angles of 10 × 10 mrad (separation angles 0.2 mrad × 0.2 mrad) and efficiency ~ 80%. To control zero order, a special 4 level design was employed causing the zero order to be defocused strongly at the work plane. In addition, the central order was designed as zero, as we expected some energy to be present there due to production tolerances. The simulation of the DOE splitter output is depicted in Fig. 1.

While all orders of the DOE are designed to have identical power (within standard deviation of 2%) mutual interference effects create a non-uniformity of the spots, calculated to have standard deviation of 28%. This results directly from the optimal ratio of beam size to DOE period for statistical area processing, defined in reference33 as 0.65.

## Results and discussion

Square shaped matrix with 51 × 51 sub-beams can be attained only in the image plane of DLITe element (Fig. 2a) with a homogeneous sub-beam intensity in a range of ± 150 µm in z-axis (Fig. 2b). Thus, prior to sample patterning, a precise alignment was performed to reach homogenous intensity between 2601 sub-beams in square shaped matrix (Fig. 2c).

After the setup alignment, the pattern acquired in the optimal image plane shows a good correspondence with the theoretical simulation of the pattern in Fig. 1.

In the following experiment, the LIPSS evolution with fluence and number of pulses (N) was studied. As depicted in Fig. 3, both LSFL and HSFL were fabricated at different fluence levels. These results indicate that the main factor affecting LIPSS fabrication is the amount of consecutive pulses.

Single pulse treatment did not result in LIPSS formation even for as high pulse energies as 4 mJ (0.49 J/cm2). By increasing the number of pulses to 5, HSFL with the periodicity of ~ 400 nm and parallel orientation to the beam polarization can be observed for all fluence levels between 0.12 – 0.37 J/cm2 (Fig. 3 a,e,i). However, as the fluence increases melting of HSFL is more pronouced decreasing the structure quality (Fig. 3 i). For the number of pulses between 10 and 15, LSFL with the periodicity of ~ 900 nm and perpendicular orientation to the beam polarization can be observed and starting to be more dominant for higher pulse counts (Fig. 3 d,h,l). LSFL are observed for a lower number of pulses when higher fluence is applied, as can be observed by coparing Fig. 3b and j. For pulse counts above 20 pulses, HSFL are visible only partially on top of LSFL and significantly melted (Fig. 3 h,l). The melted rim around the whole microcrater can be observed for the higest energy inputs resulting from high fluence and high number of pulses (Fig. 3 h,l).

IR radiometry measurements are shown in Fig. 4. After a sharp peak the signal decreases to a certain value. In some cases, the first peak is followed by a smaller second peak. This evolution is very similar to phase change plateau45, where the phase change can be solidification or LIPSS formation. The plateau level significantly increases with the number of pulses. This can be caused by increasing of the melted area on the surface and thus radiating area (see Fig. 3). Heat accumulation signal is a signal before the next laser pulse – representing residual temperature from previous laser pulses46. This signal is also increasing with the number of pulses, mainly for the higher fluences (Fig. 4 b). The increase of heat accumulation with laser fluence is shown in Fig. 5a.

The phase change plateau also changes length with fluence and number of pulses. For higher fluence above 0.22 J/cm2 even two phase change plateaus appear for a higher number of pulses (> 5). It can be a sign of two different processes, e.g. LIPSS formation and solidification. The increase of plateau length can be explained by the bigger volume (depth) of melt on the surface. Averaged values from all number of pulses for each fluence are shown in Fig. 5b. The plateau length correlates well with the laser fluence. It means that it can be used for on-line process monitoring and control of LIPSS formation during the laser process. Also, heat accumulation signal can be used for on-line process control, because it increases mainly during the first pulses, where LIPSS are formed (1 to 15 pulses for higher fluences). According to its value the pulsing can be stopped on each place.

An example of the resulting measured and calculated temperature evolutions is shown in Fig. 6. The measured temperatures are obtained from measured voltage signal by calibration based on observed solidification plateaus separately for each laser pulse. Such a dynamic calibration is a completely new approach and can have also some drawbacks. For example, the peak temperature for the first pulse rises up to 3500 °C, which may not be correct when taking into account the peak temperature of other pulses and the detector relatively slow response time compared to the laser pulse duration (60 ns compared to 2 ps). But this approach can reveal significant new insights into the problematics of the LIPSS formation. For example, in the case of 0.12 J/cm2 the phase change was observed already during the first pulse, although LIPSS was not observed on the SEM.

The presented model of heat accumulation has not included emissivity and absorptivity changes during LIPSS formation. Every laser pulse interaction is described as a heat source with constant temperature $${T}_{plateau}$$ during LIPSS formation period (more details in the methods section) and it has not included nor chemical nor structure changes of laser irradiated material. Such simplification can be responsible for the difference between calculated and measured temperature–time history. The character of experimentally detected temperature decrease after the plateau is changing from pulse to pulse, especially in the beginning of the laser surface processing (Fig. 6b). The theoretical model has repeatable law of temperature–time history changes from pulse to pulse because it has a fixed geometrical coefficient $$n$$ and time interval of LIPSS formation $${t}_{plateau}$$. The difference between the theoretical model and experimental data can help to discover the thermodynamics of multi-pulse LIPPS structure formation.

The heat accumulation temperature shown in Fig. 6 increased fast during the first pulses over 400 °C and stabilizes around 600 °C at the end of the process. The detail values of heat accumulation temperature for all laser pulses and fluences are shown in Fig. 7. There can be found two groups of heat accumulation temperatures. First has the stabilized value around 600 °C (fluences 0.07 and 0.12 J/cm2) and the second around 1000 °C (fluences 0.24 and 0.31 J/cm2). The fluence 0.17 J/cm2 is somewhere in between them. This may be explained as two different process regimes with a transition between them. However, the temperatures around 600 °C for the laser fluence of 0.07 J/cm2 were unexpected due to the very low heat accumulation signal compared to the processes with fluence 0.12 and 0.17 J/cm2 (8 and 35 times lower). In this case, the explanation may be in the very localized areas of melting or LIPSS formation for the fluence of 0.07 J/cm2. In addition, a threshold to attain a stable value of the process was observed, which might be connected to the beginning of LIPSS formation. For low fluences, it takes a certain number of pulses to attain it (13 and 8 for 0.07 and 0.12 J/cm2). On the other hand for high fluences, it is reached already after the first pulse together with some oscillation during the first three pulses. Also, the heat accumulation temperature for the fluence of 0.31 J/cm2 is more unstable than in the case of 0.24 J/cm2, although in average showing the same results.

To cover larger areas, DLITe DOE was placed before the galvanometric scanner to stitch dot matrices. The advantageous square shape of the beam matrix allows 0% overlap and thus fast beam movement over the sample. To demonstrate the throughput, the sample area of 40 × 40 mm was covered by LSFL and HSFL with parameters corresponding to Fig. 3a and d, respectively. During this experiment, the scanning speed was set to 9 m/s and the laser system was operated at 100 kHz delivering 200 W of average power. The required time for patterning of the whole area reached 503 ms and 1205 ms thus reaching 1910 cm2/min and 797 cm2/min for HSFL and LSFL respectively. The real-time video of the process can be found in the supplementary material (Supplementary video S1) and a detail of processed surface in Supplemetary Fig. 1.

## Conclusion

The unique combination of the high-quality, high-energy laser beam with M2 < 1.2 and DLITe DOE element shaped the beam into the homogeneous square-shaped matrix of 51 × 51 sub-beams with a separation distance of 20 µm. Thus, resulting in a simultaneous microstructuring in 1mm2 area, 2601 times faster compared to the single beam approach. By adjusting the fluence and number of pulses additional nanostructures can be observed inside microholes produced by DLITe element. Both, LSFL and HSFL can be produced. LSFL were observed for pulse energies between 1 to 3 mJ (fluence of 0.12 to 0.37 J/cm2) and the number of pulses above 15 for 0,12 J/cm2 and above 10 for higher fluences. HSFL were observed for all tested fluences and the number of pulses between 5 to 10. For a higher number of pulses, LSFL start to dominate especially for high fluences. In addition, these results show a correlation with fast infrared radiometry measurements. Solidification plateaus were observed after each laser pulse, confirming that LIPSS formation is connected with the melting and solidification of the material. A new approach of using a derivation of the signal was applied to automatically calculate plateau length. Hence phase change plateau length and heat accumulation signal can be used for real-time quality control during LIPSS formation and solidification and for controlling the melt volume on the surface. Observation of solidification plateaus after each laser pulse enabled dynamic calibration of the measurement for obtaining temperatures. The resulting temperatures were compared with a theoretical model of heat accumulation. Heat accumulation temperatures from 200 to 1000 °C were observed from both the measurement and the model. The model can help to increase understanding of thermodynamics of multi-pulse LIPPS structure formation when comparing it with measurement for different laser pulse numbers. The temperature measurements revealed interesting changes in the physics of the laser ablation process, mainly two levels of heat accumulation and correlation of the heat accumulation fast increase to a stable value depicting the start of LIPSS formation.

DOE placed before a standard galvo scanner enables easy integration and stitching of the multi-beam pattern, demonstrating the potential for high-speed micro/nanostructuring over larger areas. By using the parameters for LSFL and HSFL production throughputs of 797 cm2/min and 1910 cm2/min were achieved on 40 × 40 mm area, respectively. Which is, to the best of our knowledge, the highest demonstrated throughput of LIPSS nanostructuring. Thus, opening new unprecedented opportunities towards a simple and robust rapid large-scale micro and nanostructuring with real-time quality control possibilities.

## Materials and methods

Stainless steel 304L samples with dimensions of 50 mm × 50 mm × 5 mm were mirror polished (Ra ~ 0.05 µm) and treated by Ytterbium based diode pumped solid state laser system Perla (HiLASE Centre, Czech Republic) emitting ultrashort pulses with pulse duration of 1.2 ps and M2 of 1.2 at 1030 nm and power up to 200 W. Repetition rate of 100 kHz and 50 kHz were used for the purpose of this experiment resulting in a pulse energy up to 4 mJ. The generated beam was guided through DLITe element MS-805-I-Y-A for beamsplitting into 51 × 51 matrix (Holo/Or Ltd., Israel), then into a high dynamic galvo-scanner system (Scanlab GmbH, Puchheim, Germany) and focused on a sample using telecentric F-theta lens with focal length of 100 mm.

The shape of the diffraction pattern and homogeneity of sub-beams were analysed by Basler ace acA3080-10um camera with a pixel size of 1.67 µm. The surface morphology was investigated by laser scanning confocal microscope, Olympus OLS5000 and scanning electron microscope, Tescan FERA 3 at electron energy of 5 kV.

Infrared (IR) radiometry system was composed of fast IR detector HgCdTe (Fermionics PV-11–1) cooled by liquid nitrogen, two off-axis paraboloid mirrors and a germanium filter. The wavelength range of the detector sensitivity was from 2 to 12 µm and its response time was 60 ns46.

IR radiometry measurements have shown in many cases a plateau of almost stable signal after a sharp peak. The plateau length was analyzed automatically by derivation of the IR signal curve. Before and after derivation also smoothing was applied. The time between maximum and minimum derivative is taken as the length of the phase change plateau. To obtain temperatures from the measured voltage values calibration was done based on the identified plateaus in the signal. First, a theoretical calibration table was created relating temperature and the response of the detector in voltage. The theoretical calibration table was calculated by integral of Planck’s law weighted by spectral sensitivity of the detector46. The calibration table was then adjusted so that the zero level matches the room temperature – the voltage column was subtracted by voltage value at room temperature.

The calibration was performed on the basis of the known solidification temperature of the material (1427°C51). The average signal in the top part of the plateau was used as the signal value of the solidification for the calibration. A multiplicative calibration coefficient was calculated for each plateau in the signal (each laser pulse) by dividing the value at the top of the plateau by the value from the calibration table corresponding to the phase transition temperature in volts. This coefficient accounts for the emissivity of the material and geometrical view factor of the optical focusing system52 and also for the size of the radiation area (comparable to the melted area). The obtained calibration coefficient was then used to multiply the voltage values from the calibration table. In this way, each plateau generated its own calibration coefficient and each part of the signal corresponding to one laser pulse (one period between pulses) has its own calibration.

Once all the calibration coefficients were found, the signal values were converted to temperature according to the corrected calibration table for every single peak. At the end of the process, the experiment signal was fully converted to the temperature, where all plateaus were aligned to the phase transition temperature of the material used.

### Theoretical calculations

The measured temperature evolutions during the experiment were compared to theoretical calculations defined as described here. The temperature evolution under laser irradiated surface is in direct dependence on intensity distribution on the material surface. For the case of planar, linear or point instantaneous heat source the temperature changes can be described as a solution of differential equation of conduction heat37,53,54

$$\Delta T=\frac{{Q}_{n}}{\sqrt{{(\pi \cdot \alpha \cdot t)}^{n}}}\cdot {e}^{-\frac{{r}_{n}^{2}}{t\cdot 4\cdot \alpha }}$$
(1)

where $$\Delta T$$—temperature changes under surface, $${Q}_{n}$$—residual heat (which can be a function of heat source geometry), $$n$$—is a geometrical coefficient (for planar source $$n=1$$, for linear source $$n=2$$ and for point source $$n=3$$), $$\alpha$$—thermal diffusivity, $${r}_{n}^{2}$$—distance from heat source (for the central point on surface $${r}_{n}^{2}=0$$) and $$t$$—is the time period after the heat source is applied. In the case of multi-spot laser distribution on the irradiated surface, the heat source acts as an array of isolated laser spots at the very start ($$t<1 \mathrm{\mu s}$$) and later it becomes more similar to the planar source model (Fig. 8). From this point of view the temperature distribution under multi-spot laser irradiation can be defined by approximation of geometrical coefficient in the interval $$1<n<3$$ and it depends on laser spots diameter, their quantity and distance between them.

Full period of LIPSS formation includes the whole interval after the laser pulse irradiation till the moment of the plateau end $${t}_{plateau}=1-5\upmu s$$ (Fig. 4). The temperature of plateau formation can be taken as a constant value equal to the melting point. The process of LIPSS formation during plateau can be described as an uninterrupted heat source and Eq. 1 will be represented as an integral54:

$${T}_{plateau} =\frac{1}{\sqrt{{(\pi \cdot \alpha )}^{n}}}{\int }_{0}^{{t}_{plateau}}\frac{{Q}_{n}(t)}{{t}^{n/2}}dt$$
(2)

where $${T}_{plateau}$$—temperature during plateau and $${Q}_{n}(t)$$ is the quantity of heat, which is transferred from the laser spot to the material surface during LIPSS formation. The temperature of LIPSS formation can be taken equal to the melting point. During LIPSS formation the transferred heat $${Q}_{n}(t)$$ is decreasing due to decreasing of temperature gradient under laser spots. The temperature of LIPSS formation and time during plateau formation can be expressed in simple form:

$$q(t)={T}_{plateau}\cdot {t}^{n/2}$$
(3)

where $$q(t)$$ is a term that includes thermal characteristics of material and strength of heat source in LIPSS formation area. Term values $$q(t)$$ can be defined numerically in whole plateau duration and then it will be used for temperature–time history evolution after every laser pulse:

$$\Delta T={\sum }_{0}^{m}{\sum }_{0}^{k}{q}_{mk}/{t}_{mk}^{n/2}$$
(4)

where $$m$$—is number of applied laser pulse, $$k$$—is number of evaluated therm in the LIPPS formation plateau, $${t}_{mk}$$—time interval between evaluated point $$mk$$ and actual temperature–time history point. For the applied multi-beam process with 10 µm diameter spots and 20 µm distance between them, the geometrical coefficient was defined es equal to $$n=1.5$$. The input parameter for this heat accumulation model is the full plateau duration, which is needed for the determination of terms $${q}_{mk}$$. For laser beam energy of 1 mJ, it was taken $${t}_{plateau}=2 \mu s$$.


text:
## 数学代写|拓扑学代写Topology代考|MATH4204

2022年12月30日

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• Statistical Inference 统计推断
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## 数学代写|拓扑学代写Topology代考|Constructing a GIC

One may wonder how to efficiently construct the graph induced complexes in practice. Experiments show that the following procedure runs quite efficiently in practice. It takes advantage of computing nearest neighbors within a range and, more importantly, computing cliques only in a sparsified graph.

Let the ball $B(q, \delta)$ in metric d be called the $\delta$-cover for the point $q$. A graph induced complex $\mathcal{G}^\alpha(P, Q$, d) where $Q$ is a $\delta$-sparse $\delta$-sample can be built easily by identifying $\delta$-covers with a rather standard greedy (farthest point) iterative algorithm. Let $Q_i=\left{q_1, \ldots, q_i\right}$ be the point set sampled so far from $P$. We maintain the invariants (i) $Q_i$ is $\delta$-sparse and (ii) every point $p \in P$ that is in the union of $\delta$-covers $\bigcup_{q \in Q_i} B(q, \delta)$ has its closest point $v(p) \in \operatorname{argmin}{q \in Q_i} \mathrm{~d}(p, q)$ in $Q_i$ identified. To augment $Q_i$ to $Q{i+1}=Q_i \cup\left{q_{i+1}\right}$, we choose a point $q_{i+1} \in P$ that is outside the $\delta$ covers $\bigcup_{q \in Q_i} B(q, \delta)$. Certainly, $q_{i+1}$ is at least $\delta$ units away from all points in $Q_i$ thus satisfying the first invariant. For the second invariant, we check every point $p$ in the $\delta$-cover of $q_{i+1}$ and update $v(p)$ to include $q_{i+1}$ if its distance to $q_{i+1}$ is smaller than the distance $\mathrm{d}(p, v(p))$. At the end, we obtain a sample $Q \subseteq P$ whose $\delta$-covers cover the entire point set $P$ and thus is a $\delta$ sample of $(P, \mathrm{~d})$ which is also $\delta$-sparse due to the invariants maintained. Next, we construct the simplices of $\mathcal{G}^\alpha(P, Q, \mathrm{~d})$. This needs identifying cliques in $G^\alpha(P)$ that have vertices with different closest points in $Q$. We delete every edge $p p^{\prime}$ from $G^\alpha(P)$ where $v(p)=v\left(p^{\prime}\right)$. Then, we determine every clique $\left{p_1, \ldots p_k\right}$ in the remaining sparsified graph and include the simplex $\left{v\left(p_1\right), \ldots, v\left(p_k\right)\right}$ in $\mathcal{G}^\alpha(P, Q, \mathrm{~d})$. The main saving here is that many cliques of the original graph are removed before it is processed for clique computation.
Next, we focus on the second topic of this chapter, namely homology groups. They are algebraic structures to quantify topological features in a space. They do not capture all topological aspects of a space in the sense that two spaces with the same homology groups may not be topologically equivalent. However, two spaces that are topologically equivalent must have isomorphic homology groups. It turns out that the homology groups are computationally tractable in many cases, thus making them more attractive in topological data analysis. Before we introduce their definition and variants in Section 2.5, we need the important notions of chains, cycles, and boundaries given in the following section.

## 数学代写|拓扑学代写Topology代考|Algebraic Structures

First, we recall briefly the definitions of some standard algebraic structures that are used in the book. For details we refer the reader to any standard book on algebra, for example, [14].

Definition 2.18. (Group; Homomorphism; Isomorphism) A set $G$ together with a binary operation “+” is a group if it satisfies the following properties: (i) for every $a, b \in G, a+b \in G$; (ii) for every $a, b, c \in G$, $(a+b)+c=a+(b+c)$; (iii) there is an identity element denoted 0 in $G$ so that $a+0=0+a=a$ for every $a \in G$; and (iv) there is an inverse $-a \in G$ for every $a \in G$ so that $a+(-a)=0$. If the operation “+” commutes, that is, $a+b=b+a$ for every $a, b \in G$, then $G$ is called abelian. A subsèt $H \subseteq G$ is a subgroup of $(G,+)$ if $(H,+)$ is also a group.

Definition 2.19. (Free abelian group; Basis; Rank; Generator) An abelian group $G$ is called free if there is a subset $B \subseteq G$ so that every element of $G$ can be written uniquely as a finite sum of elements in $B$ and their inverses disregarding trivial cancellations $a+b=a+c-c+b$. Such a set $B$ is called a basis of $G$ and its cardinality is called its rank. If the condition of uniqueness is dropped, then $B$ is called a generator of $G$ and we also say that $B$ generates $G$.

Definition 2.20. (Coset; Quotient) For a subgroup $H \subseteq G$ and an element $a \in G$, the left coset is $a H={a+b \mid b \in H}$ and the right coset is $H a={b+a \mid b \in H}$. For abelian groups, the left and right cosets are identical and hence are simply called cosets. If $G$ is abelian, the quotient group of $G$ with a subgroup $H \subseteq G$ is given by $G / H={a H \mid a \in G}$ where the group operation is inherited from $G$ as $a H+b H=(a+b) H$ for every $a, b \in G$.

Definition 2.21. (Homomorphism; Isomorphism; Kernel; Image; Cokernel) A map $h: G \rightarrow H$ between two groups $(G,+)$ and $(H, *)$ is called a homomorphism if $h(a+b)=h(a) * h(b)$ for every $a, b \in G$. If, in addition, $h$ is bijective, it is called an isomorphism. Two groups $G$ and $H$ with an isomorphism are called isomorphic and denoted as $G \cong H$. The kernel, image, and cokernel of a homomorphism $h: G \rightarrow H$ are defined as subgroups ker $h={a \in G \mid h(a)=0}, \operatorname{Im} h={b \in H \mid \exists a \in G$ with $h(a)=b}$, and the quotient group coker $h=H /$ im $h$, respectively.

# 拓扑学代考

## 数学代写|拓扑学代写Topology代考|Constructing a GIC

Ileft{p_1, Idots p_klright} 在剩余的稀疏图中并包括单纯 $\mathcal{G}^\alpha(P, Q, \mathrm{~d})$. 这里的主要节省是原始图的许多团在处 理团计算之前被删除。

## 数学代写|拓扑学代写Topology代考|Algebraic Structures

Image; Cokernel) 地图 $h: G \rightarrow H$ 两组之间 $(G,+)$ 和 $(H, *)$ 称为同态如果 $h(a+b)=h(a) * h(b)$ 每一个 $a, b \in G$. 如果，此外， $h$ 是双射的，称为同构。两组 $G$ 和 $H$ 具有同构的称为同构并表示为 $G \cong H$. 同态的内 核、图像和上核 $h: G \rightarrow H$ 被定义为子群ker $h=a \in G|h(a)=0, \operatorname{Im} h=b \in H| \exists a \in G \$$with$\$h$ ，和商群 cokerh $h=H /$ 在里面 $h$ ，分别。

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。


text:
Data representation#

Socceraction uses a tabular action-oriented data format, as opposed to the formats by commercial vendors that describe events. The distinction is that actions are a subset of events that require a player to perform the action. For example, a passing event is an action, whereas an event signifying the end of the game is not an action. Unlike all other event stream formats, we always store the same attributes for each action. Excluding optional information snippets enables us to store the data in a table and more easily apply automatic analysis tools.

Socceraction implements two versions of this action-oriented data format: SPADL and Atomic-SPADL.

Definitions#

SPADL (Soccer Player Action Description Language) represents a game as a sequence of on-the-ball actions $$[a_1, a_2, . . . , a_m]$$, where $$m$$ is the total number of actions that happened in the game. Each action is a tuple of the same twelve attributes:

Attribute

Description

game_id

the ID of the game in which the action was performed

period_id

the ID of the game period in which the action was performed

seconds

the action’s start time

player

the player who performed the action

team

the player’s team

start_x

the x location where the action started

start_y

the y location where the action started

end_x

the x location where the action ended

end_y

the y location where the action ended

action_type

the type of the action (e.g., pass, shot, dribble)

result

the result of the action (e.g., success or fail)

bodypart

the player’s body part used for the action

Start and End Locations

SPADL uses a standardized coordinate system with the origin on the bottom left of the pitch, and a uniform field of 105m x 68m. For direction of play, SPADL uses the “home team attacks to the right” convention, but this can be converted conveniently with the play_left_to_right() function such that the lower x-coordinates represent the own half of the team performing the action.

Action Type

The action type attribute can have 22 possible values. These are pass, cross, throw-in, crossed free kick, short free kick, crossed corner, short corner, take-on, foul, tackle, interception, shot, penalty shot, free kick shot, keeper save, keeper claim, keeper punch, keeper pick-up, clearance, bad touch, dribble and goal kick. A detailed definition of each action type is available here.

Result

The result attribute can either have the value success, to indicate that an action achieved it’s intended result; or the value fail, if this was not the case. An example of a successful action is a pass which reaches a teammate. An example of an unsuccessful action is a pass which goes over the sideline. Some action types can have special results. These are offside (for passes, corners and free-kicks), own goal (for shots), and yellow card and red card (for fouls).

Body Part

The body part attribute can have 4 possible values. These are foot, head, other and none. For Wyscout, which does not distinguish between the head and other body parts a special body part head/other is used.

Example#

Socceraction currently supports converters for Opta, Wyscout, and StatsBomb event stream data. The Opta converter is also compatible with derived formats, such as the ones used by WhoScored and Stats Perform. We’ll use StatsBomb data to illustrate the API, but the API of the other converters is identical.

First, we load the event stream data of the third place play-off in the 2018 FIFA World Cup between Belgium and England.

from socceraction.data.statsbomb import StatsBombLoader

df_events = SBL.events(game_id=8657)

These events can now be converted to SPADL using the convert_to_actions() function of the StatsBomb converter.

import socceraction.spadl as spadl

The obtained dataframe represents the body part, result, action type, players and teams with numeric IDs. The code below adds their corresponding names.

df_actions = (
.merge(SBL.teams(game_id=8657))
.merge(SBL.players(game_id=8657))
)

Below are the five actions in the SPADL format leading up to Belgium’s second goal.

game_id

period_id

seconds

team

player

start_x

start_y

end_x

end_y

actiontype

result

bodypart

8657

2

2179

Belgium

Witsel

37.1

44.8

53.8

48.2

pass

success

foot

8657

2

2181

Belgium

De Bruyne

53.8

48.2

70.6

42.2

dribble

success

foot

8657

2

2184

Belgium

De Bruyne

70.6

42.2

87.4

49.1

pass

success

foot

8657

2

2185

Belgium

Hazard

87.4

49.1

97.9

38.7

dribble

success

foot

8657

2

2187

Belgium

Hazard

97.9

38.7

105

37.4

shot

success

foot

Here is the same phase visualized using the matplotsoccer package

This notebook gives an example of the complete pipeline to download public StatsBomb data and convert it to the SPADL format.

Definitions#

Atomic-SPADL is an alternative version of SPADL which removes the result attribute from SPADL and adds a few new action types. Each action is a now a tuple of the following eleven attributes:

Attribute

Description

game_id

the ID of the game in which the action was performed

period_id

the ID of the game period in which the action was performed

seconds

the action’s start time

player

the player who performed the action

team

the player’s team

x

the x location where the action started

y

the y location where the action started

dx

the distance covered by the action along the x-axis

dy

the distance covered by the action along the y-axis

action_type

the type of the action (e.g., pass, shot, dribble)

bodypart

the player’s body part used for the action

In this representation, all actions are atomic in the sense that they are always completed successfully without interruption. Consequently, while SPADL treats a pass as one action consisting of both the initiation and receival of the pass, Atomic-SPADL sees giving and receiving a pass as two separate actions. Because not all passes successfully reach a teammate, Atomic-SPADL introduces an interception action if the ball was intercepted by the other team or an out event if the ball went out of play. Atomic-SPADL similarly divides shots, freekicks, and corners into two separate actions. Practically, the effect is that this representation helps to distinguish the contribution of the player who initiates the action (e.g., gives the pass) and the player who completes the action (e.g., receives the pass).

Example#

SPADL actions can be converted to their atomic version with the convert_to_atomic() function.

import socceraction.atomic.spadl as atomicspadl

This is what Belgium’s second goal against England in the third place play-off in the 2018 FIFA world cup looks like in the Atomic-SPADL format.

game_id

period_id

seconds

team

player

x

y

dx

dy

actiontype

bodypart

8657

2

2179

Belgium

Witsel

37.1

44.8

0.0

0.0

dribble

foot

8657

2

2179

Belgium

Witsel

37.1

44.8

16.8

3.4

pass

foot

8657

2

2180

Belgium

De Bruyne

53.8

48.2

0.0

0.0

receival

foot

8657

2

2181

Belgium

De Brunne

53.8

48.2

16.8

-6.0

dribble

foot

8657

2

2184

Belgium

De Bruyne

70.6

42.2

16.8

6.9

pass

foot

8657

2

2184

Belgium

Hazard

87.4

49.1

0.0

0.0

receival

foot

8657

2

2185

Belgium

Hazard

87.4

49.1

10.6

-10.3

dribble

foot

8657

2

2187

Belgium

Hazard

97.9

38.7

7.1

-1.4

shot

foot

8657

2

2187

Belgium

Hazard

105.0

37.4

0.0

0.0

goal

foot


text:
# Image of a closed set is closed under bounded linear transformation between Banach spaces?

Let $A,B$ be Banach spaces and let $T:A \to B$ be a surjective, bounded, linear operator. Let $A_1$ be a non-empty subset of $A$, then:

$T(A_1)$ is closed if and only if $A_1+ \textrm{ker}(T)$ is closed.

I have shown that if $A_1+ \textrm{ker} (T)$ is closed, then $T(A_1)$ is closed, but am unsure of how to proceed in the other direction. Note that we know that $T(A \setminus [A_1 + \textrm{ker}(T)]) = B \setminus T(A_1)$. Figured the reverse direction could be proved using some modification of the Open Mapping Theorem or Closed Graph theorem, but not sure how to tackle this. Thanks.

Assume that $T(A_1)$ is closed. Let $\{x_n\} \subset A_1$, $\{y_n\} \subset ker(T)$ and $x_n +y_n \to z$. Since T is continuous we get $Tx_n =Tx_n +Ty_n \to Tz$. Thus $T(z)$ is the limit of the sequence $\{Tx_n\}$ which lies in $T(A_1)$. It follows that $Tz=Tw$ for some $w \in A_1$. But then $z=w+(z-w) \in A_1 +ker (T)$. This proves that $A_1 +ker (T)$ is closed.

Let

$A_1' = T^{-1}(T(A_1)) = \{ y \in A \mid T(y) \in T(A_1) \} ; \tag 1$

clearly,

$A_1 \subset A_1', \tag 2$

now if $w \in A_1$ and $u \in \ker T$, then

$T(w + u) = T(w) + T(u) = T(w) + 0 = T(w) \in T(A_1), \tag 3$

or

$w + u \in A_1', \tag 4$

which shows that in fact

$A_1 + \ker T \subset A_1'; \tag 5$

and if $z \in A_1'$ then $T(z) \in T(A_1)$, and there is some $x \in A_1$ such that $T(z) = T(x)$, whence

$T(z - x) = T(z) - T(x) = 0; \tag 6$

thus

$z - x \in \ker T, \tag 7$

or

$z \in x + \ker T \subset A_1 + \ker T, \tag 8$

whence

$A_1' \subset A_1 + \ker T; \tag 9$

(5) and (9) together yield

$A_1' = A_1 + \ker T; \tag{10}$

we now conclude by observing that since

$A_1'= T^{-1}(T(A_1)) \tag{11}$

(as (1) asserts), and $T(A_1)$ is closed, the continuity of $T$ implies $T^{-1}(T(A_1))$ is closed as well; thus (10) and (11) together yield $A_1 + \ker T$ is closed.


text:
# Why does comoving Hubble radius increase with time?

1. Feb 27, 2012

### ck99

I am looking at inflation at the moment, and it says in my textbook that (aH)^(-1) is constantly increasing in matter or radiation dominated epochs.

a is always positive and always increasing. This tells me that da/dt is positive. I think that setting the universe to MD/RD means that da/dt is decreasing with time (eg a decelerating universe as there is no cosmological constant driving expansion). So dt/da (which is another expression for comoving Hubble radius) is increasing with time.

Have I got this right?

2. Feb 27, 2012

### bapowell

da/dt decreasing in time is not a decelerating universe, it's a contracting one.

EDIT: This statement is obviously incorrect. See below for my efforts to redeem myself...

Last edited: Feb 27, 2012
3. Feb 27, 2012

### ck99

If todays value of da/dt is lower than yesterdays, but they are both greater than 0, doesn't that mean the universe is expanding, but that the rate of expansion is slowing down?

4. Feb 27, 2012

### bapowell

Errr...of course. My apologies. Yes, the comoving Hubble radius is indeed increasing in time during RD/MD, because as you say the universe is decelerating. You've undoubtedly noticed this is not the case during inflation.

5. Feb 27, 2012

### Deuterium2H

Ummm...no one has answered the question correctly so far.

Comoving Coordinates (and Comoving distances) do NOT increase with the expansion of the Universe, and do not increase in time. That is the whole point of the Comoving coordinate system.

Proper distances increase in time.

The Hubble Paramater is measuring the rate of change of Scale Factor (da/dt) divided by the Scale factor (a). The Scale factor is time-dependent, and is directly related to the Proper distance.

Last edited: Feb 27, 2012
6. Feb 27, 2012

### bapowell

I think you mean to say that comoving coordinate systems do expand with the universe, so that comoving distances have constant coordinates.

It does not follow that all quantities measured in comoving coordinates are constant -- what would be the point of using them then?? Any proper distance that is not increasing with the expansion will have non-constant comoving coordinates.

The Hubble radius, $H^{-1}$, measured with respect to comoving coordinates is the comoving Hubble radius, $(Ha)^{-1}$. It very much depends on time.

7. Feb 27, 2012

### Deuterium2H

I didn't say that "all quantities measured in comoving coordinates are constant". I specifically said that comoving distances are constant. And any equation involving the Hubble paramater (which involves the scale factor) is time-dependent, because it is based upon proper distance at a given (fixed) instant in time.

8. Feb 27, 2012

### bapowell

OK, well then what does this have to do with the OP? He's asking about the comoving Hubble parameter, which is the Hubble parameter in comoving coordinates. It is not a comoving distance!
I don't know what this has to do with the OP. Looks like you're making things more confused than they need to be. He's asking about the comoving Hubble radius. It is increasing in an RD/MD universe. So please tell me where we've gone wrong here?

9. Feb 27, 2012

### Deuterium2H

Perhaps I misunderstood the question of the OP. The term Comoving Hubble radius only makes sense when measured at a particular instant of cosmological time, and it is dependent upon the coordinate (proper) distance at the time of measuement.

And I agree, I have probably needlessly confused the question in the original post.

Last edited: Feb 27, 2012
10. Feb 27, 2012

### bapowell

Yes, it is the quantity $(aH)^{-1}$, which is a function of coordinate time, $t$.

11. Feb 27, 2012

### cepheid

Staff Emeritus
Yeah, your reasoning sounds alright to me. As you've already pointed out, just from the definition of the Hubble parameter $(aH)^{-1} = (\dot{a})^{-1}$. For the rad-dominated and matter-dominated cases, a~t1/2 and a~t2/3 respectively. (I believe that these only apply for models with no cosmological constant). Differentiating those, you get da/dt ~ t-1/2 or t-1/3 respectively. So a-dot decreases with time, which means that its reciprocal increases with time.

12. Feb 28, 2012

### ck99

Hooray! Nothing is simple in cosmology, is it? :D