text:
A Computation of the correlation function

# Quasi-long-range order in trapped two-dimensional Bose gases

## Abstract

We study the fate of algebraic decay of correlations in a harmonically trapped two-dimensional degenerate Bose gas. The analysis is inspired by recent experiments on ultracold atoms where power-law correlations have been observed despite the presence of the external potential. We generalize the spin wave description of phase fluctuations to the trapped case and obtain an analytical expression for the one-body density matrix within this approximation. We show that algebraic decay of the central correlation function persists to lengths of about 20% of the Thomas–Fermi radius. We establish that the trap-averaged correlation function decays algebraically with a strictly larger exponent weakly changing with trap size and find indications that the recently observed enhanced scaling exponents receive significant contributions from the normal component of the gas. We discuss radial and angular correlations and propose a local correlation approximation which captures the correlations very well. Our analysis goes beyond the usual local density approximation and the developed summation techniques constitute a powerful tool to investigate correlations in inhomogeneous systems.

###### pacs:
05.70.Jk, 64.60.an, 67.85.-d

With the achievement of quantum degeneracy in ultracold atom gases a new experimental platform for studying fundamental questions related to phase transitions and critical phenomena has appeared. In particular, correlation functions can be measured through interference and time-of-flight techniques Polkovnikov et al. (2006); Hadzibabic et al. (2006); Cladé et al. (2009); Tung et al. (2010); Plisson et al. (2011); Singh and Mathey (2014); Murthy et al. (2014). A crucial aspect of the experiments, however, consists in the absence of translation invariance due to the presence of the external trapping potential. Consequently, correlations between points r and are not uniquely determined by the value of . This effect should be unimportant for short-range correlations, and, in fact, thermodynamic properties of ultracold gases are often very well-captured by a local density approximation. The situation changes drastically for a system at a critical point, where the correlation length diverges and thus competes with the inhomogeneity of the trapping potential.

Whereas the experimental preparation of critical systems typically requires a highly fine-tuned setup, they are rather effortlessly realized in two-dimensional (2D) systems whose low-energy excitations can be mapped onto an XY model. Examples apart from 2D ultracold quantum gases Hadzibabic and Dalibard (2011); Desbuquois et al. (2012); Ries et al. (2015); Fletcher et al. (2015); Murthy et al. (2015) are given by thin Helium films Bishop and Reppy (1978), layered magnets Dürr et al. (1989); Ballentine et al. (1990), and 2D exciton-polariton condensates Roumpos et al. (2012). To quantify correlations in these systems we introduce the one-body density matrix , where is the creation operator for a particle at point r. For the spatially homogeneous case we then have with some function due to translation and rotation invariance. The Mermin–Wagner–Hohenberg theorem Mermin and Wagner (1966); Hohenberg (1967) forbids long-range order at any finite temperature such that . However, in the XY model an ordered low-temperature phase with infinite correlation length exists, where correlations decay with a temperature-dependent scaling exponent as for large . Since is well below unity, this behavior of correlations is named quasi-long-range order (QLRO). Above a transition temperature, vortices disorder the system and QLRO is lost Holzmann and Krauth (2008); Choi et al. (2013). The described picture has been developed by Berezinskii and Kosterlitz, Thouless (BKT) Berezinskii (1971, 1972); Kosterlitz and Thouless (1973); Kosterlitz (1974).

The presence of the trap raises several conceptual questions towards the validity of the BKT picture in ultracold quantum gas experiments: On which length scales is QLRO visible? If samples bigger than the state of Texas are needed to reach the thermodynamic limit in the homogeneous system Bramwell and Holdsworth (1994), do we need ultracold atomic clouds covering the whole continent for the local density approximation to hold? Does the formula for a Bose gas with mass , temperature , and superfluid density imply a locally varying scaling exponent due to the inhomogeneous density? Can the 2D XY-model explain the large scaling exponent recently observed in experiment and Quantum Monte Carlo (QMC) computations Murthy et al. (2015)? Can we obtain the correlation function of the trapped system from the homogeneous one by a generalization of the local density approximation?

In this Rapid Communication, we address and answer all of these questions within the spin wave approximation for the phase fluctuations of the trapped gas. The main results are (1) the generalization of the textbook spin wave theory to a discrete collective mode spectrum, (2) the derivation of explicit analytical expressions for the one-body density matrix and first-order correlation functions that are suited for practical applications, and (3) the demonstration that the trap-averaged correlation function decays algebraically with an increased exponent. Our calculations provide a simple picture to the large exponents observed in experiment and simulation. We limit the derivation to the key steps and invite the mathematically interested reader to consult the detailed Supplemental Material (SM) SOM ().

Spin wave theory. At sufficiently high temperatures, quantum corrections are small and the macroscopic properties of the trapped gas are described by the classical action

 S[Φ]=1T∫d2r[ |∇Φ|22M+(−μ+V(r))|Φ|2+g2|Φ|4] (1)

for the complex bosonic field . Herein, and are the chemical potential and coupling constant, and we use units such that . We assume an isotropic and harmonic trapping potential . The action is stationary towards small changes in if is satisfied. We neglect gradient terms of the fields and obtain a Thomas–Fermi (TF) density profile given by with and TF radius . By neglecting deviations of from the stationary configuration we then arrive at the action for the phase given by

 Sph[θ]=n02MT∫Rλd2r (1−r2R2)(∇θ)2. (2)

We explicate the length scales that delimit the validity of our description, given by the TF radius at large distances, and a microscopic scale , which is given by the thermal wavelength in cold atom experiments, or by the lattice spacing in spin models. For typical 2D ultracold quantum gases we have and , such that is large.

After partial integration the phase-only action can be written as quadratic form with self-adjoint operator

 DR=−(1−r2R2)∇2+2r⋅∇R2 (3)

and scalar product . The normalized eigenfunctions of for are given by

 θnm(r)=√2n+|m|+1πs|m|/2P(|m|,0)n(1−2s)eimϕ (4)

with , , , , and Jacobi and Legendre polynomials and , respectively Abramowitz and Stegun (1964); SOM (). The energy levels read

 εnm=2|m|+4n(n+|m|+1). (5)

These energies match collective superfluid modes of harmonically trapped 2D Bose gases Boettcher et al. (2011)1. In fact, spin waves are related to the superfluid velocity by means of , i.e., they describe the same physical excitations Edwards et al. (1996); Stringari (1996); De Rossi et al. (2016).

Since the phase-only action is quadratic, though with a nontrivial discrete spectrum, correlation functions are obtained via Gaussian integration. Within the spin wave approximation we have with and

 ⟨Δθ(r,r′)2⟩=MTn0∑n,m ′ |θnm(r)−θnm(r′)|2εnm. (6)

In the homogeneous limit () we have and the eigenfunctions are plane waves with energies , and

 ⟨Δθ(r,r′)2⟩hom =MTn0∫λ−1d2q(2π)2|θq(r)−θq(r′)|2εq ≃η0log(|r−r′|2λ2) for |r−r′|≫λ, (7)

with Herbut (2007); Altland and Simons (2010). Due to and for large and small , we recover this formula for . The homogeneous case is recalled in the SM SOM (). In the homogeneous system, spin wave theory breaks down for Kosterlitz and Thouless (1973); Kosterlitz (1974). In a trap it is applicable to the superfluid core in the center of the atomic cloud. We estimate its radius from as . For () we have (), which is sufficiently large to observe the effects studied here. Much of the elegance of BKT theory stems from replacing the integral in Eq. (7) by the logarithm, which is valid for all relevant separations, but not exact.

Evaluation of the sum. Just as the momentum integration in the homogeneous case is limited by , also Eq. (6) has an ultraviolet cutoff on possible values of , indicated by the prime. In fact, besides , the values of and are limited from above according to . In the following we choose summation cutoffs with Euler’s constant SOM (). We can then write

 ⟨Δθ(r,r′)2⟩ =η02[F(N)0(s,s)−2F(N)0(s,s′)+F(N)0(s′,s′)] +η0M∑m=1[F(Nm)m(s,s)−2cos(mΔϕ)F(Nm)m(s,s′) +F(Nm)m(s′,s′)] (8)

with and

 F(N)0(s,s′) =N∑n=12n+1n(n+1)Pn(1−2s)Pn(1−2s′), F(Nm)m(s,s′) =Nm∑n=02n+m+1m2+n(n+m+1)(ss′)m/2 ×P(m,0)n(1−2s)P(m,0)n(1−2s′). (9)

Here and in the following we denote

 r=(rcosϕrsinϕ), s=r2R2, Δϕ=ϕ−ϕ′, s>=max(s,s′), s<=min(s,s′), ^λ=λR. (10)

Equation (8) can be implemented numerically and thus allows the computation of the phase fluctuations in a trap for arbitrary values of . Note in particular that . If is large, however, analytical approximation formulas can be derived from the limits.

Evaluating Eq. (9) for relies on the observation that for a self-adjoint operator with eigenvalues and eigenfunctions , the function is the Green function of the operator due to the completeness of the . Accordingly, if we know the Green function by some other means, we also know the result of the summation Englis and Peetre (1998); Gustavsson (2001). Now, for fixed the operator is of Sturm–Liouville-type and thus its Green function is given by , where are zero modes of the operator. With this approach we find

 F(∞)0(s,s′) =−1−log[s>(1−s<)], (11) F(∞)m(s,s′) =1m(s)m/2um(s<)vm(s>) (12)

for . We have and with hypergeometric function and , such that and Abramowitz and Stegun (1964); SOM (). The key properties of the functions and are and

 limm→∞um(s)=limm→∞vm(s)=1√1−s. (13)

We define the remainder function

 R(s,s′,Δϕ) =M0∑m=11m(s)m/2cos(mΔϕ) (14) ×(um(s<)vm(s>)−1√(1−s)(1−s′)),

which converges quickly in and is well-approximated by summing up the first terms. The careful limit in Eq. (8) is presented in the SM SOM ().

Correlation functions. The phase fluctuations due to Eq. (8) for are given by

 ⟨Δθ(r,r′)2⟩ ≃η02(1+11−s>)log(¯s>^λ2(1−s>))−η02(1−11−s<)log(¯s<^λ2(1−s<)) +η0√(1−s)(1−s′)log(Δ¯r2¯s>)+η0(R(s,s,0)+R(s′,s′,0)−2R(s,s′,Δϕ)) (15)

with . The latter two definitions give a proper way to treat the singularities that occur when . Their origin is already visible in the homogeneous limit (7): Whereas the integral is well-defined for all , the logarithm expression requires a meaningful procedure of how to treat the ultraviolet singularities. The remainder term involving the -functions in Eq. (15) can be neglected for most cases of interest. The homogeneous limit is recovered for , i.e., .

The central correlation function derived from Eq. (15) is given by

 g1(r)=n0(1−s)12[1+η(s)](rλ)−η(s)e−η02R(s,s,0) (16)

with . The factor is close to unity and may be neglected. The algebraic decay with for small agrees with the central correlation functions obtained in Refs. Petrov et al. (2000); Crecchi and Vicari (2011); Ceccarelli et al. (2013). The angular correlations for read

 ⟨Δθ(r,r,Δϕ)2⟩ =η01−slog(2s(1−cosΔϕ)^λ2(1−s)) +2η0[R(s,s,0)−R(s,s,Δϕ)] (17)

for . (This corresponds to . For the correlations vanish.) In Fig. 1 we plot a selection of correlation functions.

Applications. To obtain a useful estimate of the trapped correlation function from the homogeneous result, we write Eq. (7) as

 ⟨Δθ(r,r′)2⟩hom=2G(r,r′)−G(r,r)−G(%r′,r′) (18)

with , the Green function of . In each separately we apply a local density approximation and obtain

 ⟨Δθ(r,r′)2⟩LCA =η02[11−s+11−s′]log(1^λ2) +η0√(1−s)(1−s′)log(|r−r′|2R2). (19)

We name this particular procedure local correlation approximation (LCA), and observe from Fig. 1 that it works sufficiently well to approximate the actual result. In contrast, the approximation fails considerably.

The Fourier transform of the momentum distribution defines the trap-averaged correlation function Cohen-Tannoudji and Guéry-Odelin (2011); Murthy et al. (2015)

 ¯g1(r)=∫d2x ρ(x,r+x). (20)

In a translation invariant setting we have , whereas both functions differ in the trapped case. As shown in Figs. 1 and 2, decays algebraically for with an increased exponent -, which weakly depends on . In the SM SOM () we analytically estimate , which supports the exceptionally weak -dependence. Although for infinitely large systems, it would require absurd values of .

The spin wave description only applies to the superfluid core of the cloud. Hence, applying Eq. (15) to the whole cloud becomes less accurate for increasing values of and underestimates . To estimate the effect of the normal gas on the trap-averaging we set for with approximate core radius SOM (). Assuming the correlation length in the normal gas to coincide with should give the right order of magnitude of the effect. Computing with this modification of , we find an increased scaling exponent which is slightly above the spin wave prediction for . Increasing further, we obtain exponents up to for small , whereas the algebraic decay disappears for larger in the typical fitting range . Since most experiments are in the first regime, an enhanced scaling exponent is likely to be measured. Neglecting the spin wave part and only keeping exponential correlations, no algebraic decay of is observed.

In order to relate to the enhanced exponents found in Ref. Murthy et al. (2015), we compare to the QMC results where is determined by the decay of the central correlation function Holzmann et al. (2010)2. As shown in Fig 2, we find good qualitative agreement with the QMC results when including the exponential decay of correlations in the normal component. Hence we identify the superfluid core, a significant normal component, and a ratio of as a possible explanation for the large universal transition exponent . Although it is possible to include more quantitative knowledge of the quasi-2D equation of state Holzmann et al. (2008, 2010); Prokof’ev and Svistunov (2002); Boettcher et al. (2016); Wu et al. (2015) into the LCA approximation, it is clear that our qualitative findings are robust against these refinements. A full quantitative analysis of the BKT transition in the trapped gas is left for future work.

In conclusion, we have generalized the spin wave theory of phase fluctuations towards the trapped 2D Bose gas, analytically computed correlation functions, and applied this to obtain the trap-averaged correlation function. Our analysis provides a natural explanation for the unexpected large exponents observed in Ref. Murthy et al. (2015). It should be possible to apply the method to other dimensions and polytropic equations of state, just as it is the case for superfluid hydrodynamic modes Fuchs et al. (2003); Heiselberg (2004); Boettcher et al. (2011). We proposed an LCA prescription which can be further developed by considering these other mode spectra. An improved understanding of correlations in inhomogeneous systems is also mandatory for conducting experiments on universality in disordered systems Allard et al. (2012); Beeler et al. (2012); Carleo et al. (2013) and situations far from equilibrium Mathey et al. (2011); Altman et al. (2015); Dagvadorj et al. (2015).

Acknowledgments. We gratefully acknowledge inspiring discussions with A. Böttcher, R. Ganesh, I. F. Herbut, P. A. Murthy, and M. M. Scherer. IB acknowledges funding by the European Research Council (ERC) under Advanced Grant No. 290623 and the German Research Foundation (DFG) under Grant No. BO 4640/1-1. MH was supported by ANR-12-BS04-0022-01, ANR-13-JS01-0005-01 and DIM Nano’K from Région Île-de-France.

Supplemental Material

## Appendix A Computation of the correlation function

In this section we derive the formula for the phase fluctuations in detail. The presentation provides all the intermediate steps that have been left out in the main text for reasons of brevity.

We start by solving the eigenvalue problem for the operator

 DR=−(1−r2R2)∇2+2r⋅∇R2 (S1)

appearing in the phase-only action

 Sph[θ]=n02MT∫Rλd2r (1−r2R2)(∇θ)2=n02MT(θ,DRθ). (S2)

We introduce dimensionless variables and to obtain the eigenvalue problem

 ^Dθε=εθε. (S3)

We employ a polar coordinate Ansatz to arrive at

 −(1−^r2)(d2d^r2+1^rdd^r−m2^r2)θεm+2^rdd^rθεm=εθεm. (S4)

Note that since is invariant under , the eigenvalues only depends on . With the new coordinate

 s=^r2=r2/R2 (S5)

we eventually arrive at with

 ^D(m)=−dds[4s(1−s)dds]+m21−ss. (S6)

Note that this operator is of Sturm–Liouville form, see Eq. (S114). The eigenfunctions of are easily seen to be of the form with being the Jacobi polynomials, see Eq. (S98). The corresponding eigenvalues are given by

 εnm=2|m|+4n(n+|m|+1) (S7)

with such that . For the solution reduces to the Legendre polynomial . In App. D.1 we collect relevant properties of the Legendre and Jacobi polynomials.

We define the normalized eigenfunctions

 θnm(r)=√2n+|m|+1πs|m|/2P(|m|,0)|n(1−2s)eimϕ (S8)

such that

 ^Dθnm=R2DRθnm=εnmθnm, (S9)

and, due to Eq. (S95), . Due to the completeness of the Jacobi polynomials, every regular phase configuration can be expanded according to

 θ(r)=∑n,manmθnm(r). (S10)

Since is real we have . We arrive at the regular phase-only action

 Missing dimension or its units for \hskip (S11)

The smallest and largest length scales and introduced in Eq. (S2) manifest in lower and upper boundaries on the summation (indicated by a prime). The implications of this are addressed below in detail.

We now consider the one-body density matrix

 ρ(r,r′)=⟨Φ∗(r)Φ(r′)⟩. (S12)

Note that since this correlation function coincides with the connected correlation function . We write and assume to be regular. Approximating by the stationary solution we arrive at

 ρ(r,r′) =√n(r)n(r′)⟨e−i(θ(r)−θ(r′))⟩ (S13)

with

 ⟨O⟩=∫Dθ O e−Sph[θ]∫Dθ e−Sph[θ]. (S14)

Since is quadratic and we have

 ⟨e−i(θ(r)−θ(r′))⟩=e−12⟨[θ(r)−θ(r′)]2⟩ (S15)

according to the rules of Gaussian integration.

We proceed by computing

 ⟨Δθ(r,r′)2⟩=⟨[θ(r)−θ(r′)]2⟩ (S16)

with the Gaussian action . We write the functional measure as and find

 ⟨[θ(r)−θ(r′)]2⟩ =∫Dθ (θ(r)−θ(%r′))(θ(r)−θ(r′))e−Sph[θ]∫Dθ e−Sph[θ] =∑n,m∑n′,m′(θnm(r% )−θnm(r′))(θn′m′(r)−θn′m′(r′)) ×∏ν,μ∫daνμ anman′m′e−12γνμaνμaν,−μ∫daνμ e−12γνμaνμaν,−μ (S17)

with . In the product, all terms except one yield unity. The nontrivial one is due to

 ∫da a2e−12aγa∫da e−12aγa=1γ (S18)

for a real variable . This eventually yields

 ⟨Δθ(r,r′)2⟩=MTn0∑n,m ′ |θnm(r)−θnm(r′)|2εnm. (S19)

The macroscopic description of the trapped Bose gas in terms of the phase-only action is restricted to length scales . This implies an ultraviolet energy cutoff according to . The energies are thus restricted according to

 0<εnm<εmax∼(Rλ)2 (S20)

with . For the largest term in the -sum is

 N=N0=−1+√1+εmax2≃√εmax2∼(Rλ). (S21)

The largest value of , denoted by , is attained for and is found to be

 M=εmax2∼(Rλ)2. (S22)

If , the possible values of are limited from above by . Given we have

 2m+4Nm(Nm+m+1)<εmax (S23)

for the largest value of . Thus

 Nm =−m+12+√εmax+m2+12 =−m+12+√2M+m2+12. (S24)

In the following we choose

 N=e−γ(Rλ), M=e−γ(Rλ)2 (S25)

with Euler’s constant . For large and this implies

 2HN=log(1^λ2), HM=log(1^λ2) (S26)

with harmonic number . We will see below that fixing and according to Eqs. (S26) leads to convenient expressions for the phase fluctuations by trading for . Note, however, that there is no such that Eqs. (S25) can be true at the same time. On the other hand, due to , Eqs. (S25) are close approximations to , i.e., with a proportionality factor of unity in .

We now write the summation formula like in the main text as

 ⟨Δθ(r,r′)2⟩ =η02[F(N)0(s,s)−2F(N)0(s,s′)+F(N)0(s′,s′)] +η0M∑m=1[F(Nm)m(s,s)−2cos(mΔϕ)F(Nm)


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# How to Create a Traditional Romanian Folk Inspired Pattern in Adobe Illustrator

Today’s tutorial is a little bit special, as I am going to show you how to build something based on a traditional hand-sewed Romanian decorative pattern.

The tutorial itself will rely on using the Rectangle Tool in collaboration with the Pixel Preview Mode in order to create a personalized bookmark, that you later print and use.

## On History and Culture

Before we begin, I want to tell you a little about the history of the artwork that we are going to reproduce.

As a person of Romanian nationality, the early years of my life have always been somehow drawn into what is considered traditional handmade art.

My mother, who is a women’s clothing designer, has always been keen on using elements inspired from different regions of the country.

My father, who was born in a small village, had always wanted me to see how life twists and flows differently in the countryside compared to the big cities, so in his quest he showed me things that belonged to his part of the family: clothes, carpets and sandals that were handmade and completely different from what you would wear today.

Even though I wouldn’t have imagined myself wearing such items, I was always intrigued by the intricate patterns that the older women would hand-sew into the fabrics that were worn by entire family members, patterns that later became symbols for the different Romanian administrative regions.

From a historical point of view, archaeologists have found pieces of such decorative art dating from over 6,000 years ago, casting light on what is now considered an important part of the life, mentality, and beliefs of the people that lived during that period.

Through these lines, Romanians have crafted an art that has been transformed today into an important touristic element, with thousands of people from all over the world passing through the hallways of Dimitrie Gusti’s National Village Museum in order to have a glimpse of its beauty.

The pattern that I have selected to recreate comes from the county of Muscel, located in the Muntenia Region, and was mostly used in the 19th century. The style that was used in that region was based heavily on straight lines that form geometric motifs, and darker, cooler colors (black, red, and later on purple and green).

It is worth mentioning that depending on the age of the person that was intended to wear the cloth, the patterns and colors were adapted so that this social indicator could be reflected through them. Younger people would have brighter colors, and less sophisticated patterns, while the elder would wear darker, more complicated ones.

The segment I took was slightly adapted by my own hand in order to give it a finishing touch, as I felt that the pattern needed delimiting sections for both its top and bottom sections.

Now I'll give a step by step process of recreating this pattern by using some basic tools and functions in Adobe Illustrator.

## 1. Setting Up the Document

Assuming you’ve already powered up Illustrator, create a New Document (File > New) with the following settings:

• Number of Artboards: 1
• Width: 120 px (1.67 in)
• Height: 600 px (8.33 in)
• Units: Pixels

• Color Mode: CMYK
• Raster Effects: High (300 ppi)
• Align New Objects to Pixel Grid: checked

Wow, wait a second. Did I just create a document intended to be printed and set the Units to pixels? Yup, I did just that. Why, you may wonder? Well, because the pattern we are going to build is based on square shapes, which are easier to position once we take advantage of the Pixel Preview mode. Also, I’ve already converted my Artboard’s dimensions from inches to px, so don’t worry about sizing problems once printed.

## 2. Setting Up Our Layers

Go to the Layers panel and create two layers, naming them as follows:

1. name
2. pattern

## 3. Creating the Pattern

Once we’ve set up our document and layered it, we can start building the actual pattern.

### Step 1

Position yourself on the pattern layer, making sure to lock the other one, and create a 4 x 4 square which we will color red (#E94849). Using the Transform panel, select and then place the object we’ve just created using these coordinates:

• X: 60 px
• Y: 524 px

### Step 2

Enter Pixel Preview mode, by going to View > Pixel Preview (Alt-Control-Y), and then zoom in on our little square.

Now comes the tricky part. Because the pattern itself is composed of a fairly large number of individual squares, we will have to create a base row of these objects, which we will copy and paste, positioning each duplicate row closer to the top. As we make our way up, I will tell you exactly which squares you need to remove in order to get the pattern right.

So select our original square, and create seven copies to the left and another seven to the right, by dragging to the left or right while holding down Alt. Once you have all 15 objects, select them, and then copy them to the Clipboard (Control-C).

### Step 3

Before we create our second row, we need to get rid of the third square from the left, and the third from the right, and the seventh one from both sides. Simply select them and hit Delete.

### Step 4

Create the second row by simply pasting (Control-V) and moving it on top of the first one.

### Step 5

Adjust the copy by removing the second and fourth squares from each side, and the three in the middle.

### Step 6

Create the third duplicate row by using the same process of pasting from the Clipboard (Control-V), and then remove the third and fifth squares off each side.

### Step 7

On row four, delete the fourth and six squares.

### Step 8

For row five, we need to get rid of squares five and seven.

### Step 9

On row six, we need to delete square number two, three and six from each side and the one in the middle.

### Step 10

Once you have created the seventh row, delete the third and seventh squares off both the left and right side.

### Step 11

As you can see, our pattern is slowly but surely taking shape, creating repeatable jumps in the positioning of our little squares. Because at this point we have a larger section that we can use to simplify our process, we will simply copy (Control-C) everything that we have created until this point, paste it in place (Control-F), and then flip the objects horizontally (right click > Transform > Reflect > Horizontal), making sure to move them on top of row number seven (the last one we created).

### Step 12

Once our first large section of the pattern is created, we can group it (Control-G) and then duplicate it by dragging while holding down Alt, positioning the copy with just one row down (4 px) towards the original, so that the original’s top row goes under the duplicate’s bottom row (I’ve marked the overlay using a red semi-transparent rectangle).

We do this because the pattern’s middle decoration are squares, which forces us to overlay the copies to get them right.

### Step 13

Having created our bottom section of the pattern, select and group its elements (Control-G) and then make a copy (Control-C > Control-F) and position it towards the top of our Artboard using these coordinates:

Your illustration should look something like this.

### Step 14

Because the top part of our pattern is longer than the one at the bottom, we need to repeat the same trick of duplicating while holding down Alt, and then press Control-D (repeat) once to create two more copies (I've colored the copies in slightly darker shades so you can see them better).

You should now have something like this.

### Step 15

Once we’ve created the pattern, we need to add some finishing touches so that it will look more attractive. The first thing we need to do is take care of the side stripe patterns.

Using the Rectangle Tool (M) create a smaller 2 x 2 px square, color it using the same red tint (#E94849) and then position it to the bottom-left side of our lower pattern section, at a distance of 4 px.

### Step 16

Create another 2 x 2 px square, and position it towards the top-right side of our previously created shape.

### Step 17

Create a slightly larger 4 x 2 px rectangle, aligning it to the bottom-left side of the bottom smaller square using the Align panel.

### Step 18

Adjust the previously created shape by double clicking on it to enter Isolation Mode (right click > Isolate Selected Path), and using the Direct Selection Tool (A) select its right side anchor points, moving them upwards by 2 px.

If you turn off Pixel Preview you should have something like this.

Quick tip: try to get used to going in and out (Alt-Control-Y) of Pixel Preview mode as it helps you design with higher accuracy, as you can see exactly what space your shapes occupy.

### Step 19

Group the previously created three objects, and then create a copy and position it towards the top section of the center larger pattern.

### Step 20

Select both the top and bottom elements, and then create a blend between them to fill in the blank space by going to Object > Blend > Blend Options. Here you will see a pop-up asking you what type of Blend you want to make. We need to set it to Specified Steps and enter 23 in the value input box.

Quick tip: If you’ve never worked with blends, you should now that once you’ve set up the Blend Options you need to go to the Object > Blend menu one more time and select Make in order for the blend to take shape.

### Step 21

Since the blend is actually a line, we need to expand it by selecting and then going to Object > Expand and checking both Object and Fill in the pop-up box.

This step isn’t mandatory but more like a personal preference, as I always prefer having all the elements set up as objects.

### Step 22

Copy the left side pattern to the right of our center larger one and position it at the same distance of 2 px.

### Step 23

Finish off the bottom section of our pattern by adding two 4 x 100 px lines on each side of the zigzag patterns, positioning them 2 px from them.

### Step 24

Copy the bottom element of the side zigzag pattern and create two copies, which we will then align to the bottom and top sides of the longer decoration.

### Step 25

Select both the top and bottom objects and then create another Blend, but this time use 71 for the Specified Steps. Make the blend, expand it and then create a duplicate and position it on the right side of the center pattern.

### Step 26

Add two 4 x 292 px lines on each side so that our design keeps a sense of continuity.

### Step 27

Seeing as our pattern is complete, we need to add the top, mid and bottom sections to it. Enter Pixel Preview mode, and using the Rectangle Tool (M) create a 92 x 4 px object and position it towards the top of our pattern, at a distance of 4 px using the Align panel.

### Step 28

Add a slightly smaller 84 x 2 px object, and position it 2 px from the previous one.

### Step 29

Create a diamond pattern by drawing eight smaller 4 x 4 px squares, which we will rotate at a 45° angle (select > right click > Transform > Rotate > 45), and then group into a left and right section, positioning the inner elements at a distance of 2 px from one another. Add one larger 8 x 8 px square, rotate it the same way we did with the smaller ones, and then center it both horizontally and vertically with the two groups. Select all nine diamond shaped objects, and make sure to distance them at 2 px from the nearest line underneath them.

Quick tip: you might have noticed that once you enter Pixel Preview mode, the rotated objects seem all fuzzy. That’s because their anchor points aren’t snapping to the pixel grid any more due to the rotation process. To fix this, simply isolate them, and using the Direct Selection Tool select and move each anchor to its nearest grid intersection.

### Step 30

Select the two bottom lines underneath the diamond pattern, and copy them (Control-C) and then paste them in place (Control-F). Make sure to move the duplicate towards the top of the diamond shapes, reflecting them horizontally (right click > Reflect > Horizontal). Once you’ve done this, select both the top and bottom lines and the inner diamond pattern and group them all together (Control-G).

### Step 31

Create a copy of the previously grouped objects (Control-C > Control-F) and position it just under the bottom section pattern at a distance of 4 px.

### Step 32

Move towards the middle section that stays between the two sections of our decorative pattern, and create two 92 x 2 px rectangles, positioning them 4 px from the top and bottom pattern sections.

### Step 33

Add another two 92 x 4 px rectangles, and position each one 2 px from the previously created ones.

### Step 34

Using the Direct Selection Tool (A), select and copy (Control-C) one of the larger diamond-shaped objects, aligning one copy to the left and another to the right of the line segments we just created, making sure to center them both horizontally and vertically.

At this point our illustration is almost done. All we need to do is add the name (your own name) using our specially picked font, and we’re good to go.

As the project revolves around creating something personal that each of you can use as a practical instrument, the last step of our process involves putting your name onto the face of the bookmark. I’ve carefully picked a custom font which you will need to download and install in order to finish this small project.

The name of the font is ArhaicRegular and it can be downloaded from dafont.com.

Assuming you’ve managed to load the font onto your machine, unlock the name layer, and using the Type Tool (T) click anywhere to start writing. Type in your own name, and then change the font to ArhaicRegular, setting the Font Size to 14 pt. Select the text and then align it to the center using the Paragraph top panel options, changing the color to #6B6B6B.

Lastly, position the text between the two line groups (the ones with the diamond-shaped objects on each side).

## Print!

That’s it! If you’ve followed the tutorial step by step, you should now have a unique bookmark piece ready for printing. You've also learned something about the origins of this traditional Romanian decorative art, and the process involved in recreating it using Adobe Illustrator.

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format: plain


text:
# §26.11 Integer Partitions: Compositions

A composition is an integer partition in which order is taken into account. For example, there are eight compositions of 4: , and . denotes the number of compositions of , and is the number of compositions into exactly parts. is the number of compositions of with no 1’s, where again . The integer 0 is considered to have one composition consisting of no parts:

Also,

The Fibonacci numbers are determined recursively by

26.11.5
,
,
,.

Explicitly,

Additional information on Fibonacci numbers can be found in Rosen et al. (2000, pp. 140–145).


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 Zap. Nauchn. Sem. POMI, 2009, Volume 370, Pages 132–150 (Mi znsl3535)

Estimates of deviations from exact solutions of variational problems with linear growth functionals

S. I. Repin

St. Petersburg Department of Steklov Mathematical Institute RAS, St. Petersburg, Russia

Abstract: In this paper, we derive estimates of deviations from exact solutions of variational problems with linear growth functionals. Since original variational problem may have no minimizer in a reflexive Banach space, the estimates are presented in terms of the dual problem. We prove the consistency of these estimates and obtain their computationally convenient forms. Bibl. – 36 titles.

Key words and phrases: a posteriori error estimates, duality theory, variational problems, linear growth functionals.

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English version:
Journal of Mathematical Sciences (New York), 2010, 166:1, 75–85

UDC: 517.9
Language:

Citation: S. I. Repin, “Estimates of deviations from exact solutions of variational problems with linear growth functionals”, Boundary-value problems of mathematical physics and related problems of function theory. Part 40, Zap. Nauchn. Sem. POMI, 370, POMI, St. Petersburg, 2009, 132–150; J. Math. Sci. (N. Y.), 166:1 (2010), 75–85

Citation in format AMSBIB
\Bibitem{Rep09} \by S.~I.~Repin \paper Estimates of deviations from exact solutions of variational problems with linear growth functionals \inbook Boundary-value problems of mathematical physics and related problems of function theory. Part~40 \serial Zap. Nauchn. Sem. POMI \yr 2009 \vol 370 \pages 132--150 \publ POMI \publaddr St.~Petersburg \mathnet{http://mi.mathnet.ru/znsl3535} \transl \jour J. Math. Sci. (N. Y.) \yr 2010 \vol 166 \issue 1 \pages 75--85 \crossref{https://doi.org/10.1007/s10958-010-9846-8} \scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-77949296650}

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. Fuchs M., Repin S., “A posteriori error estimates for the approximations of the stresses in the Hencky plasticity problem”, Numer. Funct. Anal. Optim., 32:6 (2011), 610–640
2. Haslinger J., Repin S., Sysala S., “A reliable incremental method of computing the limit load in deformation plasticity based on compliance: Continuous and discrete setting”, J. Comput. Appl. Math., 303 (2016), 156–170
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