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# Algorithmic Fractal Dimensions in Geometric Measure Theory

The development of algorithmic fractal dimensions in this century has had many fruitful interactions with geometric measure theory, especially fractal geometry in Euclidean spaces. We survey these developments, with emphasis on connections with computable functions on the reals, recent uses of algorithmic dimensions in proving new theorems in classical (non-algorithmic) fractal geometry, and directions for future research.

## Authors

• 10 publications
• 4 publications

Algorithmic fractal dimensions – constructs of computability theory – ha...
11/30/2019 ∙ by Jack H. Lutz, et al. ∙ 0

• ### On the close interaction between algorithmic randomness and constructive/computable measure theory

This is a survey of constructive and computable measure theory with an e...
12/08/2018 ∙ by Jason Rute, et al. ∙ 0

• ### Algorithmically Optimal Outer Measures

We investigate the relationship between algorithmic fractal dimensions a...
06/15/2020 ∙ by Jack H. Lutz, et al. ∙ 0

• ### A Survey on Approximation in Parameterized Complexity: Hardness and Algorithms

Parameterization and approximation are two popular ways of coping with N...
06/08/2020 ∙ by Andreas Emil Feldmann, et al. ∙ 0

• ### The Dimensions of Hyperspaces

We use the theory of computing to prove general hyperspace dimension the...
04/16/2020 ∙ by Jack H. Lutz, et al. ∙ 0

• ### Can one design a geometry engine? On the (un)decidability of affine Euclidean geometries

We survey the status of decidabilty of the consequence relation in vario...
12/20/2017 ∙ by J. A. Makowsky, et al. ∙ 0

• ### Generalized Nonlinear and Finsler Geometry for Robotics

Robotics research has found numerous important applications of Riemannia...
10/28/2020 ∙ by Nathan D. Ratliff, et al. ∙ 0

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## 1 Introduction

In early 2000, classical Hausdorff dimension Haus19  was shown to admit a new characterization in terms of betting strategies called martingales DCCcon . This characterization enabled the development of various effective, i.e., algorithmic, versions of Hausdorff dimension obtained by imposing computability and complexity constraints on these martingales. These algorithmic versions included resource-bounded dimensions, which impose dimension structure on various complexity classes DCC , the (constructive) dimensions of infinite binary sequences, which interact usefully with algorithmic information theory DISS , and the finite-state dimensions of infinite binary sequences, which interact usefully with data compression and Borel normality FSD . Soon thereafter, classical packing dimension Tric82 ; Sull84  was shown to admit a new characterization in terms of martingales that is exactly dual to the martingale characterization of Hausdorff dimension ESDAICC . This led immediately to the development of strong resource-bounded dimensions, strong (constructive) dimension, and strong finite-state dimension ESDAICC , which are all algorithmic versions of packing dimension. In the years since these developments, hundreds of research papers by many authors have deepened our understanding of these algorithmic dimensions.

Most work to date on effective dimensions has been carried out in the Cantor space, which consists of all infinite binary sequences. This is natural, because effective dimensions speak to many issues that were already being investigated in the Cantor space. However, the classical fractal dimensions from which these effective dimensions arose–Hausdorff dimension and packing dimension–are powerful quantitative tools of geometric measure theory that have been most useful in Euclidean spaces and other metric spaces that have far richer structures than the totally disconnected Cantor space.

This chapter surveys research results to date on algorithmic fractal dimensions in geometric measure theory, especially fractal geometry in Euclidean spaces. This is a small fraction of the existing body of work on algorithmic fractal dimensions, but it is substantial, and it includes some exciting new results.

It is natural to identify a real number with its binary expansion and to use this identification to define algorithmic dimensions in Euclidean spaces in terms of their counterparts in Cantor space. This approach works for some purposes, but it becomes a dead end when algorithmic dimensions are used in geometric measure theory and computable analysis. The difficulty, first noted by Turing in his famous correction Tur38 , is that many obviously computable functions on the reals (e.g., addition) are not computable if reals are represented by their binary expansions Wei00 . We thus take a principled approach from the beginning, developing algorithmic dimensions in Euclidean spaces in terms of the quantity in the following paragraph, so that the theory can seamlessly advance to sophisticated applications.

Algorithmic dimension and strong algorithmic dimension are the most extensively investigated effective dimensions. One major reason for this is that these algorithmic dimensions were shown by the second author and others May02 ; ESDAICC ; LM08  to have characterizations in terms of Kolmogorov complexity, the central notion of algorithmic information theory. In Section 2 below we give a brief introduction to the Kolmogorov complexity of a point in Euclidean space at a given precision .

In Section 3 we use the above Kolmorogov complexity notion to develop the algorithmic dimension and the strong algorithmic dimension of each point in Euclidean space. This development supports the useful intuition that these dimensions are asymptotic measures of the density of algorithmic information in the point . We discuss how these dimensions relate to the local dimensions that arise in the so-called thermodynamic formalism of fractal geometry; we discuss the history and terminology of algorithmic dimensions; we review the prima facie case that algorithmic dimensions are geometrically meaningful; and we discuss what is known about the circumstances in which algorithmic dimensions agree with their classical counterparts. We then discuss the authors’ use of algorithmic dimensions to analyze self-similar fractals LM08 . This analysis gives us a new, information-theoretic proof of the classical formula of Moran Mora46  for the Hausdorff dimensions of self-similar fractals in terms of the contraction ratios of the iterated function systems that generate them. This new proof gives a clear account of “where the dimension comes from” in the construction of such fractals. Section 3 concludes with a survey of the dimensions of points on lines in Euclidean spaces, a topic that has been surprisingly challenging until a very recent breakthrough by N. Lutz and Stull LutStu17 .

We survey interactive aspects of algorithmic fractal dimensions in Euclidean spaces in Section 4, starting with the mutual algorithmic dimensions developed by Case and the first author CasLut15 . These dimensions, and , are analogous to the mutual information measures of Shannon information theory and algorithmic information theory. Intuitively, and are asymptotic measures of the density of the algorithmic information shared by points and in Euclidean spaces. We survey the fundamental properties of these mutual dimensions, which are analogous to those of their information-theoretic analogs. The most important of these properties are those that govern how mutual dimensions are affected by functions on Euclidean spaces that are computable in the sense of computable analysis Wei00 . Specifically, we review the information processing inequalities of CasLut15 , which state that and hold for all computable Lipschitz functions , i.e., that applying such a function to a point cannot increase the density of algorithmic information that it contains about a point . We also survey the conditional dimensions and recently developed by the first author and N. Lutz LutLut17 . Roughly speaking, these conditional dimensions quantify the density of algorithmic information in beyond what is already present in .

It is rare for the theory of computing to be used to answer open questions in mathematical analysis whose statements do not involve computation or related aspects of logic. In Section

5 we survey exciting new developments that do exactly this. We first describe new characterizations by the first author and N. Lutz LutLut17  of the classical Hausdorff and packing dimensions of arbitrary sets in Euclidean spaces in terms of the relativized dimensions of the individual points that belong to them. These characterizations are called point-to-set principles because they enable one to use a bound on the relativized dimension of a single, judiciously chosen point in a set in Euclidean space to prove a bound on the classical Hausdorff or packing dimension of the set . We illustrate the power of the point-to-set principle by giving an overview of its use in the new, information-theoretic proof LutLut17  of Davies’s 1971 theorem stating that the Kakeya conjecture holds in the Euclidean plane Davi71 . We then discuss two very recent uses of the point-to-set principle to solve open problems in classical fractal geometry. These are N. Lutz and D. Stull’s strengthened lower bounds on the Hausdorff dimensions of generalized Furstenberg sets LutStu17  and N. Lutz’s extension of the fractal intersection formulas for Hausdorff and packing dimensions in Euclidean spaces from Borel sets to arbitrary sets. These are, to the best of our knowledge, the first uses of algorithmic information theory to solve open problems in classical mathematical analysis.

We briefly survey promising directions for future research in Section 6. These include extending the algorithmic analysis of self-similar fractals LM08  to other classes of fractals, extending algorithmic dimensions to metric spaces other than Euclidean spaces, investigating algorithmic fractal dimensions that are more effective than constructive dimensions (e.g., polynomial-time or finite-state fractal dimensions) in fractal geometry, and extending algorithmic methods to rectifiability and other aspects of geometric measure theory that do not necessarily concern fractal geometry. In each of these we begin by describing an existing result that sheds light on the promise of further inquiry.

Overviews of algorithmic dimensions in Cantor space appear in DowHir10 ; EFDAIT , though these are already out of date. Even prior to the development of algorithmic fractal dimensions, a rich network of relationships among gambling strategies, Hausdorff dimension, and Kolmogorov complexity was uncovered by reserach of Ryabko Ryab84 ; Ryab86 ; Ryab93 ; Ryab94 , Staiger Stai93 ; Stai98 ; Stai00 , and Cai and Hartmanis CaiHar94 . A brief account of this “prehistory” of algorithmic fractal dimensions appears in section 6 of DISS .

## 2 Algorithmic Information in Euclidean Spaces

Algorithmic information theory has most often been used in the set of all finite binary strings. The conditional Kolmogorov complexity (or conditional algorithmic information content) of a string given a string is

 K(x|y)=min{|π|∣∣π∈{0,1}∗ and U(π,y)=x}.

Here

is a fixed universal Turing machine, and

is the length of a binary “ program” . Hence is the minimum number of bits required to specify to , when is provided as side information. We refer the reader to any of the standard texts LiVit09 ; DowHir10 ; Nies12 ; ShUsVe17  for the history and intuition behind this notion, including its essential invariance with respect to the choice of the universal Turing machine . The Kolmogorov complexity (or algorithmic information content) of a string is then

 K(x)=K(x|λ),

where is the empty string.

Routine binary encoding enables one to extend the definitions of and to situations where and range over other countable sets such as , , , etc.

The key to “lifting” algorithmic information theory notions to Euclidean spaces is to define the Kolmogorov complexity of a set to be

 K(E)=min{K(q)|q∈Qn∩E}. (2.1)

(Shen and Vereshchagin SheVer02  used a very similar notion for a very different purpose.) Note that is the amount of information required to specify not the set itself, but rather some rational point in . In particular, this implies that

 E⊆F⟹K(E)≥K(F).

Note also that, if contains no rational point, then .

The Kolmogorov complexity of a point at precision is

 Kr(x)=K(B2−r(x)), (2.2)

where is the open ball of radius about , i.e., the number of bits required to specify some rational point satisfying , where is the Euclidean distance of from the origin.

## 3 Algorithmic Dimensions

### 3.1 Dimensions of Points

We now define the (constructive) dimension of a point to be

 dim(x)=liminfr→∞Kr(x)r (3.1)

and the strong (constructive) dimension of to be

 Dim(x)=limsupr→∞Kr(x)r. (3.2)

We note that and were originally defined in terms of algorithmic betting strategies called gales DISS ; ESDAICC . The identities (3.1) and (3.2) were subsequent theorems proven in LM08 , refining very similar results in May02 ; ESDAICC . These identities have been so convenient for work in Euclidean space that it is now natural to regard them as definitions.

Since is the amount of information required to specify a rational point that approximates to within (i.e., with bits of precision), and are intuitively the lower and upper asymptotic densities of information in the point . This intuition is a good starting point, but the fact that and are geometrically meaningful will only become evident in light of the mathematical consequences of (3.1) and (3.2) surveyed in this chapter.

It is an easy exercise to show that, for all ,

 0≤dim(x)≤Dim(x)≤n. (3.3)

If is a computable point in , then , so . On the other hand, if is a random point in (i.e., a point that is algorithmically random in the sense of Martin-Löf MarL66 ), then , so . Hence the dimensions of points range between 0 and the dimension of the Euclidean space that they inhabit. In fact, for every real number , the dimension level set

 DIMα={x∈Rn|dim(x)=α} (3.4)

and the strong dimension level set

 DIMαstr={x∈Rn|Dim(x)=α} (3.5)

are uncountable and dense in DISS ; ESDAICC . The dimensions and can coincide, but they do not generally do so. In fact, the set is a comeager (i.e., topologically large) subset of HitPav05 .

Classical fractal geometry has local, or pointwise, dimensions that are useful, especially in connection with dynamical systems. Specifically, if is an outer measure on , i.e., a function satisfying , monotonicity (), and countable subadditivity (), and if is locally finite (i.e., every has a neighborhood with ), then the lower and upper local dimensions of at a point are

 (dimlocν)(x)=liminfr→∞log(1ν(B2−r(x)))r (3.6)

and

 (Dimlocν)(x)=limsupr→∞log(1ν(B2−r(x)))r, (3.7)

respectively, where Falc14 .

Until very recently, no relationship was known between the dimensions and and the local dimensions (3.6) and (3.7). However, N. Lutz recently observed that a very non-classical choice of the outer measure remedies this. For each , let

 κ(E)=2−K(E), (3.8)

where is defined as in (2.1). Then is easily seen to be an outer measure on that is finite (i.e., ), hence certainly locally finite, whence the local dimensions and are well defined. In fact we have the following.

###### Theorem 3.1

(N. LutzNLut16 ) For all ,

 dim(x)=(dimlocκ)(x)

and

 Dim(x)=(Dimlocκ)(x).

There is a direct conceptual path from the classical Hausdorff and packing dimensions to the dimensions of points defined in (3.1) and (3.2).

The Hausdorff dimension of a set was introduced by Hausdorff Haus19  before 1920 and is arguably the most important notion of fractal dimension. Its classical definition, which may be found in standard texts such as SteSha05 ; Falc14 ; BisPer17 , involves covering the set by families of sets with diameters vanishing in the limit. In all cases, .

At the beginning of the present century, in order to formulate versions of Hausdorff dimensions that would work in complexity classes and other algorithmic settings, the first author DCC  gave a new characterization of Hausdorff dimension in terms of betting strategies, called gales, on which it is easy to impose computability and complexity conditions. Of particular interest here, he then defined the constructive dimension of a set exactly like the gale characterization of , except that the gales were now required to be lower semicomputable DISS . He then defined the dimension of a point to be the constructive dimension of its singleton, i.e., . The existence of a universal Turing machine made it immediately evident that constructive dimension has the absolute stability property that

 cdim(E)=supx∈Edim(x) (3.9)

for all . Accordingly, constructive dimension has since been investigated pointwise. As noted earlier, the second author May02  then proved the characterization (3.1) as a theorem.

Two things should be noted about the preceding paragraph. First, these early papers were written entirely in terms of binary sequences, rather than points in Euclidean space. However, the most straightforward binary encoding of points bridges this gap. (In this survey we freely use those results from Cantor space that do extend easily to Euclidean space.) Second, although the gale characterization is essential for polynomial time and many other stringent levels of effectivization, constructive dimension can be defined equivalently by effectivizing Hausdorff’s original formulation Reim04 .

### 3.2 The Correspondence Principle

In 2001, the first author conjectured that there should be a correspondence principle (a term that Bohr had used analogously in quantum mechanics) assuring us that for sufficiently simple sets , the constructive and classical dimensions agree, i.e.,

 cdim(E)=dimH(E). (3.10)

Hitchcock Hitchcock:CPED  confirmed this conjecture, proving that (3.10) holds for any set that is a union of sets that are computably closed, i.e., that are in Kleene’s arithmetical hierarchy. (This means that (3.10) holds for all sets, and also for sets that are nonuniform unions of sets.) Hitchcock also noted that this result is the best possible in the arithmetical hierarchy, because there are sets (e.g., , where is a Martin-Löf random point that is ) for which (3.10) fails.

By (3.9) and (3.10) we have

 dimH(E)=supx∈Edim(x), (3.11)

which is a very nonclassical, pointwise characterization of the classical Hausdorff dimensions of sets that are unions of sets. Since most textbook examples of fractal sets are , (3.11) is a strong preliminary indication that the dimensions of points are geometrically meaningful.

The packing dimension of a set was introduced in the early 1980s by Tricot Tric82  and Sullivan Sull84 . Its original definition is a bit more involved that that of Hausdorff dimension Falc14 ; BisPer17  and implies that for all .

After the development of constructive versions of Hausdorff dimension outlined above, Athreya, Hitchcock, and the authors ESDAICC  undertook an analogous development for packing dimension. The gale characterization of turns out to be exactly dual to that of , with just one limit superior replaced by a limit inferior. The strong constructive dimension of a set is defined by requiring the gales to be lower semicomputable, and the strong dimension of a point is . The absolute stability of strong constructive dimension,

 cDim(E)=supx∈EDim(x), (3.12)

holds for all , as does the Kolmogorov complexity characterization (3.2). All this was shown in ESDAICC , but a correspondence principle for strong constructive dimension was left open. In fact, Conidis Coni08  subsequently used a clever priority argument to construct a set for which . It is still not known whether some simple, logical definability criterion for implies that . Staiger’s proof that regular -languages satisfy this identity is an encouraging step in this direction Stai07 .

### 3.3 Self-Similar Fractals

The first application of algorithmic dimensions to fractal geometry was the authors’ investigation of the dimensions of points in self-similar fractals LM08 . We give a brief exposition of this work here, referring the reader to LM08  for the many missing details.

Self-similar fractals are the most widely known and best understood classes of fractals Falc14 . Cantor’s middle-third set, the von Koch curve, the Sierpinski triangle, and the Menger sponge are especially well known examples of self-similar fractals.

Briefly, a self-similar fractal in a Euclidean space is generated from an initial nonempty closed set by an iterated function system (IFS), which is a finite list of contracting similarities . Each of these similarities is coded by the symbol in the alphabet , and each has a contraction ratio . The IFS is required to satisfy Moran’s open set condition Mora46 , which says that there is a nonempty open set whose images , for , are disjoint subsets of .

For example, the Sierpinski triangle is generated from the set consisting of the triangle with vertices , , and , together with this triangle’s interior, by the IFS , where each is defined by

 Si(p)=vi+12(p−vi)

for . Note that and in this example. Note also that the open set condition is satisfied here by letting be the topological interior of . Each infinite sequence codes a point that is obtained by applying the similarities coded by the successive symbols in in a canonical way. (See Figure 1.) The Sierpinski is the attractor (or invariant set) of and , which consists of all points for .

The main objective of LM08  was to relate the dimension and strong dimension of each point in a self-similar fractal to the corresponding dimensions of the coding sequence . As it turned out, the algorithmic dimensions in had to be extended in order to achieve this.

The similarity dimension of an IFS with contraction ratios is the unique solution of the equation

 k−1∑i=ocsi=1. (3.13)

The similarity probability measure of

is the probability measure on

that is implicit in (3.13), i.e., the function defined by

 πS(i)=csdim(S)i (3.14)

for each . If the contraction ratios of are all the same, then is the uniform probability measure on , but this is not generally the case. We extend to the domain by setting

 πS(w)=|w|−1∏m=0πS(w[m]) (3.15)

for each . We define the Shannon -self-information of each string to be the quantity

 lS(w)=log1πS(w). (3.16)

Finally, we define the dimension of a sequence with respect to the IFS to be

 dimS(T)=liminfj→∞K(T[0..j])lS(T[0..j]). (3.17)

Similarly, the strong dimension of with respect to is

 DimS(T)=limsupj→∞K(T[0..j])lS(T[0..j]). (3.18)

The dimension (3.17) is a special case of an algorithmic Billingsley dimension Bill60 ; Wegm68 ; Caja82 . These are treated more generally in LM08 .

A set is a computably self-similar fractal if it is the attractor of some and as above such that the contracting similarities are all computable in the sense of computable analysis.

The following theorem gives a complete analysis of the dimensions of points in computably self-similar fractals.

###### Theorem 3.2

(J. Lutz and Mayordomo LM08 ) If is a computably self-similar fractal and is an IFS testifying to this fact, then, for all points and all coding sequences for ,

 dim(x)=sdim(S)dimS(T) (3.19)

and

 Dim(x)=sdim(S)DimS(T). (3.20)

The proof of Theorem 3.2 is nontrivial. It combines some very strong coding properties of iterated function systems with some geometric Kolmogorov complexity arguments.

The following characterization of continuous functions on the reals is one of the oldest and most beautiful theorems of computable analysis.

###### Theorem 3.3

(Lacombe Lac55a ; Lac55b ) A function is continuous if and only if there is an oracle relative to which is computable.

Using Lacombe’s theorem it is easy to derive the classical analysis of self-similar fractals (which need not be computably self-similar) from Theorem 3.2.

###### Corollary 1

(Moran Mora46 , Falconer Falc89 ) For every self-similar fractal and every IFS that generates ,

 dimH(F)=dimP(F)=sdim(F). (3.21)
###### Proof

Let and be as given. By Lacombe’s theorem there is an oracle relative to which is computable. It follows by a theorem by Kamo and Kawamura KK99  that the set is relative to , whence the relativization of (3.11) tells us that

 dimAH(F)=supx∈FdimA(x). (3.22)

We then have

 dimH(F) ≤ dimP(F) = dimAP(F) ≤ cDimA(F) = supx∈FDimA(x) =(???) supT∈Σ∞sdim(S)DimS,A(T) = sdim(S) = supT∈Σ∞sdim(S)dimS,A(T) =(???) supx∈FdimA(x) =(???) dimAH(F) = dimH(F),

so (3.21) holds.

Intuitively, Theorem 3.2 is stronger than its Corollary 1, because Theorem 3.2 gives a complete account of “where the dimension comes from”.

### 3.4 Dimension Level Sets

The dimension level sets and defined in (3.4) and (3.5) have been the focus of several investigations. It was shown in DISS ; ESDAICC  that, for all ,

 cdim(DIMα)=dimH(DIMα)=α

and

 cDim(DIMαstr)=dimP(DIMαstr)=α.

Hitchcock, Terwijn, and the first author HiLuTe07  investigated the complexities of these dimension level sets from the viewpoint of descriptive set theory. Following standard usage Mosc80 , we write and for the classes at the th level () of the Borel hierarchy of subsets of . That is, is the class of all open subsets of , each is the class of all complements of sets in , and each is the class of all countable unions of sets in . We also write and for the classes of the th level of Kleene’s arithmetical hierarchy of subsets of . That is, is the class of all computably open subsets of , each is the class of all complements of sets in , and each is the class of all effective (computable) unions of sets in .

Recall that a real number is -computable if there is a computable function such that .

The following facts were proven in HiLuTe07 .

1. is but not .

2. For all , is (and if is -computable) but not .

3. is and but not .

4. For all , is (and if is -computable) but not .

Weihrauch and the first author LutWei08  investigated the connectivity properties of sets of the form

 DIMI=⋃α∈IDIMα,

where is an interval. After making the easy observation that each of the sets and is totally disconnected, they proved that each of the sets and is path-connected. These results are especially intriguing in the Euclidean plane, where they say that extending either of the sets and to include the level set transforms it from a totally disconnected set to a path-connected set. This suggests that is somehow a very special subset of .

Turetsky Ture11  investigated this matter further and proved that is a connected set in . He also proved that is not a path-connected subset of .

### 3.5 Dimensions of Points on Lines

Since effective dimension is a pointwise property, it is natural to study the dimension spectrum of a set , i.e., the set . This study is far from obvious even for sets as apparently simple as straight lines. We review in this section the results obtained so far, mainly for the case of straight lines in .

As noted in section 3.4, the set of points in of dimension exactly one is connected, while the set of points in with dimension less than 1 is totally disconnected. Therefore every line in contains a point of dimension 1. Despite the surprising fact that there are lines in every direction that contain no random points LL15 , the first author and N. Lutz have shown that almost every point on any line with random slope has dimension 2 LutLut17 . Still all these results leave open fundamental questions about the structure of the dimension spectra of lines, since they don’t even rule out the possibility of a line having the singleton set as its dimension spectrum.

Very recently this latest open question has been answered in the negative. N. Lutz and Stull LutStu17  have proven the following general lower bound on the dimension of points on lines in .

###### Theorem 3.4

(N. Lutz and Stull LutStu17 ) For all ,

 dim(x,ax+b)≥dima,b(x)+min{dim(a,b),dima,b(x)}.

In particular, for almost every , .

Taking and a Martin-Löf random real relative to , Theorem 3.4 gives us two points in the line, and , whose dimensions differ by at least one, so the dimension spectrum cannot be a singleton.

We briefly sketch here the main intuitions behind the (deep) proof of Theorem 3.4, fully based on algorithmic information theory. Theorem 3.4’s aim is to connect with (i.e., a dimension in with a dimension in ). Notice that in the case the theorem’s conclusion is close to saying .

The proof is based on the property that says that under the following two conditions

1. is small

2. whenever , either is large or is close to

it holds that is close to .

There is an extra ingredient to finish this intuition.While condition (ii) can be shown to hold in general, condition (i) can only be proven in a particular relativized world whereas the conclusion of the theorem still holds for every oracle.

N. Lutz and Stull LS17  have also shown that the dimension spectrum of a line is always infinite, proving the following two results. The first theorem proves that if then the corresponding line contains a length one interval.

###### Theorem 3.5

(N. Lutz and Stull LS17 ) Let satisfy that . Then for every there is a point such that .

The second result proves that all spectra of lines are infinite.

###### Theorem 3.6

(N. Lutz and Stull LS17 ) Let be any line in . Then the dimension spectrum is infinite.

## 4 Mutual and Conditional Dimensions

Just as the dimension of a point in Euclidean space is the asymptotic density of the algorithmic information in , the mutual dimension between two points and in Euclidean spaces is the asymptotic density of the algorithmic information shared by and

. In this section, we survey this notion and the data processing inequalities, which estimate the effect of computable functions on mutual dimension. We also survey the related notion of conditional dimension.

### 4.1 Mutual Dimensions

The mutual (algorithmic) information between two rational points and is

 I(p:q)=K(p)−K(p|q).

This notion, essentially due to Kolmogorov Kolm65 , is an analog of mutual entropy in Shannon information theory Shan48 ; CovTho06 ; LiVit09 . Intuitively, is the amount of information in not contained in , so is the amount of information in that is contained in . It is well known LiVit09  that, for all and ,

 I(p:q)≈K(p)+K(q)−K(p,q) (4.1)

in the sense that the magnitude of the difference between the two sides of (4.2) is . This fact is called symmetry of information, because it immediately implies that .

The ideas in the rest of this section were introduced by Case and the first author CasLut15 . In the spirit of (2.1) they defined the mutual information between sets and to be

 I(E:F)=min{I(p:q)∣∣p∈Qm∩E and q∈Qn∩F}.

This is the amount of information that rational points and must share in order to be in and , respectively. Note that, for all and ,

 [(E1⊆E2) and (F1⊆F2)]⟹I(E1:F1)≥I(E2:F2).

The mutual information between two points and at precision is

 Ir(x:y)=I(B2−r(x):B2−r(y)).

This is the amount of information that rational approximations of and must share, merely due to their proximities (distance less than ) to and .

In analogy with (3.1) and (3.2), the lower and upper mutual dimensions between points and are

 mdim(x:y)=liminfr→∞Ir(x:y)r (4.2)

and

 Mdim(x:y)=limsupr→∞Ir(x:y)r, (4.3)

respectively.

The following theorem shows that the mutual dimensions mdim and Mdim have many of the properties that one should expect them to have. The proof is involved and includes a modest generalization of Levin’s coding theorem Levi73a ; Levi74 .

###### Theorem 4.1

(Case and J. Lutz CasLut15 ) For all and , the following hold.

1. .

2. .

3. .

4. .

5. .

6. .

7. .

8. .

9. If and are independently random, then .

The expressions and in 7 and 8 above refer to the dimensions of the point . In 9 above, and are independently random if is a Martin-Löf random point in .

More properties of mutual dimensions may be found in CasLut15 ; CasLut15b .

### 4.2 Data Processing Inequalities

The data processing inequality of Shannon information theory CovTho06  says that, for any two probability spaces and , any set , and any function ,

 I(f(X);Y)≤I(X;Y), (4.4)

i.e., the induced probability space obtained by “processing the information in through ” has no greater mutual entropy with than has with . More succintly, tells us no more about than tells us about . The data processing inequality of algorithmic information theory LiVit09  says that, for any computable partial function , there is a constant (essentially the number of bits in a program that computes ) such that, for all strings and ,

 I(f(x):y)≤I(x:y)+cf. (4.5)

The data processing inequality for the mutual dimension should say that every nice function has the property that, for all and ,

 mdim(f(x):y)≤mdim(x:y). (4.6)

But what should “nice” mean? A nice function certainly should be computable in the sense of computable analysis BC06 ; Ko91 ; Wei00 . But this is not enough. For example, there is a function that is computable and space-filling in the sense that Saga94 ; CDM12 . For such a function, choose such that , and let . Then

 mdim(f(x):y) = mdim(y:y) = dim(y) = 2 > 1 ≥ dim(x) ≥ mdim(x:y),

so (4.6) fails.

Intuitively, the above failure of (4.6) occurs because the function is extremely sensitive to its input, a property that “ nice” functions do not have. A function is Lipschitz if there is a real number such that, for all ,

 |f(x1)−f(x2)|≤c|x1−x2|.

The following data processing inequalities show that computable Lipschitz functions are “nice”.

###### Theorem 4.2

(Case and J. Lutz CasLut15 ) If is computable and Lipschitz, then, for all and ,

 mdim(f(x):y)≤mdim(x:y)

and

 Mdim(f(x):y)≤Mdim(x:y)

Several more theorems of this type and applications of these appear in CasLut15 .

### 4.3 Conditional Dimensions

A comprehensive theory of the fractal dimensions of points in Euclidean spaces requires not only the dimensions and and the mutual dimensions and , but also the conditional dimensions and formulated by the first author and N. Lutz LutLut17 . We briefly describe these formulations here.

The conditional Kolmogorov complexity , defined for rational points and , is lifted to the conditional dimensions in the following four steps.

1. For , , and , the conditional Kolmogorov complexity of at precision given is

 ^Kr(x|q)=min{K(p|q)|p∈Qm∩B2−r(x)}.
2. For , , and , the conditional Kolmogorov complexity of at precision given at precision is

 Kr,s(x|y)=max{^Kr(x|q)|q∈Qn∩B2−s(y)}.
3. For , , and , the conditional Kolmogorov complexity of given at precision is

 Kr(x|y)=Kr,r(x|y).
4. For and , the lower and upper conditional dimensions of given are

 dim(x|y)=liminfr→∞Kr(x|y)r

and

 Dim(x|y)=limsupr→∞Kr(x|y)r,

respectively.

Steps 1, 2, and 4 of the above lifting are very much in the spirit of what has been done in section 2, 3.1, and 4.1 above. Step 3 appears to be problematic, because using the same precision bound for both and makes the definition seem arbitrary and “brittle”. However, the following result shows that this is not the case.

###### Theorem 4.3

(LutLut17 ) Let . If , then, for all and ,

 dim(x|y)=liminfr→∞Kr,s(r)(x|y)r

and

 Dim(x|y)=limsupr→∞Kr,s(r)(x|y)r.

The following result is useful for many purposes.

###### Theorem 4.4

(chain rule for

) For all and ,

 Kr(x,y)=Kr(x|y)+Kr(y)+o(r). (4.7)

An oracle for a point is a function such that, for all , . The Kolmogorov complexity of a rational point relative to a point is

 Ky(p)=max{Kg(p)∣∣g is an oracle% for y},

where is the Kolmogorov complexity of when the universal machine has access to the oracle . The purpose of the maximum here is to prevent from using oracles that code more than into their behaviors. For and , the dimension relative to is defined from exactly as was defined from in Sections 2 and 3.1 above. The relativized strong dimension is defined analogously.

The following result captures the intuition that conditioning on a point is a restricted form of oracle access to .

###### Lemma 1

(LutLut17 ) For all and , and .

The remaining results in this section confirm that conditional dimensions have the correct information-theoretic relationships to dimensions and mutual dimensions.

###### Theorem 4.5

(LutLut17 ) For all and ,

 mdim(x:y)≥dim(x)−Dim(x|y)

and

 Mdim(x:y)≤Dim(x)−dim(x|y).
###### Theorem 4.6

(chain rule for dimension LutLut17 ) For all and ,

 di


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What are some real world application of fourier series?

what are some real world application of Fourier series ? particularly the complex Fourier integrals

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advanced noise cancellation and cell phone network technology uses Fourier series where digital filtering is used to minimize noise ans bandwidth demands respectively. –  Randomizer Nov 25 '13 at 6:27

It turns out that (almost) any kind of a wave can be written as a sum of sines and cosines. So for example, if I was to record your voice for one second saying something, I can find its fourier series which may look something like this for example

$$\textrm{voice} = \sin(x)+\frac{1}{10}\sin(2x)+\frac{1}{100}\sin(3x)+...$$

and this interactive module shows you how when you add sines and/or cosines the graph of cosines and sines becomes closer and closer to the original graph we are trying to approximate.

The really cool thing about fourier series is that first, almost any kind of a wave can be approximated. Second, when fourier series converge, they converge very fast.

So one of many many applications is compression. Everyone's favorite MP3 format uses this for audio compression. You take a sound, expand its fourier series. It'll most likely be an infinite series BUT it converges so fast that taking the first few terms is enough to reproduce the original sound. The rest of the terms can be ignored because they add so little that a human ear can likely tell no difference. So I just save the first few terms and then use them to reproduce the sound whenever I want to listen to it and it takes much less memory.

JPEG for pictures is the same idea.

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Most audio and image CODECs (including JPEG and mp3) actually use DCTs, which are a subset of generalized Fourier transforms. –  fluffy Nov 24 '13 at 22:59

I can say about these applications.

1. Signal Processing. it may be the best application of Fourier analysis.

2. Approximation Theory. We use Fourier series to write a function as a trigonometric polynomial.

3. Control Theory. The Fourier series of functions in the differential equation often gives some prediction about the behavior of the solution of differential equation. They are useful to find out the dynamics of the solution.

4. Partial Differential equation We use it to solve higher order partial differential equation by the method of sepertion of variables.

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can you please be more elaborate. I have gone through entire wiki page and find Fourier series quite complicated i understand what it does graphically but want to understand where i can try to apply these equations. –  Randomizer Nov 24 '13 at 17:27
I shoul have added the FT-IR Spectrometry field, in that case this tool is so essential you won't be able to obtain the specific diagrams (in such a way it is almost possible to "see" the transformation of the functions). –  Andrea L. Nov 24 '13 at 22:36
I do not know Chemistry. Thank you for this addition. –  Dutta Nov 25 '13 at 1:39

For a very specific example: One of our undergraduate students was taking data generated by a person running on a force plate. Since force exerted on your feet from running is for the most part periodic, she fit the data with a curve using Fourier analysis. The work that followed can be used to help develop better running shoes.

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%matplotlib inline
import nilearn

## Understanding neuroimaging data

### Text files: phenotype or behavior

Phenotypic or behavioral data are often provided as a text or CSV (Comma Separated Values) file. They can be loaded with the pandas package but you may have to specify some options. Here, we’ll specify the sep field, since our data is tab-delimited rather than comma-delimited.

For our dataset, let’s load participant level information:

import os
import pandas as pd

data_dir = '/home/jovyan/nilearn_data/development_fmri/development_fmri/'
participants = 'participants.tsv'

participant_id Age AgeGroup Child_Adult Gender Handedness ToM Booklet-Matched ToM Booklet-Matched-NOFB FB_Composite FB_Group WPPSI BD raw WPPSI BD scaled KBIT_raw KBIT_standard DCCS Summary Scanlog: Scanner Scanlog: Coil Scanlog: Voxel slize Scanlog: Slice Gap
0 sub-pixar052 5.46 5yo child F R 0.681818 0.625000 5.0 pass NaN NaN 27.0 130.0 2.0 3T2 32ch adult 3mm iso 0.1
1 sub-pixar109 10.38 8-12yo child F R 0.960000 0.947368 6.0 pass NaN NaN 30.0 100.0 NaN 3T1 32ch adult 3.13 mm iso None
2 sub-pixar042 5.01 5yo child F L 0.681818 0.750000 3.0 inc NaN NaN 18.0 110.0 2.0 3T2 32ch adult 3mm iso 0.1
3 sub-pixar068 7.92 7yo child F R 0.863636 0.875000 5.0 pass NaN NaN 38.0 137.0 NaN 3T2 32ch adult 3mm iso 0.1
4 sub-pixar110 10.78 8-12yo child F R 1.000000 1.000000 6.0 pass NaN NaN 38.0 122.0 NaN 3T1 32ch adult 3.13 mm iso None

### Nifti data

For volumetric data, nilearn works with data stored in the Nifti structure (via the nibabel package).

The NifTi data structure (also used in Analyze files) is the standard way of sharing data in neuroimaging research. Three main components are:

• data: raw scans in form of a numpy array:
data = img.get_data()
• affine: returns the transformation matrix that maps from voxel indices of the numpy array to actual real-world locations of the brain:
affine = img.affine
header = img.header

It is important to appreciate that the representation of MRI data we’ll be using is a big 4D matrix representing (3D MRI + 1D for time), stored in a single Nifti file.

### Niimg-like objects

Nilearn functions take as input argument what we call “Niimg-like objects”:

Niimg: A Niimg-like object can be one of the following:

• A string with a file path to a Nifti image
• A SpatialImage from nibabel, i.e., an object exposing the get_data() method and affine attribute, typically a Nifti1Image from nibabel.

Niimg-4D: Similarly, some functions require 4D Nifti-like data, which we call Niimgs or Niimg-4D. Accepted input arguments are:

• A path to a 4D Nifti image
• List of paths to 3D Nifti images
• 4D Nifti-like object
• List of 3D Nifti-like objects

Note: If you provide a sequence of Nifti images, all of them must have the same affine !

## Manipulating and looking at data

There is a whole section of the Nilearn documentation on making pretty plots for neuroimaging data ! But let’s start with a simple one.

# Let's use a Nifti file that is shipped with nilearn
from nilearn import datasets

# Note that the variable MNI152_FILE_PATH is just a path to a Nifti file
print('Path to MNI152 template: {}'.format(datasets.MNI152_FILE_PATH))

Path to MNI152 template: /opt/conda/lib/python3.6/site-packages/nilearn/datasets/data/avg152T1_brain.nii.gz

In the above, MNI152_FILE_PATH is nothing more than a string with a path pointing to a nifti image. You can replace it with a string pointing to a file on your disk. Note that it should be a 3D volume, and not a 4D volume.

from nilearn import plotting
plotting.plot_img(datasets.MNI152_FILE_PATH)

<nilearn.plotting.displays.OrthoSlicer at 0x7fee2229b438>

We can also directly manipulate these images using Nilearn ! As an example, let’s try smoothing this image.

from nilearn import image
smooth_anat_img = image.smooth_img(datasets.MNI152_FILE_PATH, fwhm=6)

# While we are giving a file name as input, the function returns
# an in-memory object:
print(smooth_anat_img)

<class 'nibabel.nifti1.Nifti1Image'>
data shape (91, 109, 91)
affine:
[[  -2.    0.    0.   90.]
[   0.    2.    0. -126.]
[   0.    0.    2.  -72.]
[   0.    0.    0.    1.]]
sizeof_hdr      : 348
data_type       : b''
db_name         : b''
extents         : 0
session_error   : 0
regular         : b'r'
dim_info        : 0
dim             : [  3  91 109  91   1   1   1   1]
intent_p1       : 0.0
intent_p2       : 0.0
intent_p3       : 0.0
intent_code     : none
datatype        : int16
bitpix          : 16
slice_start     : 0
pixdim          : [-1.  2.  2.  2.  1.  1.  1.  1.]
vox_offset      : 0.0
scl_slope       : nan
scl_inter       : nan
slice_end       : 0
slice_code      : unknown
xyzt_units      : 10
cal_max         : 8186.02001953125
cal_min         : 0.0
slice_duration  : 0.0
toffset         : 0.0
glmax           : 0
glmin           : 0
descrip         : b'FSL3.3'
aux_file        : b''
qform_code      : mni
sform_code      : mni
quatern_b       : 0.0
quatern_c       : 1.0
quatern_d       : 0.0
qoffset_x       : 90.0
qoffset_y       : -126.0
qoffset_z       : -72.0
srow_x          : [ -2.   0.   0.  90.]
srow_y          : [   0.    2.    0. -126.]
srow_z          : [  0.   0.   2. -72.]
intent_name     : b''
magic           : b'n+1'

plotting.plot_img(smooth_anat_img)

<nilearn.plotting.displays.OrthoSlicer at 0x7fee22180ac8>

We can then save this manipulated image from in-memory to disk as follows:

# smooth_anat_img.to_filename('smooth_anat_img.nii.gz')
# os.getcwd()  # We'll' check our "current working directory" (cwd) to see where the file was saved

## Visualizing neuroimaging volumes

What if we want to view not a structural MRI image, but a functional one ? No problem ! Let’s try loading one:

fmri_filename = 'downsampled_derivatives:fmriprep:sub-pixar109:sub-pixar109_task-pixar_run-001_swrf_bold.nii.gz'
# plotting.plot_epi(os.path.join(data_dir, fmri_filename))

Uh-oh, what happened ?! Let’s look back at the error message:

DimensionError: Input data has incompatible dimensionality: Expected dimension is 3D and you provided a 4D image. See http://nilearn.github.io/manipulating_images/input_output.html.

We can fix that ! Let’s take an average of the EPI image and plot that instead:

from nilearn.image import mean_img

plotting.view_img(mean_img(os.path.join(data_dir, fmri_filename)))

## Convert the fMRI volumes to a data matrix

These are some really lovely images, but for machine learning we want matrices so that we can use all of the techniques we learned this morning !

To transform our Nifti images into matrices, we’ll use the nilearn.input_data.NiftiMasker to extract the fMRI data from a mask and convert it to data series.

First, let’s do the simplest possible mask—a mask of the whole brain. We’ll use a mask that ships with Nilearn and matches the MNI152 template we plotted earlier.

brain_mask = datasets.load_mni152_brain_mask()

<nilearn.plotting.displays.OrthoSlicer at 0x7fee21fe21d0>

from nilearn.input_data import NiftiMasker

NiftiMasker(detrend=False, dtype=None, high_pass=None, low_pass=None,
standardize=True, t_r=None, target_affine=None, target_shape=None,
verbose=0)

# We give the masker a filename and retrieve a 2D array ready
# for machine learning with scikit-learn !

[[ -2.46781182e+00  -2.67007852e+00  -4.48857832e+00 ...,  -4.39526767e-01
7.21451223e-01   7.84595236e-02]
[  1.02899298e-01  -1.04104400e+00  -1.46710646e+00 ...,  -4.39009994e-01
5.83695173e-01   7.43223906e-01]
[ -9.04305649e+00  -9.81106377e+00  -8.46901321e+00 ...,   1.16129816e-01
5.13390243e-01   1.89174011e-01]
...,
[ -9.28782403e-01  -3.53061348e-01   9.24790427e-02 ...,  -2.53264427e-01
-3.80187541e-01  -2.81309426e-01]
[ -6.69653893e-01  -7.78272897e-02   4.90018964e-01 ...,   8.03511497e-03
2.88530111e-01  -2.03638941e-01]
[ -1.85315013e-01   3.53153758e-02   3.71591300e-01 ...,   4.08590078e-01
6.65539980e-01  -2.24340595e-02]]

print(fmri_masked.shape)

(168, 228453)

One way to think about what just happened is to look at it visually:

Essentially, we can think about overlaying a 3D grid on an image. Then, our mask tells us which cubes or “voxels” (like 3D pixels) to sample from. Since our Nifti images are 4D files, we can’t overlay a single grid – instead, we use a series of 3D grids (one for each volume in the 4D file), so we can get a measurement for each voxel at each timepoint. These are reflected in the shape of the matrix ! You can check this by checking the number of positive voxels in our brain mask.

There are many other strategies in Nilearn for masking data and for generating masks. I’d encourage you to spend some time exploring the documentation for these !

We can also display this time series to get an intuition of how the whole brain signal is changing over time.

We’ll display the first three voxels by sub-selecting values from the matrix. You can also find more information on how to slice arrays here.

import matplotlib.pyplot as plt

plt.title('Voxel Time Series')
plt.xlabel('Scan number')
plt.ylabel('Normalized signal')
plt.tight_layout()

## Extracting signals from a brain parcellation

Now that we’ve seen how to create a data series from a single region-of-interest (ROI), we can start to scale up ! What if, instead of wanting to extract signal from one ROI, we want to define several ROIs and extract signal from all of them ? Nilearn can help us with that, too ! 🎉

For this, we’ll use nilearn.input_data.NiftiLabelsMasker. NiftiLabelsMasker which works like NiftiMasker except that it’s for labelled data rather than binary. That is, since we have more than one ROI, we need more than one value ! Now that each ROI gets its own value, these values are treated as labels.

# First, let's load a parcellation that we'd like to use
multiscale = datasets.fetch_atlas_basc_multiscale_2015(resume=True)
print('Atlas ROIs are located at: %s' % multiscale.scale064)

Atlas ROIs are located at: /home/jovyan/nilearn_data/basc_multiscale_2015/template_cambridge_basc_multiscale_nii_sym/template_cambridge_basc_multiscale_sym_scale064.nii.gz

plotting.plot_roi(multiscale.scale064)

<nilearn.plotting.displays.OrthoSlicer at 0x7fee21e789e8>

from nilearn.input_data import NiftiLabelsMasker

NiftiLabelsMasker(background_label=0, detrend=False, dtype=None,
high_pass=None,
labels_img='/home/jovyan/nilearn_data/basc_multiscale_2015/template_cambridge_basc_multiscale_nii_sym/template_cambridge_basc_multiscale_sym_scale064.nii.gz',
memory_level=1, resampling_target='data', smoothing_fwhm=None,
standardize=True, t_r=None, verbose=0)

fmri_matrix = label_masker.fit_transform(os.path.join(data_dir, fmri_filename))
print(fmri_matrix)

[[-0.24587893 -1.62436938  1.05949652 ...,  2.83225298  0.38711333
-6.05006695]
[ 1.6175766   1.82269943  0.71587718 ...,  1.60115886  0.38711333
0.69677722]
[-6.76797295 -7.36948395 -3.4075551  ...,  0.37006465 -2.15778995
-7.06209373]
...,
[-0.55645484 -0.18809079  1.05949652 ..., -1.47657669 -0.46118778
-0.31524941]
[-0.55645484  0.09916494  1.05949652 ..., -1.06621194 -0.46118778
0.35943499]
[-0.24587893 -0.18809079  0.3722578  ..., -1.47657669 -0.46118778
0.02209279]]

print(fmri_matrix.shape)

(168, 64)

### Compute and display a correlation matrix

Now that we have a matrix, we’d like to create a connectome. A connectome is a map of the connections in the brain. Since we’re working with functional data, however, we don’t have access to actual connections. Instead, we’ll use a measure of statistical dependency to infer the (possible) presence of a connection.

Here, we’ll use Pearson’s correlation as our measure of statistical dependency and compare how all of our ROIs from our chosen parcellation relate to one another.

from nilearn import connectome
correlation_measure = connectome.ConnectivityMeasure(kind='correlation')
correlation_measure

ConnectivityMeasure(cov_estimator=LedoitWolf(assume_centered=False, block_size=1000, store_precision=False),

correlation_matrix = correlation_measure.fit_transform([fmri_matrix])
correlation_matrix

array([[[ 1.        ,  0.79118115,  0.39523318, ...,  0.52283996,
0.56544089,  0.47097546],
[ 0.79118121,  1.        ,  0.43269271, ...,  0.50233686,
0.51261008,  0.56233442],
[ 0.39523318,  0.43269268,  1.        , ...,  0.25408298,
0.26091293,  0.2559599 ],
...,
[ 0.52284002,  0.50233686,  0.25408301, ...,  1.        ,
0.62439972,  0.17996702],
[ 0.56544083,  0.51261008,  0.26091293, ...,  0.62439972,
1.        ,  0.5316267 ],
[ 0.47097546,  0.56233442,  0.25595987, ...,  0.17996702,
0.53162676,  1.        ]]], dtype=float32)

import numpy as np

correlation_matrix = correlation_matrix[0]
# Mask the main diagonal for visualization:
# np.fill_diagonal(correlation_matrix, 0)
plotting.plot_matrix(correlation_matrix)

<matplotlib.image.AxesImage at 0x7fee21cd42b0>

### The importance of specifying confounds

In fMRI, we’re collecting a noisy signal. We have artifacts like physiological noise (from heartbeats, respiration) and head motion which can impact our estimates. Therefore, it’s strongly recommended that you control for these and related measures when deriving your connectome measures. Here, we’ll repeat the correlation matrix example, but this time we’ll control for confounds.

conf_filename = 'sub-pixar109_task-pixar_run-001_ART_and_CompCor_nuisance_regressors.tsv'
confounds=os.path.join(data_dir, conf_filename))
clean_correlation_matrix = correlation_measure.fit_transform([clean_fmri_matrix])[0]
np.fill_diagonal(clean_correlation_matrix, 0)
plotting.plot_matrix(clean_correlation_matrix)

<matplotlib.image.AxesImage at 0x7fee21c170f0>

That looks a little different !

Looking more closely, we can see that our correlation matrix is symmetrical; that is, that both sides of the diagonal contain the same information. We don’t want to feed duplicate information into our machine learning classifier, and Nilearn has a really easy way to remove this redundancy !

vectorized_correlation = connectome.ConnectivityMeasure(kind='correlation',
clean_vectorized_correlation = vectorized_correlation.fit_transform([clean_fmri_matrix])[0]
clean_vectorized_correlation.shape  # Exercise: Why is this value not 64 * 64 ?

(2016,)

## Interactive connectome plotting

It can also be helpful to project these connection weightings back on to the brain, to visualize these connectomes ! Here, we’ll use the interactive connectome plotting in Nilearn.

coords = plotting.find_parcellation_cut_coords(multiscale.scale064)
view = plotting.view_connectome(clean_correlation_matrix, coords=coords)
view


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# Arrangement in motion

In the arrangement shown in figure, the mass of the body A is 4 times that of body B. The height h=20 cm . At a certain instant, the body B is released and the system is set in motion. What is the maximum height (in cm), the body B will go up? Assume enough space above B and A sticks to the ground. ($$g= 10ms^{-2}$$)

×


text:
Seminar Spectral Theory and PDE

## Seminar Spectral Theory and PDE, May 16, 17:00

Frédéric Klopp. Institut de Mathématiques Jussieu – Paris Rive Gauche, Sorbonne
Title: Exponential decay for the 2 particle density matrix of disordered many-body fermions at zero and positive temperature.
Place: Pontificia Universidad Católica de Chile, Facultad de Matemáticas (campus San Joaquin), Sala 2
Abstract:
We will consider a simple model for interacting fermions in a random background at zero and positive temperature. At zero temperature, we prove exponential decay for the 2 particle density matrix of a ground state. At positive temperature we prove exponential decay for the 2 particle density matrix of the density operator in the grand canonical ensemble.
http://www.mat.uc.cl/~graikov/seminar.html

Seminar Spectral Theory and PDE

## Seminar Spectral Theory and PDE, May 2, 17:00

Georgi Raikov. Pontificia Universidad Católica de Chile
Title: Threshold singularities of the spectral shift function for geometric perturbations of a magnetic Hamiltonian
Place: Pontificia Universidad Católica de Chile, Facultad de Matemáticas (campus San Joaquin), Sala 2
Abstract:
I will consider the 3D Schrödinger operator H0 with constant magnetic field, and its perturbations $H_+$ (resp., $H_-\,$ ) obtained from $H_0$  by imposing Dirichlet (resp., Neumann) conditions on an appropriate surface. I will introduce the Krein spectral shift function for the operator pairs $(H_0;H_+)$ and $(H_0;H_-)$, and will discuss its singularities at the Landau levels which play the role of thresholds in the spectrum of the unperturbed operator $H_0$
The talk is based on a joint work with V. Bruneau (Bordeaux).
http://www.mat.uc.cl/~graikov/seminar.html


text:
# Tag Info

5

You might want to read this: Size, Value, and Momentum in International Stock Returns by Fama and French (2011) Abstract: In the four regions (North America, Europe, Japan, and Asia Pacific) we examine, there are value premiums in average stock returns that, except for Japan, decrease with size. Except for Japan, there is return momentum ...

4

What you observe in your regression is not strange at all. The regression beta you estimated is $\beta_i = \frac {\mathrm{cov}(r_i,r_m)}{\mathrm{var}(r_m)}$ where $i$ represents the country/region (such as the USA or China) and $m$ represents the "market" (which you take to be the ACWI). Since the USA is itself such a large component of the ACWI (about ...

4

There is a rationale here and it has to do with the connection between diversification and the number of assets in a portfolio. Suppose we purchase an equal-weighted portfolio of n stocks. The variance of the return is then: $\sigma_{p}^2$ = $\sum$ $\sum$ $w_i$$w_jCov(R_i,R_j) In the above notation, the sigmas are summing over i and j ... 4 The risk-free rate is the y-intercept of the Security market line. If the risk free rate goes negative the y-intercept of the Security market line would simply be below the x-axis. So if the risk-free rate decreases the whole line shifts down. This just means people are willing to pay for safety. According to the formula for the SML: E(Ri) : expected ... 3 The CAPM states that the expected return of an asset i is related to the expected market return by$$\mathbb{E}[R_i] = r_f + \beta_i (\mathbb{E}[R_M] - r_f) $$If the CAPM is a correct description of risk and return, then the next period price Q should be given by$$ Q = P (1+r_f + \beta_i (\mathbb{E}[R_M] - r_f)) $$In your formulation, the denominator ... 3 Beta as a measure of risk has serious drawbacks, particularly in emerging markets. You need to consider alternative risk metrics (cost-of-capital build-up method or volatility, for example), or if you do use beta consider what the market index refers to and the composition of that index. This paper actually happens to touch on beta estimation and uses ... 2 Idiosyncratic volatility is NOT included in the regressors, so it should not be and actually cannot be part of your matrix X. Idiosyncratic volatility is the volatility (of Y) your matrix X (explanatory variables) cannot explain (i.e. remaining unexplained part), so it is the error term of your regression equation. Just compute the standard deviation of ... 2 Assuming those are arithmetic returns and covariances at the horizon, calculate a 9\times1 vector containing the betas with respect to the world index using the covariance matrix, call it \beta. The covariance resulting from the world index can be described as \beta\sigma_{world}^{2}\beta'. The matrix ... 2 2) you only take trading days for your analysis because taking in account days on which no price changes took place would shift results in a wrong direction. For exmple, you mostly take 250 trading days p.a. 3) Your time interval up to 2007 is okay and excludes the financial crisis, which is a non-normal circumstance. Therefore, your time interval can be ... 2 In the following paper: "On the Cross-Section of Expected Stock Returns: Fama-French Ten Years Later" (by Chou, Chou, and Wang), the authors found, using the Fama-Mac Beth two-pass regression, that the size effect becomes insignificant during the post-1981 period, and the Book/Market effect becomes insignificant during the post-1990 period. It is important ... 2 Focusing on intuition rather than theory, \beta can also be thought of as the "risk premium" of that specific asset relative to the market. In general, market risk premium links two very important aspects of the world: Consumption & Return. So if we look at the world in two states, an "Up State" & "Down State", here is what we would see: ... 2 When we discuss CAPM it assumes a lot of things about the market and investor behaviour. There is enough literature on "CAPM doesn't hold". In fact most low beta stocks plot above the security market line (SML). So it would be a mistake to take CAPM so seriously in practice and I would cross question if CAPM works as it is? In theory, if there are negative ... 1 Given that you're correctly measuring Alpha, the difference lies in the Beta exposures of the two managers. You may not be capturing certain tilts, which would show up in your error or incorrectly categorized as Beta. Consider the simple case where you have returns grouped into just technology vs. Energy for instance. R_p = B_0 + B_t$$R_m$ + ...

1

Error term The error term tells the difference between the theoretical and the observed values of the dependent variable. As such it is referred to the single observations. In your equation, as you say, $i$ stands for the $i$-th share, therefore the meaning of $\varepsilon_i$ is unclear (as it can't be the $i$-th observation) and the related equations ...

1

Basically you have two equations as follows: -Regression: $$R_i = α + β \cdot R_m + e_i$$ -CAPM equation: $$E(R_i) = r_f + β\left[E(R_m) - r_f\right]$$ In CAPM sense, there is no α . It only exists as idiosyncratic return. You would need lot more than one year of data to estimate the coefficients for regression. You can check p-values to see if the ...

1

This is in essence the idea behind Andrea Frazzini's paper 'Betting Against Beta'. There are various ETFs that aim to exploit the premium. In R, you can do just do a linear regression using the lm(Y~X) which includes an intercept or using lm(Y~X+0) which regresses without an intercept. Assuming you've saved the model in variable lm.r, then to get the ...

1

The answer is NO. It's mathematically incorrect. Simply look the correlation and covariance formulas. But here is a gedankenexperiment (thought experiment) that demonstrates that it's incorrect. Suppose, R1 = M. Then the claim Corr(M,R1) = Corr(M,R2) implies 1 = Corr(M,R2) for any R2, which is obviously wrong.

1

beta refers to the fact that on an average the stock has a degree of correlation with the movement of the index. the important thing is "on an average" because two different stocks may have the same beta but this average may have different weightages of different parts of that time period. so lets say that in the first part of the data, stock1 is not ...

1

Did the portfolio manager have the option of investing in emerging markets? If yes, use MSCI All-World. If the portfolio has holdings based in countries with "developed markets" yet has has emerging markets exposure to revenue/earnings, the convention is to use MSCI World.

1

I have never seen such an adjustment. While monthly data are irregularly sampled in time (in every way...calendar days, trading days, seconds, etc), that irregularity is likely to be a smaller effect than your choice of data frequency (monthly, weekly, daily data). That said, your question is intriguing because in other fields they do have to deal with ...

1

Normalize for trading days if possible.

1

Compounding the monthly excess returns won't provide the annual excess return. You need to compute the difference between the annual return of the portfolio and the annual return of the benchmark. To illustrate this let's look at an example. Consider the following two situations: The benchmark performs well with a $2\%$ return each month; The benchmark ...

1

Your formula is adding where you should be multiplying, and you plugged your inputs into the wrong places (your levered Beta notably). In any case, the process for un-levering/re-levering the beta goes like so: Step 1: Find benchmark company/asset/project Beta. Step 2: Un-lever the benchmark Beta: Unlevered Beta = Levered Beta * (1 / ( 1 + (1 - ...

1

you get what should get. You can't prove that strategy long $X$ short $Y$ is market neutral: is strategy long EUR/USD short USD/CHF risk neutral? I wouldn't say that. It depends, on what? On relationship between these variables, so it is perfectly hedged only if dX=dY so your task is bad stated: it should be rather: what should be $b$ to assure that ...

1

The efficient frontier should be expressed in terms of arithmetic returns since only these returns can account for cross-sectional aggregation. Hence, if you assume the log returns of the risky portfolio are $X_{p} \sim N(\mu,\sigma^{2})$, then you first have to convert it to log-normal moments before combining it with the risk-free rate, $r_{f}$. However, ...

1

I'm not sure what you mean exactly by "does it matter...", but generally speaking it should not surprise you that your alpha is not significant, as many trading strategies are more or less "transformations" of beta. In the purest sense, alpha is not easy to accomplish, and various forms of the EMH would say that it is nearly impossible to achieve it for a ...

1

No, the "low-beta" anomaly is not the result of the difference between arithmetic and geometric mean returns. Statistical tests verifying the existence of the anomaly rely on models employing the arithmetic mean returns, $$\mu_a = \mu_g + \frac{\sigma^2}{2}$$, hence the penalty excess volatility incurs when compounding returns over time does not explain the ...

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text:
Department of

# Mathematics

Seminar Calendar
for Graduate Student Colloquium events the year of Thursday, September 20, 2018.

.
events for the
events containing

Questions regarding events or the calendar should be directed to Tori Corkery.
     August 2018           September 2018          October 2018
Su Mo Tu We Th Fr Sa   Su Mo Tu We Th Fr Sa   Su Mo Tu We Th Fr Sa
1  2  3  4                      1       1  2  3  4  5  6
5  6  7  8  9 10 11    2  3  4  5  6  7  8    7  8  9 10 11 12 13
12 13 14 15 16 17 18    9 10 11 12 13 14 15   14 15 16 17 18 19 20
19 20 21 22 23 24 25   16 17 18 19 20 21 22   21 22 23 24 25 26 27
26 27 28 29 30 31      23 24 25 26 27 28 29   28 29 30 31
30

Wednesday, February 14, 2018

4:00 pm in 245 Altgeld Hall,Wednesday, February 14, 2018

#### Hyperbolicity in geometric group theory

###### Heejoung Kim (Illinois Math)

Abstract: In geometric group theory, we study a finitely generated group by looking at how the group acts on a metric space, using the topological or geometric properties of the metric space to shed light on the group. In particular, there has been lots of study about a finitely generated group acting nicely on a hyperbolic space based on properties of hyperbolic geometry. However, an arbitrary group does not admit a nice action on a hyperbolic space in general. Hence, many people have tried to generalize the techniques of hyperbolic geometry to study a more general metric space where a group might act nicely. In this talk, we will discuss hyperbolic geometry and how to use it to understand a finitely generated group. Moreover, we will talk about some generalizations of hyperbolic geometry with examples.

Monday, March 12, 2018

1:00 pm in 163 Noyes Laboratory,Monday, March 12, 2018

#### Kirwan-Ness stratifications for Cyclic quivers

###### Itziar Ochoa de Alaiza Gracia (Illinois Math)

Abstract: Mathematical and physical problems can often be simplified by making use of symmetries: for example, by taking a space with symmetries and forming a quotient (“dividing out by the symmetries”). Starting with elementary examples, I will explain difficulties that can arise in forming such quotients, and outline a scientific procedure in algebraic geometry that builds reasonable quotients by cutting up spaces into “stable” and “unstable” orbits. I will then explain that, by refining the “stable/unstable” dichotomy to measure “just how unstable each orbit is,” one can get a lot of topological/geometric information about the quotient.

Wednesday, April 4, 2018

4:00 pm in 245 Altgeld Hall,Wednesday, April 4, 2018

#### Measurable Differentiable Structures

###### Chris Gartland (Illinois Math)

Abstract: We'll discuss Lipschitz differentiable structures on a metric measure spaces. The structures consist of a Borel cover of the space, together with Lipschitz maps from the elements of the cover to $\mathbb{R}^n$, such that any real-valued Lipschitz function is differentiable almost everywhere with respect to the maps. Lipschitz differentiable structures were defined in a paper by Cheeger in '99, where he also proved the fundamental theorem that any doubling metric measure space satisfying a Poincare inequality admits such a structure. A corollary of this theorem is a criterion for the non-biLipschitz embedability of these metric spaces in $\mathbb{R}^n$ which generalizes the known results for Carnot groups and Laakso space. These structures allow for interesting constructions on the space such as the $L^\infty$ cotangent bundle and $L^\infty$ cohomology.

Wednesday, April 25, 2018

4:00 pm in 245 Altgeld Hall,Wednesday, April 25, 2018

#### Randomness in 3-Dimensional Geometry and Topology

###### Malik Obeidin (Illinois Math)

Abstract: The probabilistic method, pioneered by Paul Erdos, has proven to be one of the most powerful and versatile tools in the field of combinatorics. Mathematicians working in diverse fields, from graph theory to number theory to linear algebra, have found the probabilistic toolset valuable. However, 3-manifold topology has only been recently approached from this angle, though the field itself is full of intricate combinatorics. In this talk, I'll describe some of the ways one might define a "random 3-manifold" and the subtleties that arise in the definition. I'll also talk about how one can use these ideas to experiment computationally with 3-manifolds, to help us get a handle on what is "common", and what is "rare".

Wednesday, September 5, 2018

4:00 pm in 245 Altgeld Hall,Wednesday, September 5, 2018

#### 65 Years of Orthogonal Projections

###### Fernando Yahdiel Roman-Garcia (UIUC Math)

Abstract: In 1954 John Marstrand published a paper where he related the Hausdorff dimension of a planar set to the Hausdorff dimension of its orthogonal projections. This paper motivated so much work in subsequent years that the topic of “projection theorems” developed into its own subfield of metric geometry. Over 60 years later this continues to be a thriving area of research with many modern adaptations of the original problem solved by Marstrand. In this talk I will give an overview of many different results obtained over the years, including some of my ongoing work on intersections of horizontal projections on the Heisenberg group.

Wednesday, October 3, 2018

4:00 pm in 245 Altgeld Hall,Wednesday, October 3, 2018

#### Connecting the upper half plane, geodesic flows, and continued fractions

###### Claire Merriman (UIUC Math)

Abstract: Continued fractions are frequently studied in number theory, but they can also be described geometrically. I will talk about continued fraction expansions as dynamical systems, and connect this symbolic system to tessellations. The first part will focus on the "regular" or "simple" continued fractions, where all of the numerators are 1. Then, I will show what happens when all of the numerators are $\pm 1$ and the denominators are all even or all odd.

Wednesday, October 31, 2018

4:00 pm in 245 Altgeld Hall,Wednesday, October 31, 2018

#### Basics of Coding Theory and Some Applications

###### Xiao Li (UIUC Math)

Abstract: In this talk I will be presenting some basics of coding theory and some applications to error correcting code. I will then focus on algebraic coding theory and give some examples with interesting algebraic and combinatorial properties. If time permits, I will give some applications in distributed storage.

Wednesday, December 5, 2018

4:00 pm in 245 Altgeld Hall,Wednesday, December 5, 2018

#### Orders of infinities: asymptotic valued differential fields

###### Nigel Pynn-Coates (UIUC Math)

Abstract: Asymptotic valued differential fields go back to work of Hardy and are motivated by comparing the asymptotic behaviour of solutions to differential equations. I will introduce valued differential fields and discuss some of this history. We will see how versions of l’Hôpital’s Rule show up as key properties in the subject. Along the way, I will draw some pictures to illustrate the topology of valued fields. Time permitting, I may mention some of my results or ongoing work.


text:
485 views

## How many numbers between 0 and 1 million can be formed using 0,7,8

$1$ million = $1000000$

so digits should be in the range of $1$ to $6$

for filling one digit we have $3$ options (either $0, 7, 8$)

so total possibility for $6$ digits $= 3^6 = 729$

out of which its count $000000$ which is equal to $0,$ can’t be considered for this question.

so, total numbers $= 729 \ – 1 = 728$

Note: I am assuming numbers between $0$ and $1$ million in numbers greater than $0$ and less than $1$ million

@Yaman Sahu I realised this after publishing my comment. But could not delete because there are no such options.

OOh
I think 729 is fine, it is not safe to assume the non-inclusiveness.

1 million = 1000000.

So, we have to consider up to 6 digit numbers.

For generalization, we consider all less than 6 digit numbers as six-digit numbers with ‘0’ as their beginning digit(s).

All six digits same: (3C1)*(6C6) = 3

Five digits same: (3C1)*(2C1)*[(6!) / {(5!)*(1!)}] = 36

Four digits same and remaining two digits using the others apiece:

(3C1)*(2C2)*[(6!) / {(4!)*(1!)*(1!)}] = 90

Four digits same and remaining two digits using the anyone from the remaining two twice:

(3C1)*(2C1)*[(6!) / {(4!)*(2!)}] = 180

Three digits same and remaining three digits using anyone from the remaining two twice:

(3C1)*(2C1)*(1C1)*[(6!) / {(3!)*(2!)*(1!)}] = 360

Using any two digits thrice each = (3C2)*[(6!) / {(3!)*(3!)}] = 60.

So, total ‘generalized’ 6-digit numbers can be formed using 0, 7 and 8

= (3+36+90+180+360+60)

= 729.

But, these group of numbers include ‘000000’, which being equal to 0, can’t be considered for this question.

So, total numbers can be formed greater than 0 and less than a million using 0, 7 and 8 = 729 - 1 = 728.

Also, there is an easier method for this problem.

This time too, we’ve to consider ‘generalized’ 6-digit numbers.

Here for filling any particular digit’s place, we have 3 options.

So, total ‘generalized’ 6-digit numbers can be formed using 0, 7 and 8 = (3^6) = 729.

But, these group of ‘generalized’ 6-digit numbers include ‘000000’, which being equal to 0, can’t be considered for this question.

So, total numbers can be formed greater than 0 and less than a million using 0, 7 and 8 = 729 - 1 = 728....

If we include 0 we have

$3 + 2*3 + 2*3^{2} + 2*3^{3} + 2*3^{4} + 2*3^{5}$

= 729

Excluding 0 we get 729 -1 = 728

We can also think of this as 6 digit positions each having 3 choices, thus total number of possibilities = $3^{6}$ = 729

Excluding 0 we get $3^{6} - 1$ = 728
by


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# I Gauge Symmetry

Tags:
1. Jan 13, 2018 at 5:06 PM

### arlesterc

I have reviewed the various posts on gauge symmetry in particular this one which is now closed. In this post there is the following link:http://www.vttoth.com/CMS/physics-notes/124-the-principle-of-gauge-invariance.

This is a good read. However, there is some clarification I need.

The article has the following:
"By far the simplest gauge theory is electromagnetism. And by far the simplest way to present electromagnetism as a gauge theory is through the non-relativistic Schrödinger equation of a particle moving in empty space:

iℏ∂ψ∂t=−ℏ22m∇2ψ.

Although the equation contains the wave function ψ, we know that the actual probability of finding a particle in some state is a function of |ψ|. In other words, the phase of the complex function ψ can be changed without altering the outcome of physical experiments. In other words, all physical experiments will produce the same result if we perform the following substitution:

ψ→eip(x,t)ψ,

where p(x,t) is an arbitrary smooth function of space and time coordinates."

Then it goes on to derive the Schrödinger substituting in this varying function and after a lot of steps shows:

"iℏ∂ψ∂t=−ℏ22m{[∇+i∇p(x,t)]2−2mℏ∂p(x,t)∂t}ψ.
which is not the original Schrödinger equation. "

My question/line of thinking: Is the Schrodinger equation supposed to produce the same 'answer' for the wave function at every point in space? In other words two different observers at two different points in space calculating the wave function via the 'ordinary' Schrödinger should end up with the same answer? However this does not happen because each observer is allowed/has a different value for the phase of the wave function based on their location. Therefore to rectify this, the Schrodinger equation has to be modified so as to cancel out this local location-dependent phase and once that is done everyone at every point using this 'modified' Schrödinger equation will end up with the same wave function?

I would appreciate any feedback as to whether I am on the right track here. Thanks in advance.

2. Jan 14, 2018 at 12:40 PM

Gauge symmetry is not a real symmetry. A symmetry acts on one state and takes it to another. A gauge transformation takes a state to the same state. The correct terminology is gauge invariance. The wave function is not a gauge invariant object. It doesn’t have to be because it is not an observable. However all observables like probability amplitudes, currents, etc. must be gauge invariant. Any gauge dependence must cancel out. That’s why you have both the canonical and kinematic momentum in the presence of the magnetic field, the canonical momentum is not gauge invariant.

One thing to note is that once you choose a gauge, the wavefunction must be continuous and single valued at every point in space. However, there are times when there are topological obstructions to this. Then you need to choose different gauge covering different areas which are related by enforcing these conditions on the wavefunction. For example, if you have a magnetic monopole, you must choose a different gauge to cover the north and south pole regions. However, given the singlevaluedness of the wavefunction, these gauge can only different by some quantized amount. For a magnetic monopole, this restricts the charge to be quantized in units of \frac{\hbar c}{2e}. This gives rise to various phenomena such as the quantization of the Hall conductance. The number of quanta n is a topological invariant, you cannot change it via smooth deformations.

3. Jan 14, 2018 at 9:29 PM

### arlesterc

Thanks for the distinction between gauge symmetry and gauge invariance. I think however based on the amount that I see gauge symmetry used it will be a long battle to knock it out of use. Or is there such a thing as gauge symmetry apart from gauge invariance?

4. Jan 15, 2018 at 8:14 PM


text:
Inhomogeneous Kondo-lattice in geometrically frustrated Pr2Ir2O7
Abstract

Magnetic fluctuations induced by geometric frustration of local Ir-spins disturb the formation of long-range magnetic order in the family of pyrochlore iridates. As a consequence, Pr2Ir2O7lies at a tuning-free antiferromagnetic-to-paramagnetic quantum critical point and exhibits an array of complex phenomena including the Kondo effect, biquadratic band structure, and metallic spin liquid. Using spectroscopic imaging with the scanning tunneling microscope, complemented with machine learning, density functional theory and theoretical modeling, we probe the local electronic states in Pr2Ir2O7and find an electronic phase separation. Nanoscale regions with a well-defined Kondo resonance are interweaved with a non-magnetic metallic phase with Kondo-destruction. These spatial nanoscale patterns display a fractal geometry with power-law behavior extended over two decades, consistent with being in proximity to a critical point. Our discovery reveals a nanoscale tuning route, viz. using a spatial variation of the electronic potential as a means of adjusting the balance between Kondo entanglement and geometric frustration.

Authors:
; ; ; ; ; ; ; ; ; ; ; ; ; ;
Award ID(s):
Publication Date:
NSF-PAR ID:
10216082
Journal Name:
Nature Communications
Volume:
12
Issue:
1
ISSN:
2041-1723
Publisher:
Nature Publishing Group
National Science Foundation
##### More Like this
1. Magnetic fluctuations induced by geometric frustration of local Ir-spins disturb the formation of long-range magnetic order in the family of pyrochlore iridates. As a consequence, Pr2Ir2O7 lies at a tuning-free antiferromagnetic-to-paramagnetic quantum critical point and exhibits an array of complex phenomena including the Kondo effect, biquadratic band structure, and metallic spin liquid. Using spectroscopic imaging with the scanning tunneling microscope, complemented with machine learning, density functional theory and theoretical modeling, we probe the local electronic states in Pr2Ir2O7 and find an electronic phase separation. Nanoscale regions with a well-defined Kondo resonance are interweaved with a non-magnetic metallic phase with Kondo-destruction. These spatial nanoscale patterns display a fractal geometry with power-law behavior extended over two decades, consistent with being in proximity to a critical point. Our discovery reveals a nanoscale tuning route, viz. using a spatial variation of the electronic potential as a means of adjusting the balance between Kondo entanglement and geometric frustration.
2. Abstract

Quantum spin systems such as magnetic insulators usually show magnetic order, but such classical states can give way toquantum liquids with exotic entanglementthrough two known mechanisms of frustration: geometric frustration in lattices with triangle motifs, and spin-orbit-coupling frustration in the exactly solvable quantum liquid of Kitaev’s honeycomb lattice. Here we present the experimental observation of a new kind of frustrated quantum liquid arising in an unlikely place: the magnetic insulator Ba4Ir3O10where Ir3O12trimers form an unfrustrated square lattice. The crystal structure shows no apparent spin chains. Experimentally we find a quantum liquid state persisting down to 0.2 K that is stabilized by strong antiferromagnetic interaction with Curie–Weiss temperature ranging from −766 to −169 K due to magnetic anisotropy. The anisotropy-averaged frustration parameter is 2000, seldom seen in iridates. Heat capacity and thermal conductivity are both linear at low temperatures, a familiar feature in metals but here in an insulator pointing to an exotic quantum liquid state; a mere 2% Sr substitution for Ba produces long-range order at 130 K and destroys the linear-T features. Although the Ir4+(5d5) ions in Ba4Ir3O10appear to form Ir3O12trimers of face-sharing IrO6octahedra, we propose that intra-trimer exchange is reduced and the lattice recombines into an array of coupled 1Dmore »

3. Abstract

In situ growth of pyrochlore iridate thin films has been a long-standing challenge due to the low reactivity of Ir at low temperatures and the vaporization of volatile gas species such as IrO3(g) and IrO2(g) at high temperatures and highPO2. To address this challenge, we combine thermodynamic analysis of the Pr-Ir-O2system with experimental results from the conventional physical vapor deposition (PVD) technique of co-sputtering. Our results indicate that only high growth temperatures yield films with crystallinity sufficient for utilizing and tailoring the desired topological electronic properties and the in situ synthesis of Pr2Ir2O7thin films is fettered by the inability to grow withPO2on the order of 10 Torr at high temperatures, a limitation inherent to the PVD process. Thus, we suggest techniques capable of supplying high partial pressure of key species during deposition, in particular chemical vapor deposition (CVD), as a route to synthesis of Pr2Ir2O7.

4. Abstract

Excitonic insulators are usually considered to form via the condensation of a soft charge mode of bound electron-hole pairs. This, however, presumes that the soft exciton is of spin-singlet character. Early theoretical considerations have also predicted a very distinct scenario, in which the condensation of magnetic excitons results in an antiferromagnetic excitonic insulator state. Here we report resonant inelastic x-ray scattering (RIXS) measurements of Sr3Ir2O7. By isolating the longitudinal component of the spectra, we identify a magnetic mode that is well-defined at the magnetic and structural Brillouin zone centers, but which merges with the electronic continuum in between these high symmetry points and which decays upon heating concurrent with a decrease in the material’s resistivity. We show that a bilayer Hubbard model, in which electron-hole pairs are bound by exchange interactions, consistently explains all the electronic and magnetic properties of Sr3Ir2O7indicating that this material is a realization of the long-predicted antiferromagnetic excitonic insulator phase.

5. Abstract

The shape of 3d-orbitals often governs the electronic and magnetic properties of correlated transition metal oxides. In the superconducting cuprates, the planar confinement of the$${d}_{{x}^{2}-{y}^{2}}$$${d}_{{x}^{2}-{y}^{2}}$orbital dictates the two-dimensional nature of the unconventional superconductivity and a competing charge order. Achieving orbital-specific control of the electronic structure to allow coupling pathways across adjacent planes would enable direct assessment of the role of dimensionality in the intertwined orders. Using CuL3and PrM5resonant x-ray scattering and first-principles calculations, we report a highly correlated three-dimensional charge order in Pr-substituted YBa2Cu3O7, where the Prf-electrons create a direct orbital bridge between CuO2planes. With this we demonstrate that interplanar orbital engineering can be used to surgically control electronic phases in correlated oxides and other layered materials.


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# IV Curve

The IV curve of a solar cell is the superposition of the IV curve of the solar cell diode in the dark with the light-generated current.1 The light has the effect of shifting the IV curve down into the fourth quadrant where power can be extracted from the diode. Illuminating a cell adds to the normal "dark" currents in the diode so that the diode law becomes:

where IL = light generated current.

The effect of light on the current-voltage characteristics of a p-junction.

The equation for the IV curve in the first quadrant is:

## Ideal Solar Cell

$I={I}_{L}-{I}_{0}\left[\mathrm{exp}\left(\frac{qV}{nkT}\right)-1\right]$

The -1 term in the above equation can usually be neglected. The exponential term is usually >> 1 except for voltages below 100 mV. Further, at low voltages, the light generated current IL dominates the I0 (...) term so the -1 term is not needed under illumination.

## Ideal Solar Cell (simplified)

$I={I}_{L}-{I}_{0}\left[\mathrm{exp}\left(\frac{qV}{nkT}\right)\right]$

Plotting the above equation gives the IV curve below with the relevant points on the curve labeled and discussed in more detail on the following pages. The power curve has a a maximum denoted as PMP where the solar cell should be operated to give the maximum power output. It is also denoted as PMAX or maximum power point (MPP) and occurs at a voltage of VMP and a current of IMP.

Light Biased IV Curve Calculator

###### Results

Several important parameters which are used to characterize solar cells are discussed in the following pages. The short-circuit current (ISC), the open-circuit voltage (VOC), the fill factor (FF) and the efficiency are all parameters determined from the IV curve.

Rearranging the equation above gives the voltage in terms of current:

$$V = \frac{n k T}{q} ln \left(\frac{I_L – I}{I_0}\right)$$

When I > IL the number in side the ln( ) is negative and undefined. So what happens in reality? The solar cell goes into reverse bias (negative voltage) and either the non-idealities in the solar cell limit the voltage or the supply limits the voltage. In either case, the solar cell will dissipate power. If there is no limit on the supply then a solar cell close to ideal (very high RSHUNT in reverse bias) will be destroyed almost instantly. Other cells will be destroyed due to heating. The problem of power dissipation in solar cells in reverse bias is covered in the module chapter and in particular the use of bypass diodes.