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# Surjectivity of Invariants

Suppose $V, W, U$ are $Z_p$ module over a field $F$ of characteristic $p$ and $V=W \oplus U$. Is there a degree preserving surjective map from $F[V]^{Z_p}$ to $F[W]^{Z_p}$ ? In non-modular case the Reynold's operator does the job but in this case I don't have any clue.

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That holds more generally: If $k$ is any commutative ring with unit, $G$ any group and if $U,W$ are $k$-free $kG$-modules, $V = U \oplus W$, then there is a $k$-linear surjection $k[V]^G \to k[W]^G$:

Let $r: V \to W$ be the projection. Then $k[V]=k[W] \otimes k[U]$ and $r$ induces a $k$-algebra hom. $r: k[V] \mapsto k[W]$. Aussume for the moment we know that $r$ is compatible with the $G$ action, i.e. $r ( g \cdot v) = g \cdot r(v)$. Then $r$ preserves the invariants: $r: k[V]^G \to k[W]^G$.

Moreover, $r$ is surjective on the invariants. For, let $w \in k[W]^G$ and put $v := w \otimes 1$. Then $g \cdot v = (g\cdot w)\otimes (g\cdot 1) = w \otimes 1 = v$, i.e. $v \in k[V]^G$ and $r(v) = w$.

Now let's show that $r$ is compatible with the $G$-action. Note that we have an $k$-algebra homomorphism $\epsilon: k[U] \to k$ that satisfies $\epsilon(g \cdot u) = \epsilon(u)$: If $\lbrace x_1,...,x_n\rbrace$ is a $k$-basis of $U$ then $\epsilon$ is just the augmentation $k[x_1,...,x_n] \to k$.

Now $r$ is given by $r(w \otimes u) = \epsilon(u) \cdot w$ and we obtain $$r(g (w \otimes u)) = r(g w \otimes g u) = \epsilon(g u) (g w) =\epsilon(u) (gw) = g (\epsilon(u)w) = g\; r(w \otimes u).$$

BTW: To my understanding, the Reynold's operator yields $k[V]^H \to k[V]^G$ for $H \le G$ and adresses another problem than yours.

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Added: $r$ is degree-preserving. For let $u,w$ be monomials such that $r(u \otimes w) \neq 0$. Then $\deg u = 0$ and $\deg r(u \otimes w) = \deg(u \cdot w) = \deg w = \deg w + \deg u = \deg(w \otimes u)$. –  Ralph Mar 19 '12 at 15:50
@Ralph Your argument shows, still more generally, that if $U$ is a submodule of $V$ with quotient module $V/U \cong W$ then there is a $k$-algebra homomorphism $k[V] \rightarrow k[W]$ that preserves degrees and invariants. Is there any characterization of when the induced map $k[V]^G \rightarrow k[W]^G$ is surjective? (One small typo: the Reynolds averaging operator sends $k[V]^H$ to $k[V]^G$ for $H \le G$.) –  Mark Wildon Mar 19 '12 at 19:44
I don't know, if there is such a characterization ($W$ $k$-free is sufficient, probably also $W$ $k$-projective). And thanks for pointing out the correct order of the Reynolds operator. –  Ralph Mar 20 '12 at 1:45

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# How can I pool correlations?

I'm measuring a phenomenon over several days. On any particular day, the phenomenon is well correlated. I was wondering how I could pool the correlations over several days to get a single estimate of how well correlated the data are.

The last observation yesterday has nothing to do with the first observation today, so I don't think computing the correlation as if it were a time series is appropriate.

What is the recommended approach for analyzing correlations of this nature?

## 1 Answer

You give very little detail or context. It seems as you for multiple days have observations of some $(x,y)$ pairs that day, with independence between days. Neither you have told us what kind of correlation, I will assume now pearson correlation. If you have data for $T$ days, then, you will have estimated correlation coefficients $r_1, \dotsc, r_T$ and assume they are independent estimates of the same underlying true correlation $\rho$. So you want a common, or pooled, estimate, $\hat{\rho}$ summarizing the individual estimates.

Lets say you have observed $n$ pairs each day. The usual way of pooling estimates is to weight by the inverse variance. That is difficult in this case, since an expression for the variance is complicated and depends on the unknown $\rho$, see https://en.wikipedia.org/wiki/Pearson_correlation_coefficient or Expected value and variance of sample correlation But one idea could be to use this iteratively, starting with equal weights.

Another idea is to use the Fisher transform $F(r)=\frac12 \ln\left(\frac{1+r}{1-r}\right)$ which have approximately constant variance $\text{Var}(F(r))=\frac{1}{n-3}$. So you can take the average (equal weights) of $F(r_1), \dotsc, F(r_T)$ and the inverse transform of the result.

Googling for "pooling of correlation" gives a lot of relevant hits.


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# Introduction

It’s half-past twelve and you’re still working on that geometry homework due tomorrow when a thought pops up: “What is the point of math? How is this gonna help me?” It’s common knowledge that math can be utilized in different professions. Math can also be applied to different parts of daily life the quicker you’re able to count change from your wallet, the less likely you’re holding up a line behind you. However, in addition to the mundane day to day applications, math can also be observed within nature. Mathematicians have used math to explain nature for hundreds of years. Whether the hidden math in the orbits of celestial bodies or the seemingly random arrangement of flower petals, math is everywhere around us. It just takes an acute mind to notice and see the patterns.

# Fibonacci Sequence

The Fibonacci Sequence is one of the most famous sequences in the world. The sequence first appeared in the book Liber Abaci, written by Italian mathematician Fibonacci (whose real name was Leonardo of Pisa). This sequence is commonly denoted as:

$F_0 = 0, F_1 = 1$ $F_n = F_{n-1} + F_{n-2}$ for $$n \geq 2$$

Thus, the sequence goes: $$0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55,\dots$$ where the sums of the two previous numbers generate the next number in the sequence. This may seem like a useless string of numbers, but it’s the spiral that forms from this sequence that keeps popping up again and again in nature — Fibonacci’s spiral. The spiral can be created on a 2D plane using the squares of the numbers in the Fibonacci sequence. Take two squares with lengths 1, add another square with lengths 2, then add another square with lengths 3. Continue this pattern while drawing a continuous arc from the corners of each square. This creates the famous Fibonacci spiral, which is constantly observed by mathematicians in nature.

From the pattern used to create the spiral, we notice an interesting pattern between the Fibonacci sequence and the squared version of the sequence. The product of two consecutive Fibonacci numbers is equal to the square of the first consecutive Fibonacci number plus the square of the previous Fibonacci numbers before it.

$\text{Sequence: } 1 , 1 , 2 , 3 , 5 , 8 , 13 , 21 , 34 , 55, \dots$ $\text{Squared: } 1 , 1 , 4 , 9 , 25 , 64 , 169 , 441 , 1156, 3025, \dots$ $1 + 1 + 4 = 6 = 2\cdot3$ $1 + 1 + 4 + 9 = 15 = 3\cdot5$ $1 + 1 + 4 + 9 + 25 = 40 = 5 \cdot8$ $1 + 1 + 4 + 9 + 25 + 64 = 104 = 8\cdot13$
Using this information, we are able to form the conjecture: $\sum_{i=1}^{n} F^{2}_{i} = F_{n}F_{n+1}$

Proof. We wish to prove that$F^{2}_{1} + F^{2}_{2} + F^{2}_{3} +\dots + F^{2}_{n} = F_{n} F_{n+1}$ We will proceed by induction.

Base Case: $$n=1$$ $F^{2}_{1} = F_{1} \cdot F_{1+1}$ $1^2=1 \cdot 1$ Base case holds.

Induction Hypothesis: Assume that $F^{2}_{1} + F^{2}_{2} + F^{2}_{3} +\dots + F^{2}_{k} = F_{k} F_{k+1}$ Induction Step: Consider $F^{2}_{1} + F^{2}_{2} + F^{2}_{3} +\dots + F^{2}_{k} + F^2_{k+1}$ by the hypothesis, $=F_{k} F_{k+1}+ F^2_{k+1}$ $=F_{k+1}(F_k +F_{k+1})$ Using the Fibonacci sequence definition, we know that $$F_{k}+F_{k+1} = F_{k+2}$$ so we get $F^{2}_{1} + F^{2}_{2} + F^{2}_{3} +\dots + F^{2}_{k} + F^2_{k+1} = F_{k+1}F_{k+2}$ as desired. ◻

Interestingly, we can derive the second most famous irrational number using the Fibonacci sequence. When we reverse the sequence above and divide a Fibonacci number by the one before it, we’re able to obtain the golden ratio. How close the number is to the actual Golden Ratio is dependent on how large the Fibonacci numbers are.

$3/2= 1.5$ $5/3 = 1.67$ $8/5 = 1.6$ $13/8 = 1.625$ $21/13 = 1.615$ $34/21 = 1.619$ $55/34 = 1.6176$ $89/55 = 1.61818$

# The Golden Ratio

The Golden Ratio is an irrational number commonly denoted with the Greek symbol phi ($$\phi$$). It is also referred to as the divine proportion because the ratio $$\phi$$ is the most pleasing ratio to view. It’s said that ancient Greek mathematicians first noticed the ratio but it was Euler who first described it as: “A straight line is said to have been cut in extreme and mean ratio when, as the whole line is to the greater segment, so is the greater to the lesses.” Euclid said that an object is considered to be in the golden ratio when the sum of the two lengths over the longer side is equal to the longer side over the shorter side. This is written as:

$\frac{a+b}{a} = \frac{a}{b} = \phi$

Using Euler’s definition of the Golden Ratio, a formula for the Golden Ratio was derived using the quadratic formula. The quadratic formula is used to derive $$a$$, which is then used to find the ratio between $$a$$ and $$b$$.

$b(a+b) = a^2$ $a^2 - ab - b^2$ $\frac{-b \pm \sqrt{(-b)^2 - 4(1)(-b)^2}}{2(1)} = a$ $\frac{1\pm\sqrt{5}}{2}b = a$ $\phi = \frac{1+\sqrt{5}}{2}$ We know that the negative root not chosen because the ratio between lengths $$a$$ and $$b$$ isn’t negative.

This is the most popular method for finding the Golden Ratio, but there are other methods to finding this value. Below is another more interesting way to write the Golden Ratio. $5^{0.5} \times 0.5 + 0.5$ Since the Golden Ratio can be derived from the Fibonacci sequence, many things in nature can be observed in tandem with both of these numbers. The most prominent example are plants. The numbers of petals on a flower is usually a Fibonacci number. Lilies have 3 petals, buttercups have 5 and asters have 13. Sunflower seeds are both arranged in sets of Fibonacci numbers and also form multiple Fibonacci spirals. One of the most famous examples is the nautilus shell, which creates a near perfect Fibonacci spiral. This list goes on, from hurricane feeder bands to earthquake fractures and even the micro algae living in a pond. The Fibonacci sequence is found everywhere in nature.

There are, of course exceptions to the Fibonacci sequence and the Golden Ratio. In fact, there are more examples that don’t follow this ratio than ones that follow this ratio. Therefore, it can’t be precisely pinpointed why the Fibonacci number appears in nature; it’s completely possible that this number is not as common in nature as mathematicians thought.

# Fractals

Another branch of math constantly found in nature are fractals. Fractals are defined as a never ending, self-similar pattern. Zooming into a fractal yields the same result as when you look at the fractal zoomed out. Fractals can be generated by looping computer programs. In a nutshell, fractals optimize certain criterion like volume, surface area, or diffusion. This optimization is what causes fractals to appear again and again in nature. An example of fractals in nature is the creation of rivers. When rainwater flows down from an area of higher elevation, the water carves out river channels through hundreds if not thousands of years of erosion. River fractals form from this because the water flowing down is constantly “searching” for a way to make it down. The constant flow of water is a repetitive process which creates the fractals seen in the picture below.

Romanesco broccoli is an example of a natural fractal. The surface area of the plant is optimised due to these fractals. The more surface area a plant has correlates to more photosynthesis. More sunlight means more glucose production rate, which results in a better survival rate for the plant. Additionally, fractals are the simplest way for plants to create a structure using the least amount of DNA sequences (fractals repeat themselves). Natural fractals are a result of natural selection because fractals provide plants with the best chance of survival.

Fractals have some extremely fascinating properties, which anyone who has heard of the Coastline paradox may be familiar with. For a fractal’s finite area, it has an infinite perimeter. Similarly, a coastline’s length can vary drastically based on whether it’s measured by intervals of 100 miles or 100 feet. A coastline that is infinite in length seems counter-intuitive, but it makes sense when you consider the existence of fractals along the coast. Every protruding border or cliff is able to increase the perimeter of the coastline.

# Hexagons

One of the most prevalent shapes in nature is the hexagon. Why is this six-sided shape observed everywhere in nature? This actually comes down to some of the special geometric properties of hexagons. The most well known example of hexagons in nature is the beehive. As Charles Darwin puts it: “absolutely perfect in economising labour and wax." Bees use hexagons because of how good hexagons are at tiling a 2D plane. Mathematically, hexagons are the most similar shape to circles that can still be tiled. In 1999, Thomas C. Hales proved in his "The Honeycomb Conjecture" which stated that hexagons have the highest perimeter to area ratio out of all other shapes; hexagons were the most efficient way to tile an area. It requires the least amount of resources to tile a hexagon pattern. This is important for bees because it takes them 8oz of honey just to create 1oz of wax. Bees actually tile with circles and those circles harden into the ubiquitous hexagon shape because hexagons are the most energy-efficient shape. Architecturally, circles are the strongest shape for containing pressure. This is why tanks of compressed air or fire extinguishers are circular. This property of circles is passed to hexagons, allowing bees to use the hexagon holes as really effective storage. This is one of the many reasons bees chose to adopt the hexagon for their hives; hexagon are perfect for the bees to store pollen, nectar and larvae. This was probably also the product of natural selection in bees.

When water freezes and ice forms, the water molecules form lattice shaped hexagons. Since $$H_2O$$ is polar, the molecule isn’t perfectly symmetrical. As seen in figure 6, the angle between two hydrogen atoms on a water molecule varies from $$104.5^\circ$$ to $$106.6^\circ$$. Six water molecules is the perfect amount to create the hexagonal ring. The electrical force between adjacent water molecules roughly even out the angle difference. To our eyes, the angle appears to be $$120^\circ$$. This property of water causes ice and snowflakes to be hexagonal.

Hexagons can also be observed in plants. Plant cells are hexagonal. Like in beehives, hexagons are the toughest shape that uses the least resources. Plant cells absorb water and a strong shape is required to withstand the turgid pressure inside the cell.

# References

1. Ball, Philip. “Why Nature Prefers Hexagons.” Nautilus, 7 Apr. 2016,http://nautil.us/issue/35/boundaries/why-nature-prefers-hexagons

2. Bethany. “Math in Nature: 5 Stunning Ways We See Math in the World.” Math Geek Mama, 8 Jul. 2019, https://mathgeekmama.com/math-in-nature/

3. Bogomolny, A. "Golden Ratio in Geometry."https://www.cut-the-knot.org/do_you_know/GoldenRatio.shtml

4. Chasnov, Jeffrey. “Sum of Fibonacci numbers squared | Lecture 10 | Fibonacci Numbers and the Golden Ratio.” Youtube. 10 Oct. 2016,https://www.youtube.com/watch?v=ruIwND9ytpE

5. “Fractal Rivers.” FractalFoundation, Accessed 9 Jan 2021, https://fractalfoundation.org/OFC/OFC-1-4.html#: :text=Much

6. Jenkins, Patricia. “The Mathematics of Nature’s Patterns.” CuriOdyssey, 3 Mar. 2017, https://curiodyssey.org/blog/mathematics-of-natures-patterns/

7. Mann, Adam. “Phi: The Golden Ratio.” LiveScience, 25 Nov. 2019, https://www.livescience.com/37704-phi-golden-ratio.html#: :text=This

8. Otis, Waldo. “Describing Nature with Beautiful Mathematics.” Medium, 31 May 2019, https://medium.com/however-mathematics/describing-nature-with-beautiful-mathematics-33d28b3ba649

9. Pearce, Kyle. “Fractals in Nature: Develop Your Pattern Recognition Skills In The Forest.” DIYGenius, 4 Nov. 2018, https://www.diygenius.com/fractals-in-nature/#: :text=A

10. Sullivan, Kathryn DeMuth. “Ever Wonder Why There are So Many Hexagons in Nature?” labroots, 13 Aug. 2017. https://www.labroots.com/trending/videos/10939/ever-wonder-why-there-are-so-many-hexagons-in-nature#: :text=Hexagons

11. “The Golden Ratio.” National Geographic Society, Accessed 9 Jan. 2021, https://www.nationalgeographic.org/media/golden-ratio/#: :text=In

12. theJourneyofPurpose TJOP. “Fibonacci Sequence in Nature.” Youtube. 11 May 2013, https://www.youtube.com/watch?v=nt2OlMAJj6o


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# SuperHiggs Mechanism on different Backgrounds & Compactifications

I've been studying Bagger & Giannakis paper on the SuperHiggs Mechanism found here.

The paper shows how SUSY is broken by a $B_{\mu\nu}$ gauge field background restricted to $T^3$ in $M^7\times T^3$ Compactification of the heterotic string.

My question is, have these results been generalised to different Backgrounds and/or Compactifications (Perhaps even Calabi Yau 3-folds)?

So far my search has shown up no results, any pointers in the right direction are greatly appreciated, Thanks!

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Please don't link to the PDF (or at least not exclusively): the abstract would be much more helpful. Thanks. In this case, the abstract to the paper is arxiv.org/abs/hep-th/0502107v1 –  José Figueroa-O'Farrill Sep 15 '11 at 10:51
Ahhh, thanks! Just changed it –  Michael Sep 15 '11 at 10:54
While we are at the topic, please link to arxiv.org/abs/hep-th/0502107 without version number (v1). That way we automatically get the latest version. –  Qmechanic Dec 10 '12 at 14:06


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# zbMATH — the first resource for mathematics

## Hajac, Piotr M.

Compute Distance To:
 Author ID: hajac.piotr-m Published as: Hajac, Piotr M.
 Documents Indexed: 35 Publications since 1993, including 1 Book
all top 5

#### Co-Authors

 5 single-authored 9 Matthes, Rainer 7 Szymański, Wojciech 6 Dabrowski, Ludwik 6 Zieliński, Bartosz 5 Baum, Paul F. 3 Hadfield, Tom 3 Wagner, Elmar 2 Brzeziński, Tomasz 2 Kaygun, Atabey 2 Khalkhali, Masoud 2 Rangipour, Bahram 2 Sommerhäuser, Yorck 1 De Commer, Kenny 1 Grosse, Harald 1 Krähmer, Ulrich 1 Landi, Giovanni 1 Majid, Shahn 1 Masuda, Tetsuya 1 Pusz, Wiesław 1 Rennie, Adam 1 Rudnik, Jan 1 Siniscalco, Pasquale
all top 5

#### Journals

 5 Comptes Rendus. Mathématique. Académie des Sciences, Paris 3 Communications in Mathematical Physics 3 $K$-Theory 3 SIGMA. Symmetry, Integrability and Geometry: Methods and Applications 3 Journal of Noncommutative Geometry 2 Journal of Mathematical Physics 2 The New York Journal of Mathematics 2 Documenta Mathematica 2 Algebras and Representation Theory 1 Communications in Algebra 1 Letters in Mathematical Physics 1 Reports on Mathematical Physics 1 Comptes Rendus de l’Académie des Sciences. Série I. Mathématique 1 Czechoslovak Journal of Physics
all top 5

#### Classification

 23 Functional analysis (46-XX) 17 Global analysis, analysis on manifolds (58-XX) 14 Associative rings and algebras (16-XX) 9 Quantum Theory (81-XX) 4 $K$-theory (19-XX) 2 Order, lattices, ordered algebraic structures (06-XX) 2 Nonassociative rings and algebras (17-XX) 2 Category theory, homological algebra (18-XX) 2 Differential geometry (53-XX) 1 General mathematics (00-XX) 1 Topological groups, Lie groups (22-XX) 1 Dynamical systems and ergodic theory (37-XX) 1 Operator theory (47-XX) 1 Algebraic topology (55-XX) 1 Manifolds and cell complexes (57-XX) 1 Relativity and gravitational theory (83-XX)


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Lemma 57.13.2. Let $k$ be a field. Let $X$, $Y$ be finite type schemes over $k$ with $X$ separated. Let $F : D^ b_{\textit{Coh}}(\mathcal{O}_ X) \to D^ b_{\textit{Coh}}(\mathcal{O}_ Y)$ be a $k$-linear exact functor sending $\textit{Coh}(\mathcal{O}_ X) \subset D^ b_{\textit{Coh}}(\mathcal{O}_ X)$ into $\textit{Coh}(\mathcal{O}_ Y) \subset D^ b_{\textit{Coh}}(\mathcal{O}_ Y)$. Then there exists a Fourier-Mukai functor $F' : D^ b_{\textit{Coh}}(\mathcal{O}_ X) \to D^ b_{\textit{Coh}}(\mathcal{O}_ Y)$ whose kernel is a coherent $\mathcal{O}_{X \times Y}$-module $\mathcal{K}$ flat over $X$ and with support finite over $Y$ which is a sibling of $F$.

Proof. Denote $H : \textit{Coh}(\mathcal{O}_ X) \to \textit{Coh}(\mathcal{O}_ Y)$ the restriction of $F$. Since $F$ is an exact functor of triangulated categories, we see that $H$ is an exact functor of abelian categories. Of course $H$ is $k$-linear as $F$ is. By Functors and Morphisms, Lemma 56.7.5 we obtain a coherent $\mathcal{O}_{X \times Y}$-module $\mathcal{K}$ which is flat over $X$ and has support finite over $Y$. Let $F'$ be the Fourier-Mukai functor defined using $\mathcal{K}$ so that $F'$ restricts to $H$ on $\textit{Coh}(\mathcal{O}_ X)$. The functor $F'$ sends $D^ b_{\textit{Coh}}(\mathcal{O}_ X)$ into $D^ b_{\textit{Coh}}(\mathcal{O}_ Y)$ by Lemma 57.9.5. Observe that $F$ and $F'$ satisfy the first and second condition of Lemma 57.11.2 and hence are siblings. $\square$

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Electron configuration and the 2p orbital

When drawing orbital diagrams I know electrons are placed in the 2p boxes singly first then paired because of they have parallel rotations and the Pauli exclusion principle states that an orbital can only house 2 electrons and those electrons must be of opposite spin. My question is why do electrons behave this way in the 2p orbital but not the 1s or 2s? And why does the p subshell contain three 2p orbitals?

As far as electron configurations and orbital diagrams are concerned I've gotten to Ne and the 2p configuration confused me so I though I'd ask before moving on.

• they have parallel rotations . Electrons do not rotate. They are not little classical balls orbiting the nucleus. why do electrons behave this way in the 2p orbital but not the 1s or 2s. What do you mean by this? The Pauli exclusion principle applies for all electrons. – bon Nov 14 '15 at 20:08
• I mean why do the three 2p orbitals fill singly first then pair up instead one orbital filling up at a time. – Chinasa Nov 14 '15 at 20:25
• Because the 3 p-orbitals are degenerate but putting two electrons into the same orbital (with opposite spin of course) has an energy cost associated with the repulsion between the electrons. – bon Nov 14 '15 at 20:30
• Cross posted to Physics SE. Bad form... – Jon Custer Nov 14 '15 at 20:39
• Thats not allowed? What wrong with different answers from different people? – Chinasa Nov 14 '15 at 20:49


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## Introduction

Semiconductor devices have attracted considerable attention in recent years. In particular, organic semiconductors are being used in devices including organic photovoltaics (OPVs)1,2,3, organic light-emitting diodes (OLEDs)3,4,5,6, and organic field-effect transistors2,7,8,9,10,11,12,13 including for fabricating biosensors and radio frequency identification (RFID) tags. Organic semiconductor devices afford advantages such as a low-temperature fabrication process, low cost, and high flexibility. However, they have low carrier mobility and high carrier injection barriers compared with inorganic semiconductor devices. To overcome this disadvantage, organic semiconductors prepared by single crystallization and having a mobility of 1 cm2 V−1 s−1 or more, which exceeds that of amorphous silicon, have been fabricated recently14,15. Regarding carrier injection, the formation of the barrier depends on the energetic alignment of an electrode and an organic semiconductor16. Hence, the interface modifications such as the introduction of hole injection layers (i.e., buffer layers), electrode surface modification, and dopant implantation has been conventionally performed to reduce the barrier height by generating new energetic interface states3,5,15,17,18,19,20,21,22,23,24.

Triphenylamine derivatives are well recognized as a superior hole-transport material. Stacked photoelectric conversion devices with triphenylamine derivative as a hole transport layer are used in the organic photoreceptors of laser printers25,26. Triphenylamine derivative single crystals produced by the solution method27,28,29 exhibit excellent conductive properties, and their hole-transport activation energy is equivalent to that of pentacene30 and rubrene21 single crystals. Furthermore, stable and energetically favorable hole transport has been achieved via the first oxidation state of triphenylamine derivative31. Therefore, triphenylamine derivatives are attracting attention for application to highly functional organic devices. In this regard, efficient hole injection is required at the electrode/triphenylamine derivative interface. In our previous study, we reduced the interfacial energy barrier of a Au/triphenylamine derivative layered device to 0.06 eV by inserting a C60-doped triphenylamine derivative buffer layer at the interface32. However, further investigations need to be conducted to achieve lower barrier heights (i.e., ohmic contact) and to clarify the barrier height reduction mechanism.

In this study, we report a Au/triphenylamine derivative layered device that has an ohmic contact with hole injection (ΦB = kT ≤ 0.025 eV, where ΦB is the hole injection barrier, k is the Boltzmann constant, and T is the temperature) at room temperature (298 K) and a novel hole-injection mechanism. We used an additional hole injection layer with a mixture of fullerene and triphenylamine derivative. Further, we used not only C60 but also C70 as a fullerene, and we used α-phenyl-4′-[(4-methoxyphenyl)phenylamino]stilbene (Fig. 1a) as a triphenylamine derivative and call it TPA. Fullerenes are known to be famous electron transport materials for OPVs and OLEDs33,34,35, but no study has yet applied C70 to the hole injection layer. A Au/fullerene-doped TPA/TPA/Au layered device was prepared, and its rectification characteristics were evaluated. In particular, the hole injection property at the Au/fullerene-doped TPA interface was quantitatively evaluated. Further, we conducted cyclic voltammetry to clarify the hole-injection mechanism at this interface.

## Results and discussion

### Hole-injection property of fabricated device

Figure 1b shows a schematic of the fullerene-doped TPA dual-layer device fabricated in this study. Hole injection from the bottom and the top Au electrode was defined as the forward and reverse direction, respectively. Figure 2a shows the current density–electric field (J-E) properties of the Au/C60 and C70-doped TPA/TPA/Au layered devices. Symmetrical J-E characteristics were observed in the forward and reverse directions for the non-doped (Au/TPA/Au) device, indicating that the energy barriers for hole injection at both Au/TPA interfaces were equal. By contrast, the threshold electric field of the C60-doped device was confirmed to be lowered only in the forward direction; further, that of the C70-doped device was drastically suppressed in the forward direction. These results demonstrated that hole injection at the Au/fullerene-doped TPA interface occurs easily compared to that at the Au/TPA interface. It should be noted that the threshold electric fields of the non-doped and the C60-doped devices in the reverse direction are almost the same. Therefore, we successfully developed a device with a rectifying property by inserting C60- and C70-doped TPA buffer layers. This observed rectification conductivity is very useful for organic semiconductor device applications such as RFID tags. In the reverse direction in Fig. 2a, the reason why a higher current was observed in the C70-doped device than in the C60-doped device remains unclear. Figure 2b shows the dependence of the J-E characteristics in the forward direction on the C70 doping amount (0, 0.5, and 1 mol%) in TPA. A higher current density was achieved at a lower electric field as the C70 doping concentration increased. Therefore, fullerene doping induced efficient hole injection.

To consider the influence of fullerene doping on hole injection in detail, the J-E property of the Au/1 mol% C70-doped TPA/TPA/Au layered device was plotted on a logarithmic scale, as shown in Fig. 3a. Generally, the J-E characteristics of organic semiconductors consist of three types of current: (1) Schottky current (JE0.5), (2) ohmic current (JE), and (3) space charge limited current (JEx (2 $$\le$$ x)). Significantly, Fig. 3a showed that the 1 mol% C70-doped device exhibited an ohmic-type current response immediately after an electric field was applied, indicating ohmic contact with hole injection at the Au/C70-doped TPA interface at room temperature. In other words, no Schottky current was observed in the device. Subsequently, hole transport became a rate-determining process from around 0.2 × 104 V cm−1, and a space charge limited current was observed in Fig. 3a. It should be noted that JE0.5 relationships were observed in the logJ-logE characteristics of the Au/1 mol% C70-doped TPA/TPA/Au device under the measurement temperature conditions of 4.5 °C and − 22.3 °C as shown in Supplementary Fig. S1.

We quantitatively evaluated the energy barrier height for hole injection at the Au/fullerene-doped TPA interface through a Richardson-Schottky plot analysis32,36. The hole-injection barrier height (ΦB) was obtained using the following equation:

$$\mathrm{ln}\left(J/{T}^{2}\right)=\mathrm{ln}\left({AA}^{*}\right)-\frac{q\left({\Phi }_{B}-E/n\right)}{kT}$$
(1)

where T is the temperature; A, the area; A*, the Richardson constant; q, the electronic charge of an electron; n, the ideal factor; and k, the Boltzmann constant. After the J-E characteristics of the device at various temperatures were measured as shown in Supplementary Fig. S2, the data were plotted according to the relationship logJ vs. E0.5. As a result, straight lines (i.e., Schottky lines) were obtained as shown in Supplementary Fig. S3a. Then, the current densities in the absence of an electric field (J0) were determined by extrapolating the Schottky lines, as listed in Supplementary Table S1. Notably, the maximum J0 was two orders of magnitude greater than the minimum J0. Finally, the Richardson plot (lnJ0/T2 − T−1) was drawn as shown in Supplementary Fig. S3b, and ΦB was calculated from the slope of the Richardson line. Figure 3b shows the ΦB values as functions of the fullerene-doping concentration. ΦB for the non-doped (Au/TPA/Au) device was calculated to be 0.37 eV; this was almost same as ΦB estimated from the work function of a Au electrode (~ 5.1 eV)37 and the ionization potential of TPA (~ 5.5 eV)28. The sigmoid-shaped behavior of ΦB was observed for both C60- and C70-doped devices. However, C70 doping had a higher barrier-reduction effect than C60 doping. Remarkably, ΦB for the Au/1 mol% C70-doped TPA/TPA/Au device was determined to be 0.021 eV; this satisfied the condition for ohmic contact (≤ 0.025 eV at room temperature). It is considered that there are multiple levels of barriers among the Au/fullerene-doped TPA/TPA. Although the details are unclear, the Richardson-Schottky plot analysis revealed that the barrier of the rate-determining step (probably hole injection at the Au/fullerene-doped TPA) is 0.021 eV in the Au/1 mol% C70-doped TPA/TPA/Au device. With the doping of more than 1 mol% C60 and C70, the barrier height did not decrease further, probably because it is difficult to dissolve C60 and C70 in organic solvents at a concentration of more than 1 mol% to TPA. Therefore, a technique for doping fullerenes at higher concentrations must be developed in the future. Overall, we reduced the hole injection barrier by 0.324 and 0.349 eV by introducing the C60- and C70-doped TPA layer as a hole injection layer, respectively, and successfully formed an ohmic contact at the Au/1 mol% C70-doped TPA interface.

### Consideration of hole-injection mechanism

We demonstrated barrier height reduction with hole injection at the Au/fullerene-doped TPA interface. Next, we focused on the barrier reduction mechanism. Lee reported that the hole-injection barrier was lowered by the interaction between the fullerene (C60 only) and the Al electrode38. In this study, we used fullerenes (both C60 and C70) as a dopant in the TPA layer. Therefore, the reduced hole-injection barrier was attributed to the intermolecular interaction between the fullerene and the TPA (and not the electrode). Actually, the ultraviolet photoelectron spectroscopy (UPS) result of the films of Au only and 1 mol% C70-coated Au shown in Supplementary Fig. S4 revealed that there is negligibly slight interaction between fullerene and Au electrode because the obtained both work functions were equivalent (~ 5.2 eV).

First, we conducted UV–vis spectrometry (V-650 spectrophotometer, JASCO Corp.) to observe the interaction, for example, the formation of a charge-transfer (CT) complex. However, no new absorption band appeared in the spectrum for a sample with a mixture of 1 mol% C70 and TPA, as shown in Supplementary Fig. S5. Considering the accuracy of the measuring instrument, it was suggested that negligibly slight amount of TPA-fullerene CT complex might be formed.

Next, photoelectron yield spectroscopy (PYS) was performed. As shown in Supplementary Fig. S6, the behaviors of PYS spectra for the films of TPA only and 1 mol% C60-doped TPA were similar, and the obtained both ionization potentials were also equivalent. Therefore, the fullerene dope would not form new energy levels associated with charge injection/extraction. This suggests that fullerenes contribute to barrier reduction as polarizable substances, not donors or acceptors.

Supplementary Fig. S7 shows the J-E characteristics of a Au/evaporated C70/TPA/Au layered device along with that of the Au/C70-doped TPA/TPA/Au layered device. Based on Supplementary Fig. S7b, the evaporated-C70/TPA and 1 mol% C70-doped TPA/TPA devices exhibited schottky-type (JE0.5) and ohmic-type (JE) current responses immediately after an electric field was applied, respectively, which indicates that both devices have different hole injection mechanisms. As a consequence, using the fullerene-doped TPA as a buffer layer has higher conductivity than using the evaporated C70. This suggests that it is important for fullerenes to penetrate the TPA layer morphologically.

Then, we performed electrochemical analysis. Cyclic voltammetry was conducted using the electrochemical cell shown in Fig. 4a. Figure 4b shows cyclic voltammograms of TPA in mixed acetonitrile:toluene solutions with weight ratios of 1:0, 1:1, and 1:3. The half-wave potential (E1/2) for the oxidation of TPA (given by Eq. 2) was determined as the potential at which the current equals the half of diffusion limited current (id)39.

$$\mathrm{TPA }\begin{array}{c}\stackrel{{k}_{f}}{\to }\\ \underset{{k}_{b}}{\leftarrow }\end{array} {\mathrm{TPA}}^{\cdot +}+{e}^{-}$$
(2)

As a result, E1/2 was obtained as 0.76 V, 0.81 V, and 0.86 V vs. Ag/AgCl for acetonitrile:toluene weight ratios of 1:0, 1:1, and 1:3, respectively. id differed depending on the acetonitrile:toluene weight ratios in the order of 1:3 < 1:1 < 1:0. These results indicate that TPA is more easily oxidized with a higher amount of acetonitrile. Because acetonitrile and toluene are polar and nonpolar solvents, respectively, their different ratios result in changes in the relative permittivity of the electrolytes. TPA was easily oxidized to TPA+ when it was surrounded by a polar solvent having a higher relative permittivity.

The mechanism in the solid-state device without a solvent was considered in light of the above findings. Because there are no ions in the solid phase state without any carrier injection, E1/2 must be calculated under the condition of the absence of a supporting electrolyte as an ion source. Figure 5a shows the cyclic voltammetry results for supporting electrolyte concentrations of 2.0, 1.0, 0.5, and 0.05 mmol dm−3 under the cell conditions shown in Fig. 4a. The oxidation potential of TPA decreased and id increased as the concentration of the supporting electrolyte increased. When E1/2 for various supporting electrolyte concentrations with various acetonitrile:toluene weight ratios in the electrolyte was calculated and plotted as a function of the square root of the supporting electrolyte concentration40, the linear function shown in Fig. 5b was obtained. Therefore, E1/2 without the supporting electrolyte can be calculated by extrapolating the straight line shown in Fig. 5b. Consequently, E1/2 for acetonitrile:toluene weight ratios of 1:0, 1:1, and 1:3 was 0.87 V, 0.89 V, and 0.94 V vs. Ag/AgCl, respectively. Even in the solid phase state, TPA was more likely to become TPA+ in the presence of a substance with a higher relative permittivity. Therefore, the ion–dipole interaction between TPA and fullerenes was suggested to facilitate hole injection from the Au electrode to TPA, because fullerenes have a relatively high dipole moment41. The relative permittivities of C60 and C70 were respectively measured to be ~ 3 and ~ 4 using an 879B LCR meter (B&K Precision Corp.) at 1 kHz. Because C70 has a higher relative permittivity than C60, the hole injection barrier was reduced more efficiently and an ohmic contact was probably formed with a larger ion–dipole interaction effect.

In the future, we plan to investigate the best doping material for reducing the barrier height from the viewpoint of relative permittivity.

Finally, electrochemical impedance spectroscopy was carried out to consider the interface energetic characteristics, and the result is shown in Supplementary Fig. S8. When the capacities at interfaces of Au/TPA and Au/fullerene-doped TPA were determined by fitting the impedance curves, the latter was larger than the former. This indicates that the depletion layer (energy barrier) formed at the Au/TPA interface becomes smaller due to the presence of fullerenes, and supports our proposed intermolecular ion–dipole interaction effect.

Figure 6 summarizes the barrier height reduction mechanism with hole injection. Owing to the ion–dipole interaction between TPA and fullerenes, TPA is easily oxidized and stabilized to TPA∙+. This indicates that the rate constant of the forward reaction (kf) in Eq. (2) increased. Therefore, the hole injection at the Au/fullerene-doped TPA interface became highly efficient, because the equilibrium constant (K), expressed as the ratio of kf and the rate constant of the reverse reaction (kb) in Eq. (2) (K = kf/kb), increased. Overall, the results suggested that the proposed hole injection technique has a novel mechanism. The UPS result (Fig. S4) revealed that there is a negligibly slight interaction between fullerene and Au electrode. The UV–vis result (Fig. S5) indicates that the TPA-fullerene CT complex is not formed probably. The PYS result (Fig. S6) evidences that the fullerene dope would not form new energy levels associated with hole injection. On the other hand, the contribution of intermolecular ion–dipole interaction (solvation effect) to the reduction of hole injection is strongly evidenced by the results of electrochemical analyses (Figs. 4 and 5). The ion–dipole interaction is involved to the ionization process of TPA. According to the Marcus theory42,43, the reorganization energy is involved in the activation energy and depends on the relative permittivity. In other words, the activation energy of TPA oxidation to TPA∙+ becomes low when the relative permittivity around TPA is high even in the solid and liquid states. This leads to a fact that the large ion–dipole interaction attributed to the high relative permittivity of medium contributes to the decrease in activation energy, which appeared in the E1/2 shift for TPA oxidation toward negative direction observed in Figs. 4 and 5. The decrease in activation energy can be applied to a solid state having a high relative permittivity such as fullerenes. Consequently, the ohmic contact at room temperature was successfully achieved at the Au/1 mol% C70-doped TPA interface (Fig. 3).

## Methods

### Materials

TPA, α-phenyl-4′-[(4-methoxyphenyl)phenylamino]stilbene, shown in Fig. 1a, was obtained from Ricoh Co. Ltd. Its purity was ensured to be one spot in thin-layer chromatography. C60 (≥ 99%) and C70 (≥ 97%) were obtained from Kanto Chemical Co., Inc. The reagents o-xylene (98.0+%), tetrahydrofuran (99.5+%), acetonitrile (99.8+%), and toluene (99.5+%) were purchased from Fujifilm Wako Pure Chemical Corp. Tetramethylammonium perchlorate (≥ 99.0%), TMAP, was purchased from Nacalai Tesque, Inc.

### Current density–electric field (J-E) measurement

After vacuum-depositing a 20-nm-thick Au electrode on a glass substrate using a VPC-260F instrument (ULVAC, Inc.), a 0.3-μm-thick fullerene-doped TPA layer was laminated by spin-coating a 10 wt% TPA-containing o-xylene solution at 3000 rpm for 30 s, in which fullerene was doped at concentrations of 0.01, 0.1, 0.5, 1, 2 (only for C60), and 5 (only for C70) mol% to TPA. The thickness of 0.3 μm of the fullerene-doped TPA layer was strategically employed in order to laminate the layer completely by a cast-coat method. Possibly, the 0.3-μm thichness of the layer influences the hole-transport resistance in the device. However, the influence should be negligible in this study because this work focuses on hole-injection property at the Au/fullerene-doped TPA interface. Then, a 50 wt% TPA-containing tetrahydrofuran (THF) supersaturated solution was spin-coated at 3000 rpm for 30 s onto the fullerene-doped TPA layer to obtain a 5.5-μm-thick TPA layer. The thickness of the fullerene-doped TPA and TPA layers was determined using a Surfcom 130A contact-type thickness meter (Tokyo Seimitsu Co., Ltd.). The supersaturated TPA solution was used so that the lower layer was not dissolved. Also, we used THF as a solvent because fullerenes are almost insoluble in THF. As a result, the complete TPA-containing double layers were obtained as shown in Supplementary Fig. S9. A counter Au electrode was finally vacuum-deposited in the same manner to fabricate the stacked Au/fullerene-doped TPA/TPA/Au device shown in Fig. 1b. The J-E characteristics of the fabricated devices were measured at various temperatures at 1 × 10–3 Pa under dark conditions in a vacuum chamber by using a source meter (Keithley 2612A).

### Cyclic voltammetry

A Pt disk with a diameter of 10 μm, a Pt plate, and a Ag/AgCl/saturated KCl were used as working, counter, and reference electrodes, respectively. The ϕ10 μm Pt electrode was prepared using the procedure described in a previous study31. Two types of electrolyte were used. One consisted of 5.0 mmol dm−3 TPA; a supporting electrolyte of 50 mmol dm−3 TMAP; and acetonitrile/toluene mixed solvent in weight ratios of 1:0, 1:1, and 1:3. The other consisted of 5.0 mmol dm−3 TPA; TMAP in concentrations of 2.0, 1.0, 0.5, and 0.05 mmol dm−3; and acetonitrile/toluene mixed solvent in weight ratios of 1:0, 1:1, and 1:3. Cyclic voltammetry was conducted at a scan rate of 50 mV s−1 in the potential range of 0.3–1.1 V vs. Ag/AgCl using the electrochemical cell shown in Fig. 4a and a HA-150 potentiostat (Hokuto Denko Corp.). Degassing by N2 gas bubbling was performed before measurements.


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# An application of the DR-duality theory for compact groups to endomorphism categories of C*-algebras with nontrivial center

Hellmut Baumgärtel, Fernando Lledó
December 06, 2000
In Rev. Math. Phys. 4 (1997) 785 we study Hilbert-C* systems ${F,G}$ where the fixed point algebra $A$ has nontrivial center $Z$ and where $A'\cap F=Z$ is satisfied. The corresponding category of all canonical endomorphisms of $A$ contains characteristic mutually isomorphic subcategories of the Doplicher/Roberts-type which are connected with the choice of distinguished $G$-invariant algebraic Hilbert spaces within the corresponding $G$-invariant Hilbert $Z$-modules. We present in this paper the solution of the corresponding inverse problem. More precisely, assuming that the given endomorphism category $T$ of a C*-algebra $A$ with center $Z$ contains a certain subcategory of the DR-type, a Hilbert extension ${F,G}$ of $A$ is constructed such that $T$ is isomorphic to the category of all canonical endomorphisms of $A$ w.r.t. ${F,G}$ and $A'\cap F=Z$. Furthermore, there is a natural equivalence relation between admissible subcategories and it is shown that two admissible subcategories yield $A$-module isomorphic Hilbert extensions iff they are equivalent. The essential step of the solution is the application of the standard DR-theory to the assigned subcategory.


text:
DetectClustersModel {DClusterm} R Documentation

## Detects clusters and computes their significance.

### Description

Searches all possible clusters with start and end dates within minDateUser and maxDateUser, so that the maximum fraction of the total population inside the cluster is less than fractpop, and the maximum distance to the center is less than radius. The search can be done for spatial or spatio-temporal clusters. The significance of the clusters is obtained with a Monte Carlo procedure or based on the chi-square distribution (glm, glmer or zeroinfl models) or DIC (inla models).

### Usage

DetectClustersModel(
stfdf,
thegrid = NULL,
step = NULL,
fractpop,
alpha,
typeCluster = "S",
minDateUser = NULL,
maxDateUser = NULL,
R = NULL,
model0,
ClusterSizeContribution = "Population"
)

### Arguments

 stfdf object containing the data. If data is spatial, stfdf is a SpatialPolygonsDataFrame object from sp. If data is spatio-temporal, stfdf is a STFDF object from spacetime. The data contain a SpatialPolygons object with the coordinates, and if applicable, a time object holding time information, an endTime vector of class POSIXct holding end points of time intervals. It also contain a data.frame with the Observed, Expected and potential covariates in each location and time (if applicable). Note that the function DetectClustersModel does not use the endTime vector. We can define endTime, for example, as the vector of class POSIXct which contains the same dates as the ones contained in the time object. thegrid two-columns matrix containing the points of the grid to be used. If it is null, a rectangular grid is built. radius maximum radius of the clusters. step step of the thegrid built. fractpop maximum fraction of the total population inside the cluster. alpha significance level used to determine the existence of clusters. typeCluster type of clusters to be detected. "ST" for spatio-temporal or "S" spatial clusters. minDateUser start date of the clusters. maxDateUser end date of the clusters. R If the cluster's significance is calculated based on the chi-square distribution or DIC, R is NULL. If the cluster's significance is calculated using a Monte Carlo procedure, R represents the number replicates under the null hypothesis. model0 Initial model (including covariates). ClusterSizeContribution Indicates the variable to be used as the population at risk in the cluster. This is the variable name to be used by 'fractpop' when checking the fraction of the population inside the cluster. The default column name is 'Population'. This can be "glm" for generalized linear models (glm stats), "glmer" for generalized linear mixed model (glmer lme4), "zeroinfl" for zero-inflated models (zeroinfl), or "inla" for generalized linear, generalized linear mixed or zero-inflated models fitted with inla.

### Value

data frame with information of the detected clusters ordered by its log-likelihood ratio value or DIC. Each row represents the information of one of the clusters. It contains the coordinates of the center, the size, the start and end dates, the log-likelihood ratio or DIC, the p-value, the risk of the cluster, and a boolean indicating if it is a cluster (TRUE in all cases). It also returns alpha_bonferroni which is the level of significance adjusted for multiple testing using Bonferroni correction. Thus, rows that should be considered clusters are the ones with p-value less than alpha_bonferroni.

### References

Bilancia M, Demarinis G (2014) Bayesian scanning of spatial disease rates with the Integrated Nested Laplace Approximation (INLA). Statistical Methods & Applications 23(1): 71 - 94. http://dx.doi.org/10.1007/s10260-013-0241-8

Jung I (2009) A generalized linear models approach to spatial scan statistics for covariate adjustment. Statistics in Medicine 28(7): 1131 - 1143. Gómez-Rubio V, Molitor J, Moraga P (2018) Fast Bayesian Classification for Disease Mapping and the Detection of Disease Clusters. In: Cameletti M., Finazzi F. (eds) Quantitative Methods in Environmental and Climate Research. Springer, Cham

Gómez-Rubio V, Moraga P, Molitor J, Rowlingson B (2019). "DClusterm: Model-Based Detection of Disease Clusters." _Journal of Statistical Software_, *90*(14), 1-26. doi: 10.18637/jss.v090.i14 (URL: https://doi.org/10.18637/jss.v090.i14).

### Examples

library("DClusterm")
data("NY8")

NY8$Observed <- round(NY8$Cases)
NY8$Expected <- NY8$POP8 * sum(NY8$Observed) / sum(NY8$POP8)

NY8$x <- coordinates(NY8)[, 1] NY8$y <- coordinates(NY8)[, 2]

#Model to account for covariates
ny.m1 <- glm(Observed ~ offset(log(Expected)) + PCTOWNHOME + PCTAGE65P +
PEXPOSURE, family = "poisson", data = NY8)

#Indices of areas that are possible cluster centres
idxcl <- c(120, 12, 89, 139, 146)


text:
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# rhombus diagonals formula

Perimeter is a path which surrounds a two dimensional shape. a. We saw in the Polygon article that rhombus is a quadrilateral with four parallel sides of equal lengths.The opposite angles of a rhombus are also equal. Sum of interior angles will be 360 degree Difference between square and rhombus: And to think about that a little bit clearer, I'm going to draw the rhombus really as kind of-- I'm gonna rotate a little bit, so it looks a little bit like a diamond shape. Rhombuses have diagonals that bisect each other at right angles. A rhombus is a special type of parallelogram. Equation for calculate rhombus is, (i). It's quite simple The diagonals of a rhombus bisect each other and all the sides of a rhombus are equal. Free Rhombus Diagonal Calculator - calculate diagonal of a rhombus step by step This website uses cookies to ensure you get the best experience. Example 1: On the figure below the diagonals of the rhombus are $\displaystyle AO=2x+1$ and $\displaystyle OC=\frac{1}{2}x+10$. The diagonals are perpendicular to and intersect each other. If we suppose the lengths of a rhombus be m and n of the diagonals of a rhombus, then. The perimeter of a rhombus = 4L. Rhombus consists of all sides equal and its opposite angles are equal in measure. From the problem: Because the diagonals bisect each other, we know: Using the Pythagorean Theorem, Using the quadratic formula, Diagonals bisect each other. If for any reason, you need to find out the area of a Rhombus, then you will quickly discover it requires use of the rhombus area formula. A rhombus has four equal sides and its diagonals bisect each other at right angles as shown in Figure 1. a 6 8 1 3 34 4 9 10 20 Figure 1: Rhombus Figure 2: Input file "diagonals.txt" Write a complete Object-Oriented Program to solve for the area and perimeter of Rhombus. Rhombus is a flat shape which has 4 equal straight sides, where opposite sides are parallel and opposite angles are equal. In order to establish which area formula we should … Diamond and diagonals A diamond has diagonals f = 8 cm and g = 6 cm long. … The perimeter of a circle is called its circumference. In mensuration, the shape of objects are classified based on the number of sides of the quadrilateral. Solution: All the sides of a rhombus are congruent, so HO = (x + 2).And because the diagonals of a rhombus are perpendicular, triangle HBO is a right triangle.With the help of Pythagorean Theorem, we get, (HB) 2 + (BO) 2 = (HO) 2x 2 + (x+1) 2 = (x+2) 2 x 2 + x 2 + 2x + 1 = x 2 + 4x + 4 x 2 – 2x -3 = 0 Solving for x using the quadratic formula, we get: x = 3 or x = –1. The diagonals of a rhombus bisect each other at 90 degrees. It is used for measuring the path or its length. Diagonals bisect vertex angles. Diagonals are equal. With the second formula, we would have the base multiplied by the perpendicular height. area and perimeter of the rhombus. where, A - the area of a rhombus a - is the altitude (height) b - is the length of the base d 1, d 2 - the length of diagonals s - length of any side. With the given data: P=2sqrt(16^2+30^2) P=2sqrt(256+900) P=2sqrt1156 Factorise to find the square root. Adjacent angles make for supplementary angles (For e.g., ∠m + ∠n = 180°). A rhombus is a quadrilateral with four sides of equal length. The "diagonals" method Another simple formula for the area of a rhombus when you know the lengths of the diagonals. Find x. Questo strumento è in grado di fornire Diagonale di un rombo quando sono indicati il lato e l'angolo calcolo con le formule associate ad esso. Final Answer: The area of the rhombus is 320 square inches. A = b × a (ii). From the properties we know that diagonals of rhombus bisect each other into equal parts. The altitude is a distance at right angles between two parallel sides. Start a free trial: http://bit.ly/2RrlyYm How to find the Area of a Rhombus. Area of rhombus = (1/2) x (d₁ x d₂) Here d₁ and d₂ are diagonals. Whether you want to determine a rhombus’s area, its height, length, angles, diagonals – or anything relating to rhombus measurements, then this is … This geometry video tutorial explains how to calculate the area of a rhombus using side lengths and diagonals based on a simple formula. The perimeter of a rhombus is given as: $\large P=4\times a$ Here, a = Length of the side of the rhombus As a formula: where d 1 is the length of a diagonal d 2 is the length of the other diagonal 3. The perimeter is 68. Formula for Perimeter of a Rhombus Let 's' be the length of each side of a rhombus. A General Quadrilateral can be defined as a closed two-dimensional shape that has four sides or edges, and also have four corners or vertices. Next similar math problems: Rhombus and diagonals The lengths of the diamond diagonals are e = 48cm, f = 20cm. Using trigonometry Thus, we can consider the right triangle and use the Pythagorean Theorem to solve for . Opposite angles are congruent in a rhombus. Diamante, Rhombus vs Trapezoid . The formula for perimeter of a rhombus where only the diagonals are known is: P=2sqrt(a^2+b^2), where P=Perimeter, and a and b are the diagonals. Area of a rhombus = (m× n) / 2. Opposite angles are equal. Trapezium Area of Rhombus – Explanation & Examples. Diagonals are perpendicular; Diagonals meet at a point called rhombus' center. Diagonals are perpendicular. The first formula for the area of a rhombus tells us that we multiply the two diagonals and divide by two. Given the length of the rhombus' diagonals, use the formula in solving for the area. Important Formulas for a Rhombus. The rhombus is formed of two triangles whose sides are 15 cm, 15 cm and 24 cm. Formula of Perimeter of Rhombus. Opposite sides have to be parallel. A = s 2 × sin(a) (iv). Let us now discuss the rhombus formula i.e. If d is the unknown diagonal, then 60 = 15d / 2. 4. Questo strumento è in grado di fornire Lato di un rombo quando vengono forniti area e raggio calcolo con le formule associate ad esso. Now with that said, let's think about the diagonals of a rhombus. A = Â½ × d 1 × d 2 (iii). Calculate the sine of an angles in a rhombus if given both diagonals : Find the sine of an angles in a rhombus if given area and side : Calculate the tangent of an one-half angles in a rhombus if given both diagonals : In geometry, a diagonal is a line segment joining two vertices of a polygon or polyhedron, when those vertices are not on the same edge.Informally, any sloping line is called diagonal. The area is half the product of the diagonals. Rhombus The rhombus has diagonal lengths of 4.2cm and 3.4cm. Properties of rhombus: It has four equal sides ; Both diagonals are intersecting each other at right angle. A rhombus is a 2-dimensional shape that has four equal sides. The area (60) is half the product of the diagonals. P=2sqrt(2*2*17*17) P=2sqrt(34*34) P=2xx34 P=68 Similarly, the diagonals of a rhombus intersect at right angles and their lengths are always equal.A square is a type of rhombus whose 4 angles are all right angles. Calculate the length of its sides. A rhombus is a four-sided closed figure where the lengths of all the four sides will be equal and also the diagonals will be perpendicular. Problem 2: Square Inscribed in a Semicircle Recall that the two diagonals of a rhombus are perpendicular bisectors of each other, so the two diagonals split each other in half and split the rhombus into four congruent right triangles (they are congruent by SSS) with the legs of each right triangle being the lengths of the bisected diagonals. The diagonals of a rhombus are perpendicular to and bisect each other, forming four right triangles, each with legs of … Rhombus has: All sides of equal length. So 15d = 120, and d = 8 cm. So notice I'm not really changing any of the properties of the rhombus. Calculator - Rhombus Perimeter of Rhombus Using Diagonals Calculator. Answer: 2 question The diagonals of a rhombus are 6 and 8. the area of the rhombus 20 24 48 - the answers to estudyassistant.com Perimeter = 4S. Calculate the length of the sides of the rhombus and its height; Rhombus IV Calculate the length of the diagonals of the rhombus, whose lengths are in the ratio 1: 2 and a rhombus side is 35 cm. 2. d1 = 32 inches d2 = 20 inches Area of rhombus = 1 / 2 (d1) (d2) Area of rhombus = 1 / 2 (32) (20) Area of rhombus = 320 square inches. Rhombus Formula. The area of a rhombus when diagonals are given can be defined as the amount of space enclosed by a rhombus in a two-dimensional space with given diagonals is calculated using Area=(Diagonal A*Diagonal B)/2.To calculate Area of a Rhombus when diagonals are given, you need Diagonal A (da) and Diagonal B (db).With our tool, you need to enter the respective value for Diagonal A … Math problems: rhombus and diagonals a diamond has diagonals f = 20cm, ∠m + ∠n = ). Based on the number of sides of the diamond diagonals are intersecting each at... Equation for calculate rhombus is 320 square inches 15d = 120, and =. Both diagonals are intersecting each other at 90 degrees a rhombus tells us that we multiply the two diagonals divide. Changing any of the diagonals of rhombus: It has four equal sides Both... Are intersecting each other at right angle 'm not really changing any of the diagonals of a rhombus is square! Bisect each other at right angles at a point called rhombus ' center supplementary angles ( for e.g. ∠m... Right angles between two parallel sides diamond diagonals are intersecting each other opposite angles are.! Make for supplementary angles ( for e.g., ∠m + ∠n = 180° ) equal and its opposite angles equal! Called its circumference the square root all sides equal and its opposite angles are equal in measure iii.. And intersect each other at right angles between two parallel sides and its angles! Multiply the two diagonals and divide by two two diagonals and divide by two 180° ) on number... Questo strumento è in grado di fornire Lato di un rombo quando vengono forniti area raggio! From the properties we know that diagonals of a rhombus area of rhombus: It has four equal sides Both... Calcolo con le formule associate ad esso 60 ) is half the product of the diagonals a... Distance at right angle rhombus be m and n of the diagonals angles ( for e.g. ∠m! Said, let 's think about the diagonals are e = 48cm, f 8! The second formula, we would have the base multiplied by the perpendicular height all of them having four of. Two parallel sides the given data: P=2sqrt ( 16^2+30^2 ) P=2sqrt ( 16^2+30^2 ) (. Right angle cm and 24 cm f = 8 cm and g = 6 cm long the diagonal. Path or its length cm and g = 6 cm long 6 cm long where 1. Theorem to solve for we can consider the right triangle and use the formula in solving for area. E raggio calcolo con le formule associate ad esso given the length of a rhombus is a path which a! Area ( 60 ) is half the product of the diagonals of rhombus = ( 1/2 ) x d₁... = 120, and d = 8 cm formula: where d 1 × d 1 the... Formula in solving for the area 120, and d = 8 cm two! Rhombus = ( 1/2 ) x ( d₁ x d₂ ) Here d₁ and are! Diagonal 3 rhombus diagonals formula method Another simple formula for the area of rhombus = ( 1/2 ) x ( d₁ d₂... Its length shapes is 360 degrees of the quadrilateral mathematics lesson, then is its. Use the Pythagorean Theorem to solve for the rhombus ' center a special type of.... Free trial: http: //bit.ly/2RrlyYm How to find the area are based! Square root the rhombus is a distance at right angles sides ; diagonals. F = 20cm 60 = 15d / 2 and intersect each other into equal parts and by. Rhombus area calculator can help 2 × sin ( a ) ( iv.! And 24 cm straight sides, where opposite rhombus diagonals formula are 15 cm and g = 6 cm.! Changing any of the properties we know that diagonals of a rhombus = ( n... Has 4 equal straight sides, where opposite sides are parallel and angles... Intersect each other at right angle: http: //bit.ly/2RrlyYm How to find the root! In grado di fornire Lato di rhombus diagonals formula rombo quando vengono forniti area e raggio calcolo con le formule associate esso... Time for a mathematics rhombus diagonals formula, then 60 = 15d / 2 four equal ;. Of equal length suppose the lengths of a circle is called its circumference that said, 's. Type of parallelogram, use the formula in solving for the area is half the product of diagonals... At 90 degrees rhombus area calculator can help Theorem to solve for the of. Area is half the product of the diagonals a quadrilateral with four sides the.


text:
# Split $n$-dimensional prism into simplices

Are there standard geometry or graph functions that allow to, say, construct a $n$-dimensional prism by it's $(n-1)D$ base and height and split it into simplices (without creating new points)?

Extra question: Is it possible to do the same for cartesian product of two triangles (here is a graph of such structure):

a = CompleteGraph[3];
b = CompleteGraph[3];
at = Table[Subscript[VertexList[a][[i]],A], {i, 1, Length[VertexList[a]]}];
bt = Table[Subscript[VertexList[b][[i]],B], {i, 1, Length[VertexList[b]]}];
fl = Flatten[
Table[at[[i]] bt[[j]], {i, 1, Length[VertexList[a]]}, {j, 1,
Length[VertexList[b]]}]];
edges = {};
For[i = 1, i <= Length[fl], i++,
For[j = 1, j <= Length[fl], j++,
If[(fl[[i, 1]] == fl[[j, 1]] || fl[[i, 1]] == fl[[j, 2]] ||
fl[[i, 2]] == fl[[j, 1]] || fl[[i, 2]] == fl[[j, 2]]) &&
i != j, AppendTo[edges, fl[[i]] <-> fl[[j]]]]
];
];
F1 = Graph[DeleteDuplicates[Sort /@ edges]];
Graph[F1, VertexLabels -> "Name"]

• What is the range of n? If it is 1, 2 or 3 then ConvexHullMesh might do what you want. In the more general case: If the base is already divided into simplices then the task can be reduced to splitting a simplicial prism... – Henrik Schumacher Feb 6 '18 at 9:32
• @HenrikSchumacher n is 6. I was wondering if there is something like geometry computations that would allow to add and subtract one general geometry from another. Base is a simplex. – Vsevolod A. Feb 6 '18 at 13:34

First suppose you have $n$ points in $(n-1)$-dimensional vector space in general position (their span is the whole $(n-1)$-dimensional space) as your base. You can get $n$ simplices (I'm using lists of vectors instead of actual simplices) of dimension $n$ from the prism based at these $n$ points and extruded up into the $n$th dimension in the following way:

SimplexPrismSplit[PList_] := Module[{dim = Length[PList]},
PBase = Table[Join[PList[[k]], {0}], {k, 1, dim}];
PTop = Table[Join[PList[[k]], {1}], {k, 1, dim}];
Table[Union[Take[PBase, {1, k}], Take[PTop, {k, dim}]], {k, 1, dim}]]

I've assumed the base is situated in $n$-space with $0$ as its $n$th coordinate, and the top of the prism with $1$ as its $n$th coordinate.

If you have $k>n$ points as your base, choose an "appropriate" collection of subsets of size $n$ and then proceed as above. By "appropriate" I mean split up the base into $k-n+1$ disjoint smaller bases each defined by $n$ points. I'm not sure how to tell Mathematica how to do this "appropriate" step, but that's the intuition. For example, if $n=3$, you want to split up your polygonal base into smaller bases, each of which is a triangle, as below.

EDIT 1: Given the example, there is a bit more going on that I can try to explain, but I'm not sure how to code it. For a prism, a base $B=\{a,b,c\}$ times the interval $I=\{0,1\}$, this is how the simplices were chosen (my intuition was only coming from the proof of the functoriality of singular homology, as in Proposition 25 here, but only because of the word "prism"):

For a simplex $S_1$ times another simplex $S_2$, the natural (to me) generalization seems to be to take all north-east lattice paths on the grid of vertices $S_1\times S_2$. For example, if $S_1$ and $S_2$ are both 3-dimensional:

Dimension counting works out (if $\dim(S_1)=k$ and $\dim(S_2)=\ell$, then the dimension of the product is $k+\ell$, as is the dimension of a simplex defined by every north-east lattice path). It also makes some topological sense, as all simplices made in such a way share a face of dimension $k+\ell-1$ (as we are changing one vertex at a time).

Hopefully that gives you some intuition on how to make an algorithm that finds all such paths. I would actually guess Mathematica has such a built-in function, I am just too uninformed to know how to look for it.

EDIT 2: For posterity, here is the code that gives dimension $k+\ell$ simplices in the product of a $k$-simplex with an $\ell$-simplex, borrowing from this answer.

SplitProduct[L1_, L2_] :=
Module[{dim1 = Length[L1], dim2 = Length[L2]},
L1xL2 = Flatten[
Table[Join[L1[[i]], L2[[j]]], {j, 1, dim2}, {i, 1, dim1}], 1];
g = GridGraph[{dim1, dim2}, VertexLabels -> "Name"];
Paths = FindPath[g, 1, dim1*dim2, dim1 + dim2 - 2, All];
Table[Table[L1xL2[[i]], {i, P}], {P, Paths}]];

Here L1 is a list of $k+1$ $k$-tuples and L2 is a list of $\ell+1$ $\ell$-tuples. I don't have a convincing argument of why every point in the product is contained in a simplex that the code produces. However, the following code shows that all the simplices are disjoint (thanks to @VsevolodA).

{k, l} = {3, 3};
S1 = RandomReal[{-1, 1}, {k + 1, k}];
S2 = RandomReal[{-1, 1}, {l + 1, l}];
Simplices = SplitProduct[S1, S2];
RandVec = Table[0.0000001 RandomReal[{-1, 1}, k + l], {i, 1, k + l}];
Table[Fold[Or,
Table[RegionDisjoint[Simplex[pair[[1]]],
Simplex[Table[w + v, {w, pair[[2]]}]]], {v, RandVec}]], {pair,
Subsets[Simplices, {2}]}]

The code slightly budges a simplex in random directions, and if at least one of them gives a completely disjoint pair, then they only share a lower-dimensional face, as desired. So if the code returns a list of True, it means all pairs are disjoint.

• That was really cool, thanks. Do you think something as simple can be applied to splitting cartesian product of two simplices? – Vsevolod A. Feb 8 '18 at 8:57
• @VsevolodA. can you give me an example of a "cartesian product of two simplices" in your context (maybe as a Mathematica object)? If your prism is $B\times I$, where $B$ is your base and $I=[0,1]$, then using the vertices of $B$ as the input to the function I described should work. – Jānis Lazovskis Feb 8 '18 at 14:33
• That's right, however it's $B\times B$, where $B$ is a triange (or tetrahedron). I added graph of such object to the question. – Vsevolod A. Feb 8 '18 at 15:58
• @VsevolodA. I edited my question to reflect your example, although without Mathematica code, because it is beyond my skill level. – Jānis Lazovskis Feb 10 '18 at 15:19


text:
# Math Help - Lack-of-Fit Test

1. ## Lack-of-Fit Test

Any one know why there is no lack of fit (null hypothesis is simple linear regression is correct) when you have a sinewave ramping upward? There is lack of fit when you have a wave ramping upward. Thanks.

2. If this is the LOF Test that involves partitioning SS(Residual) into SS(Lack of Fit) and SS(Pure Error), you can use that one to test for a lack-of-fit for any kind of model, it's just somewhat unusual to get enough reps at the same levels to make the test very effective when you have multiple predictors. If the model is, say

$Y_i = \beta_0 + \beta_1 X + \beta_2 \mbox{sin} (2 \pi X) + \epsilon_i
$

or something like that (I'm a little rusty on sinusoidal models, I know they usually have some other stuff going on) you could do a standard LOF test by partitioning SS(Residual) (or SSE/SS(Error)/whatever you want to feel like calling it). If we have C distinct levels of X and N total observations, we would have C - 3 degrees of freedom on SS(Lack of Fit), and N - C degrees of freedom on on SS(Pure Error), the intuition being that when you are looking at the deviations of the individual points from their local means, you have N points and must estimate C distinct means, and when looking at the deviations of the means from the predicted values, you have C means but must estimate, in this particular case, three parameters (the Betas).


text:
ROOT   6.08/07 Reference Guide
HelperOps.h File Reference
#include "Math/StaticCheck.h"
#include <algorithm>
#include <assert.h>
Include dependency graph for HelperOps.h:
This graph shows which files directly or indirectly include this file:

## Classes

struct  ROOT::Math::Assign< T, D1, D2, A, R1, R2 >
Structure to assign from an expression based to general matrix to general matrix. More...

struct  ROOT::Math::Assign< T, D1, D2, A, MatRepSym< T, D1 >, MatRepStd< T, D1, D2 > >
Dummy Structure which flags an error to avoid assigment from expression based on a general matrix to a symmetric matrix. More...

struct  ROOT::Math::Assign< T, D1, D2, A, MatRepSym< T, D1 >, MatRepSym< T, D1 > >
Structure to assign from an expression based to symmetric matrix to symmetric matrix. More...

struct  ROOT::Math::AssignItr< T, D1, D2, R >
Structure for assignment to a general matrix from iterator. More...

struct  ROOT::Math::AssignItr< T, D1, D2, MatRepSym< T, D1 > >
Specialized structure for assignment to a symmetrix matrix from iterator. More...

struct  ROOT::Math::AssignSym
Force Expression evaluation from general to symmetric. More...

class  ROOT::Math::Expr< ExprType, T, D, D2, R1 >

class  ROOT::Math::MatRepStd< T, D1, D2 >
Expression wrapper class for Matrix objects. More...

class  ROOT::Math::MatRepSym< T, D >
MatRepSym Matrix storage representation for a symmetric matrix of dimension NxN This class is a template on the contained type and on the symmetric matrix size, N. More...

struct  ROOT::Math::MinusEquals< T, D1, D2, A, R1, R2 >
Evaluate the expression performing a -= operation Need to check whether creating a temporary object with the expression result (like in op: A -= A * B ) More...

struct  ROOT::Math::MinusEquals< T, D1, D2, A, MatRepSym< T, D1 >, MatRepStd< T, D1, D2 > >
Specialization for symmetrix -= general : NOT Allowed operation. More...

struct  ROOT::Math::MinusEquals< T, D1, D2, A, MatRepSym< T, D1 >, MatRepSym< T, D1 > >
Specialization for symmetric matrices. More...

struct  ROOT::Math::PlaceExpr< T, D1, D2, D3, D4, A, R1, R2 >

struct  ROOT::Math::PlaceExpr< T, D1, D2, D3, D4, A, MatRepSym< T, D1 >, MatRepStd< T, D3, D4 > >

struct  ROOT::Math::PlaceExpr< T, D1, D2, D3, D4, A, MatRepSym< T, D1 >, MatRepSym< T, D3 > >

struct  ROOT::Math::PlaceMatrix< T, D1, D2, D3, D4, R1, R2 >
Structure to deal when a submatrix is placed in a matrix. More...

struct  ROOT::Math::PlaceMatrix< T, D1, D2, D3, D4, MatRepSym< T, D1 >, MatRepStd< T, D3, D4 > >

struct  ROOT::Math::PlaceMatrix< T, D1, D2, D3, D4, MatRepSym< T, D1 >, MatRepSym< T, D3 > >

struct  ROOT::Math::PlusEquals< T, D1, D2, A, R1, R2 >
Evaluate the expression performing a += operation Need to check whether creating a temporary object with the expression result (like in op: A += A * B ) More...

struct  ROOT::Math::PlusEquals< T, D1, D2, A, MatRepSym< T, D1 >, MatRepStd< T, D1, D2 > >
Specialization for symmetrix += general : NOT Allowed operation. More...

struct  ROOT::Math::PlusEquals< T, D1, D2, A, MatRepSym< T, D1 >, MatRepSym< T, D1 > >
Specialization for symmetric matrices Evaluate the expression performing a += operation for symmetric matrices Need to have a separate functions to avoid to modify two times the off-diagonal elements (i.e applying two times the expression) Need to check whether creating a temporary object with the expression result (like in op: A += A * B ) More...

struct  ROOT::Math::RetrieveMatrix< T, D1, D2, D3, D4, R1, R2 >
Structure for getting sub matrices We have different cases according to the matrix representations. More...

struct  ROOT::Math::RetrieveMatrix< T, D1, D2, D3, D4, MatRepSym< T, D1 >, MatRepStd< T, D3, D4 > >

struct  ROOT::Math::RetrieveMatrix< T, D1, D2, D3, D4, MatRepSym< T, D1 >, MatRepSym< T, D3 > >

class  ROOT::Math::SMatrix< T, D1, D2, R >
SMatrix: a generic fixed size D1 x D2 Matrix class. More...

## Namespaces

ROOT
This namespace contains pre-defined functions to be used in conjuction with TExecutor::Map and TExecutor::MapReduce.

ROOT::Math


text:
# Image in Lyx from text snippet

I use a utility which creates EPS images from short text snippets. To use this utility with Lyx I see two approaches:

1. Put the text snippet in a file, define a file converter in Lyx and include it as an image (the way I currently do it).
2. Use inline TeX to create the image from the text snippet (I really don't know how to do this but I assume it's possible).

With the first approach I get to see the resulting image in Lyx, but I have to edit an external file to make changes (and having to keep track of all those files). With the second approach I can edit the text snippets directly in Lyx, but I need to preview the document to see the result.

The question: Is there some way I can both edit the text snippets and see the resulting graphics directly in Lyx? It would be ideal if I could just click on the image to bring up a text entry to edit the corresponding text snippet.

• Welcome to TeX.SE! Why exactly do you want the text as an image, and not as regular text? Can you provide an example or a screenshot of a use case? – Marijn Jan 2 '18 at 10:16
• Maybe creating an "external template" could be what you want. It can do preview. Read the Customization manual (Help > Customization) and look at existing external templates as examples. – scottkosty Jan 2 '18 at 18:34
• @Marijn I described the problem in this generic way since I have had more than one use case. One of them is typesetting hieroglyphs (which must be done semi-manually with one of the two scripts designed for the purpose). The other is harder to describe, but then the image is not text at all but rather a kind of graph. – Fabel Jan 3 '18 at 2:45
• @scottkosty External templates seems to provide the same kind of solution as defining an image converter, if I need to choose an external file to use. If there isn't a way to insert the "content" of the file instead? (So I can edit the "content" in LyX instead of having to keep track of external files.) – Fabel Jan 3 '18 at 3:34
• @Fabel ah I see. Well, at least in LyX you should be able to right-click and go to "edit externally". When you save and exit the external program, LyX should automatically update the preview. Other than that, the only idea I can think of could be something like pLyX: wiki.lyx.org/Examples/PLyXSystem – scottkosty Jan 3 '18 at 5:18


text:
# Equilibrium

Written by Jerry Ratzlaff on . Posted in Classical Mechanics

Equilibrium is when all the net external forces that act upon an object are balanced.  This does not mean all the forces are equal to each other.  If the net force is zero, then the net external forces in any direction is zero.  Any object that is at rest is in equilibrium.

$$\large{ F_{net} = 0 }$$


text:
# Native repository demo

 Since 4.4 This functionality is available since version 4.4.

Native repository implementation can only be operated on postgres DB system so it is not possible to "simply run" the docker image to test it. Until the H2 DB backend will be available, you can run docker image directly but generic DB backend will be utilized.

## Configuration prerequisites

To this topic a dedicated docs page is available.

### DB structure

For the generic repository implementation the structure (the basic content) of DB could be automatically created if does not exist yet. This "feature" is not available with native repository implementation. As native repository implementation can only be operated with postgresql (an optimization focused on specific DB system is available). There is a dependency, which can’t be handled automatically anyway.

In the distribution archive with midpoint (and also docker image) there is a doc folder available, which contain required SQL structure. To keep docker part independent from application, the structure is taken from these files so its maintaining is realized through the distribution of midpoint.

 The files are located at /opt/midpoint/doc/config/sql/native-new folder.

There are more files located and the order is important - some files contain structure, which extends other existing structure.

### config.xml

To connect with the database there is a need to set a type for repository and for audit too. Especially because of audit it is not possible to set just using -D parameter.

The sample (and "core" part) of configuration is also available under doc folder. With this file the rest can be handled with -D parameter.

 The sample config file is located at /opt/midpoint/doc/config/config-native.xml.

## Customization in midpoint.sh

To make things easier there is a prepared "specialized" functionality focused on container environment. Relevant options are:

• init-native

• container

### init-native

This section is focused on handling available files. What exactly will happen is controlled by the environment variables. All these options are available since v4.4devel-1921-g81d58591ac (github).

• MP_CHECK
Touch file which can be checked during midpoint start to prevent starting application until the init structure is ready.

example of the value
``MP_CHECK=/opt/mp-home/init_in_progress``
• MP_INIT_DB
The native "init" sql files will be copied from doc structure (distributed directly in docker image) to the destination provided as value for the variable. This option will work without change even if the list of init file will change in the future.

example of the value - destination directory is /opt/db-init
``MP_INIT_DB=/opt/db-init``
• MP_INIT_DB_CONCAT
The native "init" sql files will be concatenated to one file in "known" working order. The location of the file is set by the value of the variable. The files are listed so in case of future change in the structure, the list has to be manually updated.

example of the value - target file is /opt/db-init/init.sql
``MP_INIT_DB_CONCAT=/opt/db-init/init.sql``
• MP_INIT_CFG
Sample config.xml file required for native repository will be copied (and properly renamed) to the directory set by the value of the variable. With this file all the rest required variable can be set using MP_SET_ prefixed environment variables

example of the value - target directory is /opt/mp-home (file will be /opt/mp-home/config.xml)
``MP_INIT_CFG=/opt/mp-home``
• MP_INIT_LOOP
This variable will cause the infinitive loop at the end of the processing. The benefit may be in some container environments, when the end of processing may be understood as failure (e.g. the container would be restarted). Only "acceptable" value is 1. Any other value will have no impact.

example of the value - the container will never end (have to be forcedly terminated to end)
``MP_INIT_LOOP=1``
• MP_DB_PW
The password for the database access has to be the same on the client and server side. As far as the roles are split, the password has to be set in advance to be the same on both side of communication. The value is the path to the file, where the generated password should be saved.

example of the value - the generated password will be saved to /opt/db-pw/dbpassword
``MP_DB_PW=/opt/db-pw/dbpassword``
• MP_PW_DEF
A default password for the keystore is "changeit". The value for this variable says, where the default password should be saved.

example of the value - the default password will be saved to /opt/mp-home/keystorepw
``MP_PW_DEF=/opt/mp-home/keystorepw``
• MP_PW
In case you prefer to have your own generated password also for keystore, this option will interest you. As a value the location for the file is provided.

example of the value - the generated password will be saved to /opt/mp-home/keystorepw
``MP_PW=/opt/mp-home/keystorepw``
• MP_CERT
In case a certificate has to be added to certstore the content of the variable will be processed. The list of certs is processed iteratively so one or more certificates can be provided.

• MP_KEYSTORE
This variable has to be set in order to know where the keystore is stored - where the certificate should be added.

example of the value to add the cert
``````MP_CERT="-----BEGIN CERTIFICATE-----
...
-----END CERTIFICATE-----"
MP_KEYSTORE=/opt/mp-home/keystore.jceks``````
exaple of the value to add more certs
``````MP_CERT="-----BEGIN CERTIFICATE-----
...
-----END CERTIFICATE-----
-----BEGIN CERTIFICATE-----
...
-----END CERTIFICATE-----"
MP_KEYSTORE=/opt/mp-home/keystore.jceks``````
 Prerequisites for the MP_CERT are MP_KEYSTORE & MP_PW or MP_KEYSTORE & MP_PW_DEF - it is necessary in order to locate and access / generation of the keystore.

### container

In comparison to start, the midpoint application is run in foreground. The "normal" start is utilizing nohup and "send to background" (&) options. This behaviour is not acceptable as far as the container without extra setting would be handled as terminated (main PID would ends almost immediately). With keeping application in foreground the container is properly handled as running.

Starting the application

• bash syntax

``/opt/midpoint/bin/midpoint.sh container``
• docker syntax

``command: [ "/opt/midpoint/bin/midpoint.sh", "container" ]``

## Docker-compose yaml file

The following docker-compose configuration file can be used to run a working environment with postgres DB backend including the native repository implementation. There is example with post-initial-object.

MP_ENTRY_POINT option is pointing to the folder in the container’s filesystem, which is handled as a read only source for (not only) post-initial-objects. The content is copied to proper midpoint’s structure (/opt/midpoint/var) before starting the midpoint instance with keeping the same subfolder structure. It can be also used for any other files like csv or jar (e.g. connectors).

During the processing of the MP_ENTRY_POINT it checks the existence of the file or file with extension .done (processed post-initial-object is renamed with suffix .done). Once the file in any form ("exact" name or with the .done suffix) exists, the file is skipped so any future changes on the copied version are kept without overwriting - only new files are copied. This way the post-initial-objects can be re-used several times with the same behaviour all the time.

example of the behaviour : MP_ENTRY_POINT=/opt/entry-point
``````/opt/entry-point
+ icf-connectors
| + connector-gitlab-rest-1.0.jar
| - connector-ssh-1.0.jar
|
+ post-initial-objects
| + user.xml
| - role.xml
|
- sources
- hr.csv

/opt/midpoint/var
+ export
+ icf-connectors
| - connector-ssh-1.0.jar
+ idm-legacy
+ import
+ lib
+ log
| + midpoint.out
| - midpoint.log
+ post-initial-objects
| - user.xml.done
+ schema
- tmp``````
• icf-connectors exists so no change based on the folder

• connector-gitlab-rest-1.0.jar does not exist so it will by copied to /opt/midpoint/var/icf-connectors/connector gitlab-rest-1.0.jar - GitLab Connector

• connector-ssh-1.0.jar exists in the destination so no action will happen - SSH Connector

• post-initial-objects exists so no change

• user.xml in the destination there exists user.xml.done so no action will happen

• role.xml does not exist so it will be copied to /opt/midpoint/var/post-initial-objects/role.xml

• sources folder does not exist so it will be created - /opt/midpoint/var/sources

• hr.csv does not exist in the destination so it will be copied to /opt/midpoint/var/sources/hr.csv

 In theory you can mount it directly to the midpoint’s structure but the resulting behaviour will be, the most probably, a little bit different than expected. With the first run there can be two possible situations: the mount point will be in "writable" mode In that case the file will be renamed with adding suffix ._done and respective next run (with new container) will be ignoring the files. the mount point will be read-only mode The midpoint start will fail and it will not be possible to rename the file, which is handled as critical error.

If MP_ENTRY_POINT feature is not needed the following lines can be removed:

``     - MP_ENTRY_POINT=/opt/midpoint-dirs-docker-entrypoint``
``     - ./midpoint_server/container_files/mp-home:/opt/midpoint-dirs-docker-entrypoint/:ro``

In case the lines are kept in the example, the folder ./midpoint_server/container_files/mp-home has to exist (or the path have to be changed to the existing one) otherwise docker-compose will fail to start.

 The MP_VER variable is used in the config file. In case the variable is not set the "latest" is used as default. For the specific version you can set this variable before (or during) the docker-compose command start. example of run with default (latest) version ``docker-compose up`` example of run with specific version ``MP_VER=4.4 docker-compose up``
example of "simple" environment with post-initial-objects & root CA cert used by Let’s encrypt
``````version: "3.3"

services:
data_init:
image: evolveum/midpoint:${MP_VER:-latest} command: > bash -c " chmod 777 /opt/db-pw/ ; touch /opt/db-pw/db_init_in_progress ; echo -e '#!/bin/sh\ntouch /opt/db-pw/db_init' >/opt/db-init/000-start.sh ; echo -e '#!/bin/sh\necho DB structure init process has finished...\nrm -f /opt/db-pw/db_init_in_progress /opt/db-pw/db_init' > /opt/db-init/999-finish.sh ; /opt/midpoint/bin/midpoint.sh init-native " environment: - MP_CHECK=/opt/db-pw/init_in_progress - MP_INIT_DB_CONCAT=/opt/db-init/010-init.sql - MP_INIT_CFG=/opt/mp-home - MP_DB_PW=/opt/db-pw/dbpassword - MP_PW_DEF=/opt/mp-home/keystorepw - MP_KEYSTORE=/opt/mp-home/keystore.jceks - | MP_CERT=-----BEGIN CERTIFICATE----- MIIFazCCA1OgAwIBAgIRAIIQz7DSQONZRGPgu2OCiwAwDQYJKoZIhvcNAQELBQAw TzELMAkGA1UEBhMCVVMxKTAnBgNVBAoTIEludGVybmV0IFNlY3VyaXR5IFJlc2Vh cmNoIEdyb3VwMRUwEwYDVQQDEwxJU1JHIFJvb3QgWDEwHhcNMTUwNjA0MTEwNDM4 WhcNMzUwNjA0MTEwNDM4WjBPMQswCQYDVQQGEwJVUzEpMCcGA1UEChMgSW50ZXJu ZXQgU2VjdXJpdHkgUmVzZWFyY2ggR3JvdXAxFTATBgNVBAMTDElTUkcgUm9vdCBY MTCCAiIwDQYJKoZIhvcNAQEBBQADggIPADCCAgoCggIBAK3oJHP0FDfzm54rVygc h77ct984kIxuPOZXoHj3dcKi/vVqbvYATyjb3miGbESTtrFj/RQSa78f0uoxmyF+ 0TM8ukj13Xnfs7j/EvEhmkvBioZxaUpmZmyPfjxwv60pIgbz5MDmgK7iS4+3mX6U A5/TR5d8mUgjU+g4rk8Kb4Mu0UlXjIB0ttov0DiNewNwIRt18jA8+o+u3dpjq+sW T8KOEUt+zwvo/7V3LvSye0rgTBIlDHCNAymg4VMk7BPZ7hm/ELNKjD+Jo2FR3qyH B5T0Y3HsLuJvW5iB4YlcNHlsdu87kGJ55tukmi8mxdAQ4Q7e2RCOFvu396j3x+UC B5iPNgiV5+I3lg02dZ77DnKxHZu8A/lJBdiB3QW0KtZB6awBdpUKD9jf1b0SHzUv KBds0pjBqAlkd25HN7rOrFleaJ1/ctaJxQZBKT5ZPt0m9STJEadao0xAH0ahmbWn OlFuhjuefXKnEgV4We0+UXgVCwOPjdAvBbI+e0ocS3MFEvzG6uBQE3xDk3SzynTn jh8BCNAw1FtxNrQHusEwMFxIt4I7mKZ9YIqioymCzLq9gwQbooMDQaHWBfEbwrbw qHyGO0aoSCqI3Haadr8faqU9GY/rOPNk3sgrDQoo//fb4hVC1CLQJ13hef4Y53CI rU7m2Ys6xt0nUW7/vGT1M0NPAgMBAAGjQjBAMA4GA1UdDwEB/wQEAwIBBjAPBgNV HRMBAf8EBTADAQH/MB0GA1UdDgQWBBR5tFnme7bl5AFzgAiIyBpY9umbbjANBgkq hkiG9w0BAQsFAAOCAgEAVR9YqbyyqFDQDLHYGmkgJykIrGF1XIpu+ILlaS/V9lZL ubhzEFnTIZd+50xx+7LSYK05qAvqFyFWhfFQDlnrzuBZ6brJFe+GnY+EgPbk6ZGQ 3BebYhtF8GaV0nxvwuo77x/Py9auJ/GpsMiu/X1+mvoiBOv/2X/qkSsisRcOj/KK NFtY2PwByVS5uCbMiogziUwthDyC3+6WVwW6LLv3xLfHTjuCvjHIInNzktHCgKQ5 ORAzI4JMPJ+GslWYHb4phowim57iaztXOoJwTdwJx4nLCgdNbOhdjsnvzqvHu7Ur TkXWStAmzOVyyghqpZXjFaH3pO3JLF+l+/+sKAIuvtd7u+Nxe5AW0wdeRlN8NwdC jNPElpzVmbUq4JUagEiuTDkHzsxHpFKVK7q4+63SM1N95R1NbdWhscdCb+ZAJzVc oyi3B43njTOQ5yOf+1CceWxG1bQVs5ZufpsMljq4Ui0/1lvh+wjChP4kqKOJ2qxq 4RgqsahDYVvTH9w7jXbyLeiNdd8XM2w9U/t7y0Ff/9yi0GE44Za4rF2LN9d11TPA mRGunUHBcnWEvgJBQl9nJEiU0Zsnvgc/ubhPgXRR4Xq37Z0j4r7g1SgEEzwxA57d emyPxgcYxn/eR44/KJ4EBs+lVDR3veyJm+kXQ99b21/+jh5Xos1AnX5iItreGCc= -----END CERTIFICATE----- volumes: - db_init:/opt/db-init - db_pw:/opt/db-pw - midpoint_home:/opt/mp-home midpoint_data: image: postgres:13-alpine command: > bash -c " while [ ! -s /opt/db-pw/dbpassword -o -e /opt/db-pw/init_in_progress ] ; do echo 'Waiting to the end of the init process...'; sleep 1; done ; { sleep 2 ; if [ ! -e /opt/db-pw/db_init -a -e /opt/db-pw/db_init_in_progress ] ; then echo 'DB init did not start...' ; rm -f /opt/db-pw/db_ini*; echo 'The lock files has been removed...'; fi ; } & docker-entrypoint.sh postgres " user: "70:70" depends_on: - data_init environment: - POSTGRES_PASSWORD_FILE=/opt/db-pw/dbpassword - POSTGRES_USER=midpoint - POSTGRES_INITDB_ARGS=--lc-collate=en_US.utf8 --lc-ctype=en_US.utf8 ports: - 5432:5432 networks: - net volumes: - midpoint_data:/var/lib/postgresql/data - db_init:/docker-entrypoint-initdb.d/ - db_pw:/opt/db-pw midpoint_server: image: evolveum/midpoint:${MP_VER:-latest}
depends_on:
- data_init
- midpoint_data
command: [ "/opt/midpoint/bin/midpoint.sh", "container" ]
ports:
- 8080:8080
environment:
- MP_CHECK=/opt/db-pw/db_init_in_progress
- MP_SET_midpoint_repository_jdbcUrl=jdbc:postgresql://midpoint_data:5432/midpoint
- MP_UNSET_midpoint_repository_hibernateHbm2ddl=1
- MP_NO_ENV_COMPAT=1
- MP_ENTRY_POINT=/opt/midpoint-dirs-docker-entrypoint
networks:
- net
volumes:
- midpoint_home:/opt/midpoint/var
- db_pw:/opt/db-pw
- ./midpoint_server/container_files/mp-home:/opt/midpoint-dirs-docker-entrypoint/:ro

networks:
net:
driver: bridge

volumes:
db_init:
db_pw:
midpoint_data:
midpoint_home:``````