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## Hibridization

$sp, sp^{2}, sp^{3}, dsp^{3}, d^{2}sp^{3}$

Valeria Viera 1B
Posts: 60
Joined: Fri Apr 06, 2018 11:05 am

### Hibridization

Can someone explain to me how to determine hybridization? What are some key things to remember?

KateCaldwell 1A
Posts: 43
Joined: Fri Apr 06, 2018 11:02 am

### Re: Hibridization

Hybridization is the mixing of orbitals into new orbitals to accommodate electrons to form new chemical bonds of the valence-bond theory. To solve for hybridization, look at the amount of electrons/atoms bonded to the central atom. For linear, there is 2 atomic orbitals as well as two hybrid orbitals, so the hybridization is sp. There’s a full table on page 125 showing all the hybridization of atoms.

Johanna Caprietta 1E
Posts: 30
Joined: Fri Apr 06, 2018 11:02 am

### Re: Hibridization

I would say for the final we need to know how to identify how many electron bonds a molecule has and memorizing what the hybridization would be.

404975170
Posts: 68
Joined: Thu Jul 27, 2017 3:00 am

### Re: Hibridization

To find hybridization you must determine the number of regions of electron density present.

A region of electron density is either a pair of lone pair electrons or a bond (every single, double and triple bonds just counts as one region). The number of regions correspond to a hybrid orbital (s, sp, sp2 etc.)
1 region - s
2 regions- sp
3 regions- sp2
4 regions- sp3
5 regions- sp3d
6 regions-sp3d2

Count the number of regions of electron density around the particular atom you are focusing on and then it’s hybridization is shown above which should be memorized.


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# Counting the number of edges between two graph partitions

Consider the following:

g = RandomGraph[WattsStrogatzGraphDistribution[1000, 0.1, 5]]
comm = FindGraphPartition[g, 2];
CommunityGraphPlot[g, comm]

FindGraphPartition[g,2] will find a partition of vertices such that the number of edges having endpoints in different parts is minimized. How can I get Mathematica to tell me the number of edges that connect these two groups? It seems like something I should be able to get at easily, but looking through the documentation I just can't find anything.

Thank you!

If you just need the number of edges, you can use

EdgeCount[g, UndirectedEdge[Alternatives @@ comm[[1]], Alternatives @@ comm[[2]]]]

 376

If you need the list of edges:

el = EdgeList[g, UndirectedEdge[Alternatives @@ comm[[1]], Alternatives @@ comm[[2]]]]

Length @ el

376

Short[el, 5]

format: plain


text:
# NEET Physics Kinetic Theory of Gases Questions Solved

The root mean square speed of the molecules of a diatomic gas is v. When the temperature is doubled, the molecules dissociate into two atoms. The new root mean square speed of the atom is

1.  $\sqrt{2}\mathrm{v}$

2.  $\mathrm{v}$

3.  $2\mathrm{v}$

4.  $4\mathrm{v}$

Explanation is a part of a Paid Course. To view Explanation Please buy the course.

Difficulty Level:


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Zbl 0957.49029
Ambrosio, Luigi
Minimizing movements.
(Italian. English summary)
[J] Rend. Accad. Naz. Sci. XL, Mem. Mat. Appl. (5) 19, 191-246 (1995). ISSN 0392-4106

The notion of minimizing movement has been introduced by De Giorgi and it unifies under a natural framework many problems and techniques in the calculus of variations, differential equations and geometric measure theory. Minimizing movements cover, in particular, as typical cases, the steepest descent method, the approximation of solutions to the heat equation, the mean curvature flow and they are closely related to some penalization methods and $\Gamma$-convergence theory. Roughly speaking, the minimizing movement is defined as the set of pointwise limits (as $\lambda \to +\infty$) of sequences of functions $\{ w^{\lambda} \}_{\lambda}$ which are obtained in the following two steps: \par for each $\lambda \in (1, +\infty)$, we define reccurently a sequence of minimizers $\{ w(\lambda,k) \}_{k \in \Bbb Z}$ in some topological space $S$, for an appropriately defined sequence of functionals ${\cal F}: (1,+\infty) \times \Bbb Z \times S^{2} \to \overline{\Bbb R}$; \par using these minimizers we define a sequence of functions $\{ w^{\lambda} \}_{\lambda}$ on $\Bbb R$ by formula $\Bbb R \ni s \mapsto w^{\lambda}(s) = w(\lambda, [\lambda s]) \in S$ ($w^{\lambda}$ are piecewise constant on intervals of length $1/\lambda$).\par The paper deals with problems which can be treated by exploiting the notion of minimizing movements: the Cauchy problem for the gradient inclusion ${\dot{u}}(t) \in - {\overline {\partial}} f(u(t))$ in a Hilbert space and the problem of evolution of surfaces by mean curvature. Some exercises on minimizing movements and their solutions are also provided.
[S.Migorski (Krakow)]
MSC 2000:
*49Q20 Variational problems in geometric measure-theoretic setting
49J45 Optimal control problems inv. semicontinuity and convergence
35K90 Abstract parabolic evolution equations
47J20 Inequalities involving nonlinear operators

Keywords: gradient equation; maximal monotone; variational evolution problem; mean curvature flow; perimeter; discretization; minimizing movement

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Finitary Languages

# Finitary Languages

Krishnendu Chatterjee IST Austria (Institute of Science and Technology, Austria)    Nathanaël Fijalkow IST Austria (Institute of Science and Technology, Austria) ÉNS Cachan (École Normale Supérieure de Cachan, France)
###### Abstract

The class of -regular languages provides a robust specification language in verification. Every -regular condition can be decomposed into a safety part and a liveness part. The liveness part ensures that something good happens “eventually”. Finitary liveness was proposed by Alur and Henzinger as a stronger formulation of liveness [AH98]. It requires that there exists an unknown, fixed bound such that something good happens within transitions. In this work we consider automata with finitary acceptance conditions defined by finitary Büchi, parity and Streett languages. We give their topological complexity of acceptance conditions, and present a regular-expression characterization of the languages they express. We provide a classification of finitary and classical automata with respect to the expressive power, and give optimal algorithms for classical decisions questions on finitary automata. We (a) show that the finitary languages are -complete; (b) present a complete picture of the expressive power of various classes of automata with finitary and infinitary acceptance conditions; (c) show that the languages defined by finitary parity automata exactly characterize the star-free fragment of -regular languages; and (d) show that emptiness is -complete and universality as well as language inclusion are -complete for finitary automata.

## 1 Introduction

Classical -regular languages: strengths and weakness. The class of -regular languages provides a robust language for specification for solving control and verification problems (see, e.g, [PR89, RW87]). Every -regular specification can be decomposed into a safety part and a liveness part [AS85]. The safety part ensures that the component will not do anything “bad” (such as violate an invariant) within any finite number of transitions. The liveness part ensures that the component will do something “good” (such as proceed, or respond, or terminate) in the long-run. Liveness can be violated only in the limit, by infinite sequences of transitions, as no bound is stipulated on when the “good” thing must happen. This infinitary, classical formulation of liveness has both strengths and weaknesses. A main strength is robustness, in particular, independence from the chosen granularity of transitions. Another main strength is simplicity, allowing liveness to serve as an abstraction for complicated safety conditions. For example, a component may always respond in a number of transitions that depends, in some complicated manner, on the exact size of the stimulus. Yet for correctness, we may be interested only that the component will respond “eventually”. However, these strengths also point to a weakness of the classical definition of liveness: it can be satisfied by components that in practice are quite unsatisfactory because no bound can be put on their response time.

Stronger notion of liveness. For the weakness of the infinitary formulation of liveness, alternative and stronger formulations of liveness have been proposed. One of these is finitary liveness [AH98]: finitary liveness does not insist on a response within a known bound  (i.e, every stimulus is followed by a response within transitions), but on response within some unknown bound (i.e, there exists such that every stimulus is followed by a response within transitions). Note that in the finitary case, the bound may be arbitrarily large, but the response time must not grow forever from one stimulus to the next. In this way, finitary liveness still maintains the robustness (independence of step granularity) and simplicity (abstraction of complicated safety) of traditional liveness, while removing unsatisfactory implementations.

Finitary parity and Streett conditions. The classical infinitary notion of fairness is given by the Streett condition: it consists of a set of pairs of requests and corresponding responses (grants) and requires that every request that appears infinitely often must be responded infinitely often. Its finitary counterpart, the finitary Streett condition requires that there is a bound such that in the limit every request is responded within steps. The classical infinitary parity condition consists of a priority function and requires that the minimum priority visited infinitely often is even. Its finitary counterpart, the finitary parity condition requires that there is a bound such that in the limit after every odd priority a lower even priority is visited within steps.

Results on classical automata. There are several robust results on the languages expressible by automata with infinitary Büchi, parity and Streett conditions, as follows: (a) Topological complexity: it is known that Büchi languages are -complete, whereas parity and Streett languages lie in the boolean closure of and  [MP92]; (b) Automata expressive power: non-deterministic automata with Büchi conditions have the same expressive power as deterministic and non-deterministic parity and Streett automata [Cho74, Saf92]; and (c) Regular-expression characterization: the class of languages expressed by deterministic parity is exactly defined by -regular expressions (see the handbook [Tho97] for details).

Our results. For finitary Büchi, parity and Streett languages, topological, automata-theoretic, regular-expression and decision problems studies were all missing. In this work we present results in the four directions, as follows:

1. Topological complexity. We show that finitary Büchi, parity and Streett conditions are -complete.

2. Automata expressive power. We show that finitary automata are incomparable in expressive power with classical automata. As in the infinitray setting, we show that non-deterministic automata with finitary Büchi, parity and Streett conditions have the same expressive power, as well as deterministic parity and Streett automata, which are strictly more expressive than deterministic finitary Büchi automata. However, in contrast to the infinitary case, for finitary parity condition, non-deterministic automata are strictly more expressive than the deterministic counterpart. As a by-product we derive boolean closure properties for finitary automata.

3. Regular-expression characterization. We consider the characterization of finitary automata through an extension of -regular languages defined as -regular languages by [BC06]. We show that languages defined by non-deterministic finitary Büchi automata are exactly the star-free fragment of -regular languages.

4. Decision problems. We show that emptiness is -complete and universality as well as language inclusion are -complete for finitary automata.

Related works. The notion of finitary liveness was proposed and studied in [AH98], and games with finitary objectives was studied in [CHH09]. A generalization of -regular languages as -regular languages was introduced in [BC06] and variants have been studied in [BT09] (also see [Boj10] for a survey); a topological characterization has been given in [HST10]. Our work along with topological and automata-theoretic studies of finitary languages, explores the relation between finitary languages and -regular expressions, rather than identifying a subclass of -regular expressions. We identify the exact subclass of -regular expressions that corresponds to non-deterministic finitary parity automata.

## 2 Definitions

### 2.1 Languages topological complexity

Let be a finite set, called the alphabet. A word is a sequence of letters, which can be either finite or infinite. A language is a set of words: is a language over finite words and over infinite words.

Cantor topology and Borel hierarchy. Cantor topology on is given by open sets: a language is open if it can be described as where . Let denote the open sets and denote the closed sets (a language is closed if its complement is open): they form the first level of the Borel hierarchy. Inductively, we define: is obtained as countable union of sets; and is obtained as countable intersection of sets. The higher a language is in the Borel hierarchy, the higher its topological complexity.

Since the above classes are closed under continuous preimage, we can define the notion of Wadge reduction [Wad84]: reduces to , denoted by , if there exists a continuous function such , where is the preimage of by . A language is hard with respect to a class if all languages of this class reduce to it. If it additionally belongs to this class, then it is complete.

For , let be the set of finite prefixes of words in . The following property holds:

###### Proposition 1

For all languages , is closed if and only if, for all infinite words , if all finite prefixes of are in , then .

Classical liveness conditions. We now consider three classes of languages that are widespread in verification and specification. They define liveness properties, i.e, intuitively say that something good will happen “eventually”. For an infinite word , let denote the set of letters that appear infinitely often in . The class of Büchi languages is defined as follows, given :

 B¨uchi(F)={w∣Inf(w)∩F≠∅}

i.e, the Büchi condition requires that some letter in appears infinitely often. The class of parity languages is defined as follows, given a priority function that maps letters to integers (representing priorities):

 Parity(p)={w∣min(p(Inf(w))) is even}

i.e, the parity condition requires that the lowest priority the appears infinitely often is even. The class of Streett languages is defined as follows, given , where are request-grant pairs:

 Streett(R,G)={w∣∀i,1≤i≤d,Inf(w)∩Ri≠∅⇒Inf(w)∩Gi≠∅}

i.e, the Streett condition requires that for all requests , if it appears infinitely often, then the corresponding grant also appears infinitely often.

The following theorem presents the topological complexity of the classical languages:

###### Theorem 2.1 (Topological complexity of classical languages [Mp92])
• For all , the language is -complete.

• The parity and Streett languages lie in the boolean closure of and .

### 2.2 Finitary languages

The finitary parity and Streett languages have been defined in [CHH09]. We recall their definitions, and also specialize them to finitary Büchi languages. Let , where , the definition for uses distance sequence as follows:

 distjk(w,(R,G))={0wk∉Rjinf{k′−k∣k′≥k,wk′∈Gj}wk∈Rj

i.e, given a position where is requested, is the time steps (number of transitions) between the request and the corresponding grant . Note that . Then and:

 FinStreett(R,G)={w∣limsupkdistk(w,(R,G))<∞}

i.e, the finitary Streett condition requires the supremum limit of the distance sequence to be bounded.

Since parity languages can be considered as a particular case of Streett languages, where , the latter allows to define . The same applies to finitary Büchi languages, which is a particular case of finitary parity languages where the letters from the set have priority and others have priority . We get the following definitions. Let a priority function, we define:

 distk(w,p)=inf{k′−k∣k′≥k,p(wk′) is even and p(wk′)≤p(wk)}

i.e, given a position where is odd, is the time steps between the odd priority and a lower even priority. Then . We define similarly the finitary Büchi language: given , let:

 nextk(w,F)=inf{k′−k∣k′≥k,wk′∈F}

i.e, is the time steps before visiting a letter in . Then .

### 2.3 Automata, ω-regular and finitary languages

###### Definition 1

An automaton is a tuple , where is a finite set of states, is the finite input alphabet, is the set of initial states, is the transition relation and is the acceptance condition.

An automaton is deterministic if it has a single initial state and for every state and letter there is at most one transition. The transition relation of deterministic automata are described by functions . An automaton is complete if for every state and letter there is a transition. This is the case when the transition function is total.

Runs. A run is a word over , where . The run is accepting if it is infinite and . We will write to denote . An infinite word induces possibly several runs of : a word induces a run if for all . The language accepted by , denoted by , is:

 L(A)={w∣there exists an accepting run ρ induced % by w}.

Acceptance conditions. We will consider various acceptance conditions for automata obtained from the last section by considering as the alphabet. For example, given , the languages and define Büchi and finitary Büchi acceptance conditions, respectively. Automata with finitary acceptance conditions are referred as finitary automata; classical automata are those equipped with infinitary acceptance conditions.

###### Notation 1

We use a standard notation to denote the set of languages recognized by some class of automata. The first letter is either or , where stands for “non-deterministic” and stands for “deterministic”. The last letter refers to the acceptance condition: stands for “Büchi”, stands for “parity” and stands for “Streett”. The acceptance condition may be prefixed by for “finitary”. For example, denotes non-deterministic parity automata, and denotes deterministic finitary Streett automata. We have the following combination:

 {ND}⋅{Fε}⋅⎧⎪⎨⎪⎩BPS⎫⎪⎬⎪⎭

We denote by the class of languages accepted by deterministic parity automata. The following theorem summarizes the results of expressive power of classical automata [Büc62, Saf92, Cho74, GH82]:

###### Theorem 2.2 (Expressive power results for classical automata)
 DB⊂Lω≐NB=DP=NP=DS=NS

## 3 Topological complexity

In this section we define a finitary operator that allows us to describe finitary Büchi, finitary parity and finitary Streett languages topologically and to relate them to the classical Büchi, parity and Streett languages; we then give their topological complexity.

Union-closed-omega-regular operator on languages. Given a language , the language is the union of the languages that are subsets of , -regular and closed, i.e, .

###### Proposition 2

For all languages we have .

###### Proof

Since the set of finite automata can be enumerated in sequence, it follows that is countable. So for all languages , the set is described as a countable union of closed sets. Hence .

We present a pumping lemma for -regular languages that we will use to prove the topological complexity of finitary languages.

###### Lemma 1 (A pumping lemma)

Let be an -regular language. There exists such that for all words , for all positions , there exist such that for all we have .

###### Proof

Given is a -regular language, let be a complete and deterministic parity automata that recognizes , and let be the number of states of . Consider a word such that , and let be the unique run induced by in . Consider a position in such that . Then there exist such that , this must happen as has states. For , if we consider the word , then the unique run induced by in is . Since the parity condition is independent of finite prefixes and the run is accepted by , it follows that is accepted by . Since recognizes , we have .

The following lemma shows that is obtained by applying the operator to .

###### Lemma 2

For all , where , we have

 UniCloOmg(Streett(R,G))=FinStreett(R,G).
###### Proof

We present the two directions of the proof.

1. We first show that . Let such that is closed and -regular. Let , and assume towards contradiction, that . Hence for all , there exists such that and . Let given by the pumping lemma on , from above given we obtain such that and . By the pumping lemma we obtain the witness . Let , and . Since , by the pumping lemma for all we have . This entails that all finite prefixes of the infinite word are in . Since is closed, it follows that . Since it follows that there is some request in position , and there is no corresponding grant for the next steps. Hence there is a position in such that there is request at and no corresponding grant in , and thus it follows that the word . This contradicts that . Hence it follows that .

2. We now show the converse: . We have:

 FinStreett(R,G)={w∣limsupkdistk(w,(R,G))<∞}=⋃B∈N{w∣limsupkdistk(w,(R,G))≤B}=⋃B∈N⋃n∈N{w∣∀k≥n,distk(w,(R,G))≤B}

The language is closed, -regular, and included in . Hence .

The result follows.

Lemma 2 naturally extends to finitary parity and finitary Büchi languages:

###### Corollary 1

The following assertions hold:

• For all , we have ;

• For all , we have .

Büchi languages are a special case of parity languages, and parity languages are in turn a special case of Streett languages. Since distance sequences for parity and Büchi languages have been defined as a special case of Streett languages, Corollary 1 follows from Lemma 2.

The following lemma states that finitary Büchi languages are -complete.

###### Theorem 3.1 (Topological characterization of finitary languages)

The finitary Büchi, finitary parity and finitary Streett are -complete.

###### Proof

We show that if , then is -complete. It follows from Corollary 1 that . We now show that is -hard. By Theorem 2.1 we have that is -complete, hence is -complete. We present a topological reduction to show that . Let be the stuttering function defined as follows:

 w=w0w1…wn…b(w)=w0w1w12…wnwn…wn2n…

The function is continuous. We check that the following holds:

 Inf(w)⊆F  iff  ∃B∈N,∃n∈N,∀k≥n,nextk(b(w),F)≤B.

Left to right direction: assume that from the position of , letters belong to . Then from the position , letters of belong to , then for .
Right to left direction: let and be integers such that for all we have . Assume and , then the letter in position in is repeated times, thus is either or higher than . The latter is not possible since it must be less than . It follows that the letter in position in belongs to . Hence we get , so is -complete. From this we deduce the two other claims as special cases.

## 4 Expressive power of finitary automata

In this section we consider the finitary automata, and compare their expressive power to classical automata. We then address the question of determinization. Deterministic finitary automata enjoy nice properties that allows to describe languages they recognize using the operator. As a by-product we get boolean closure properties of finitary automata.

### 4.1 Comparison with classical automata

Finitary conditions allow to express bounds requirements:

###### Example 1 (DFB⊈Lω)

Consider the finitary Büchi automaton shown in Fig. 1, the state labeled 0 being its only final state. Its language is . Indeed, -labeled state is visited while reading the letter , and the -labeled state is visited while reading the letter . An infinite word is accepted iff the -labeled state is visited infinitely often and there is a bound between two consecutive visits of the -labeled state. We can easily see that is not -regular, using proof ideas from [BC06]: its complement would be -regular, so it would contain ultimately periodic words, which is not the case.

However, finitary automata cannot distinguish between “many b’s” and “only b’s”:

###### Example 2 (DB⊈NFB)

Consider the language of infinitely many ’s, i.e, . The language is recognized by a simple deterministic Büchi automaton. However, we can show that there is no finitary Büchi automata that recognizes . Intuitively, such an automaton would, while reading the infinite word , have to distinguish between all b’s, otherwise it would accept a word with only b’s at the end. Assume towards contradiction that there exists a non-deterministic finitary Büchi automaton with states recognizing . Let us consider the infinite word . Since must be accepted by , there must be an accepting run , represented as follows:

 q0a→p0b→q1…qna→pnbn+1−−→qn+1…

and

 pn−1b→qn,1b→qn,2…b→qn,n−1b→qn,n=qn…

Since is accepting, there exists , and , such that for all we have . Let be the lowest priority infinitely visited in . As is accepting, is even. The state is in position in . Let be an integer such that (a)  and (b) . Let us consider the set of states . Since the distance function is bounded by from the -th position, the priority appears at least once in each set of consecutively visited states of size . Since and is the state following , the latter holds from . Since , it appears at least times in . Since there is states in , at least one state has been reached twice. We can thus iterate: the infinite word , and the word is accepted by . However, and hence we have a contradiction.

We summarize the results in the following theorem.

###### Theorem 4.1

The following assertions hold: (a) ; (b) .

### 4.2 Deterministic finitary automata

Given a deterministic complete automaton with accepting condition , we will consider the language obtained by using as acceptance condition. Treating the automaton as a transducer, we consider the following function: which maps an infinite word to the unique run of on (there is a unique run since is deterministic and complete). Then:

 L(A)={w∣CA(w)∈Acc}=C−A(Acc).

We will focus on the following property: , which follows from the following lemma. Deterministic complete automata, regarded as transducers, preserve topology and -regularity. Hence applying the finitary operator to the input (the language ) or to the acceptance condition is equivalent.

###### Lemma 3

For all deterministic complete automaton, we have:

1. for all , is closed closed ( is continuous).

2. for all , is closed closed ( is closed).

3. for all , is -regular -regular.

4. for all , is -regular -regular.

###### Proof

We prove all the cases below.

1. Let such that is closed. Let be such that for all we have . We define the run and show that . Since is closed, we will show for all we have . From the hypothesis we have , and then there exists an infinite word such that . Let , then we have . Since is deterministic, we get , and hence .

2. Let such that is closed. Let such that for all we have . Then for all , there exists a word such that , and . We define by induction on an infinite nested sequence of finite words . We denote by the limit of this nested sequence of finite words. We have that . Since is closed, .

3. Let such that recognized by a Büchi automaton . We define the Büchi automaton , where iff in and in . We now show the correctness of our construction. Let accepted by , then there exists an accepting run , as follows:

 (q0,p0)w0−→(q1,p1)w1−→(q2,p2)…(qn,pn)wn−→(qn+1,pn+1)…

where the second component visits infinitely often. Hence:

 (†)⎧⎨⎩q0w0−→q1→w1q2…qnwn−→qn+1…in Ap0q0−→p1q1−→p2…pnqn−→pn+1…in B

Hence from , we have , and it follows that . Conversely, let , then we have . Then the above statement holds, which entails that is accepted by . It follows that recognizes .

4. Let such that is recognized by a Büchi automaton . We define the Büchi automaton , where iff there exists , such that in and in . A proof similar to above show that recognizes .

The desired result follows.

###### Theorem 4.2

For any deterministic complete automaton recognizing a language , the finitary restriction of this automaton recognizes .

###### Proof

A word is accepted by iff .

Theorem 4.2 allows to extend all known results on deterministic classes to finitary deterministic classes: as a corollary, we have and .

We now show that non-deterministic finitary parity automata are more expressive than deterministic finitary parity automata. However, for every language there exists such that recognizes , and by Theorem 4.2 the deterministic finitary parity automaton recognizes .

###### Corollary 2

For every language there is a deterministic finitary parity automata such that .

###### Example 3 (DFP⊂NFP)

As for Example 1 we consider the languages and . It follows from Example 1 that both and belong to , hence to . A finitary parity automaton, relying on non-determinism, is easily built to recognize , hence . We can show that we cannot bypass this non-determinism, as by reading a word we have to decide well in advance which sequence will be bounded: a’s or b’s, i.e, . To prove it, we interleave words of the form and , and use a pumping argument to reach a contradiction. Assume towards contradiction that , and let be a deterministic complete finitary parity automaton with states that recognizes . Let be the starting state. Since belongs to , its unique run on is accepting, and can be decomposed as follows: where is the lowest priority visited infinitely often while reading . Then, belongs to this , its unique run on is accepting, and has the following shape: where is the lowest priority visited infinitely often while reading . Repeating this construction and by induction we have, as shown in Fig 2:

where is the lowest priority visited infinitely often while reading and is the lowest priority visited infinitely often while reading . There must be , such that . Let and , we have:

 q0u→tiani+1−−−→si+1v→tj=ti

Consider the words and

 w∗=u⋅(bp′iani+piv)⋅(b2p′iani+2piv)…(bkp′iani+kpiv)…

must be accepted by since it belongs to . Hence is accepted as well, but does not belong to . We have a contradiction, and the result follows.

###### Theorem 4.3

We have .

Observe that Theorem 4.2 does not hold for non-deterministic automata, since we have but .

### 4.3 Non-deterministic finitary automata

We can show that non-deterministic finitary Streett automata can be reduced to non-deterministic finitary Büchi automata, and this would complete the picture of expressive power comparison. We first show that non-deterministic finitary Büchi automata are closed under intersection, and use it to show Theorem 4.4.

###### Lemma 4

is closed under intersection.

###### Proof

Let and be two non-deterministic finitary Büchi automata. Without loss of generality we assume both and to be complete. We will define a construction similar to the synchronous product construction, where a switch between copies will happen while visiting or . The finitary Büchi automaton is . We define the transition relation below:

 δ={((q1,q2,k),σ


text:
# Orientations of ring spectra

String orientations on KO and TMF

Yuqing Shi and I are organising a reading seminar on orientations of structured ring spectra. The goal will be to understand the construction of the Spin and String orientations on real K-theory $\mathrm{KO}$, and the String orientation on topological modular forms $\mathrm{tmf}$. We (usually) meet on Tuesdays, from 13:30 to 15:30; any deviations are indicated in the schedule below. Note that the location changes a couple of times.

We have prepared a list of abstracts for the talks.

## Schedule

• March 1, 13:30-15:30, in Minnaert 0.16.
1. Introduction, Lennart Meier.

• March 8, 13:30-15:30, in Minnaert 0.16.
2. Prerequisites, Miguel Barrero.

• March 15, 13:30-15:30, in Minnaert 0.16.
3. Thom spectra, Max Blans.

• March 22, 13:30-15:30, in Minnaert 0.16.
4. The $\widehat{A}$-genus, Yuqing Shi.

• March 29, 13:30-15:30, in Minnaert 0.16.
5. The Atiyah–Bott–Shapiro orientation as a map of spectra, Miguel Barata.

• April 5, 13:30-15:30, in HFG 610.
6. Elliptic curves and elliptic genera, Sven van Nigtevecht.

• April 12, 13:30-15:30, in BBG 161.
7. Calculations with elliptic genera and properties.

• April 19, 13:30-15:30, in BBG 161.
8. Introduction to $\mathrm{TMF}$, Jack Davies.

• April 26, 13:30-15:30, in BBG 161.
9. Introduction to the Witten genus, Luuk Stehouwer.

• May 3, 13:30-15:30, in BBG 161.
10. $\mathrm{TMF}$ and String bordism.


text:
# Why do lattice cubes in odd dimensions have integer edge lengths?

This is a spinoff from Characterization of Volumes of Lattice Cubes. That question claims a number of facts as being proven, but doesn't include the full proofs. That's fine for the question as it stands, but I find the subject interesting enough to wonder about the details. So here I'm asking.

Let's start by defining the set of squared lattice cube lengths in dimension $n$ as

$$S(n)=\left\{s\in\mathbb N\;\middle|\;\exists M\in\mathbb Z^{n\times n}:M^TM=s\,I_n\right\}$$

In other words, a number $s$ is part of that set iff there exists some $n\times n$ matrix $M$ with integer coordinates whose vectors are orthogonal to one another and all have Euclidean norm $\sqrt s$. They can be interpreted as edge vectors of a cube whose corners are all lattice points in $\mathbb Z^n$ and whose edges are of length $\sqrt s$. Hence the name lattice cube.

Notes about notation: I use $I_n$ to denote the $n\times n$ identity matrix, and I consider $0\in\mathbb N$.

The referenced question claims that for $k\in\mathbb N$,

\begin{align*} S(2k+1)&=\bigl\{a^2\;\big|\;a\in\mathbb N\bigr\} \\ S(4k)&=\bigl\{a\;\big|\;a\in\mathbb N\bigr\}=\mathbb N \qquad\text{(}k>0\text{ of course)}\\ S(4k+2)&\supseteq\bigl\{a^2+b^2\;\big|\;a,b\in\mathbb N\bigr\} \\ \end{align*}

While that question asks for proof that the last point is in fact an identity as well, here I'm concerned with proofs for the rest of the points. Some of them are easy, others are harder.

• $S(2k+1)\supseteq\bigl\{a^2\;\big|\;a\in\mathbb N\bigr\}$:
Simply choose $M=a\,I_n$.
• $S(2k+1)\subseteq\bigl\{a^2\;\big|\;a\in\mathbb N\bigr\}$:
I don't have a proof here yet. This is the core of my question.
• $S(4k)\supseteq\bigl\{a\;\big|\;a\in\mathbb N\bigr\}$:
According to Lagrange's four-square theorem, any natural number $a$ can be described as the sum of four squares, i.e. $a=b^2+c^2+d^2+e^2$. For $n=4$ you can e.g. choose $$M=\begin{pmatrix}b&-c&-d&-e\\c&b&e&-d\\d&-e&b&c\\e&d&-c&b\end{pmatrix}$$ For $k>1$ you can let $M$ be a block diagonal matrix with blocks shaped as above.
• $S(4k)\subseteq\bigl\{a\;\big|\;a\in\mathbb N\bigr\}$:
This one is easy: since the elements of $M$ are integers, the squared norm $s$ must be a non-negative integer.
• $S(4k+2)\supseteq\bigl\{a^2+b^2\;\big|\;a,b\in\mathbb N\bigr\}$:
Choose $\displaystyle M=\begin{pmatrix}a&-b\\b&a\end{pmatrix}$ or a block diagonal matrix formed from blocks of this form.

So essentially what I'm asking for is a proof of the second point in the above list, i.e. a proof of $S(2k+1)\subseteq\bigl\{a^2\;\big|\;a\in\mathbb N\bigr\}$. Or, as the title of this question states, I want to know why every lattice cube in odd dimensions must have integer edge lengths.

If you have an idea for an alternate proof which also covers one of the other aspects, I'll welcome alternate views on these as well. But make sure to include an answer for the main question as well, or post a comment instead.

OP of the linked post here. Your proofs are identical to mine. The one you are missing is as follows: A parallelepiped volume can be calculated as the determinant of a matrix for which the rows are the vectors defining the paralelipiped. Thus if all the coordinates are integers, then the determinant must be as well. Thus the volume of any paralelipiped in $\mathbb{R}^n$ for any $n$ must be an integer. If we let the side length of the cube be $\ell=\sqrt{s}$, then $V=\ell^n=s^\frac{n}{2}$ must be an integer which forces $s$ to be a perfect square when $n$ is odd, making $\ell$ an integer.


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February 8, 2012

Higher Algebraic and Geometric Structures: Modern Methods in Representation Theory

Posted by Alexander Hoffnung

This past October Alistair Savage and I organized a workshop on Category Theoretic Methods in Representation Theory at the University of Ottawa. The event was generously supported by the Fields Institute.

Following the success of the October workshop, Oded Yacobi, Chris Dodd, and I decided to hold another workshop, this time with a focus on researchers who are still very early on in their careers. The Fields Institute has again offered funding and this time will host the event as well.

We would like to draw your attention to the upcoming Young Researchers Workshop on Higher Algebraic and Geometric Structures: Modern Methods in Representation Theory to be held May 7-9, 2012 at the Fields Institute in Toronto.

Keep reading below the fold and see the workshop website for more on the content of the workshop, registration information, and applications for financial support.

This three day workshop will bring together young (mainly postdoctoral and graduate student) researchers in representation theory and related fields with a focus on exciting new developments in algebraic and geometric methods.

The meeting will incorporate both a graduate student workshop and a research level conference. The workshop component will consist of a series of three lectures by Ben Webster (Northeastern) explaining S-duality. In addition to these lectures, there will be research talks given by the leading young researchers in representation theory and related fields.

The goals of this workshop are:

• to highlight algebraic-geometric and categorical methods in representation theory which have made important contributions in recent years,
• to learn about the program on S-duality initiated by Braden, Licata, Proudfoot, and Webster,
• to provide an opportunity for young researchers in the field to learn of and present cutting-edge research, and to foster interaction between Canadian and American researchers working in this area.

There will be approximately 12 talks, each 50 minutes in length, spread out over Monday, Tuesday and Wednesday, with ample time in between talks for discussion. Three of these talks will be reserved for Ben Webster’s lectures.

Here is the current list of speakers:

• Ben Cooper (University of Virginia)
• Ben Elias (MIT)
• David Jordan (UT Austin)
• Carl Mautner (Harvard)
• Weiwei Pan (Göttingen) *
• Jaimie Thind (University of Toronto)
• Peter Tingley (MIT) *
• Ben Webster (Northeastern)
• Xinwen Zhu (Harvard) *

(* tentative)

We will be able to provide limited support for some graduate student and postdoc participants. Registration and applications for support will be open soon at the website. Please don’t hesitate to contact one of us if you have any questions.

We hope to see you there!

Posted at February 8, 2012 7:28 PM UTC

TrackBack URL for this Entry:   http://golem.ph.utexas.edu/cgi-bin/MT-3.0/dxy-tb.fcgi/2493

Re: Young Researchers Workshop on Higher Algebraic and Geometric Structures: Modern Methods in Representation Theory

to learn about the program on S-duality initiated by Braden, Licata, Proudfoot, and Webster,

I have only briefly looked at the slides

Ben Webster, Representation theory and a strange duality for symplectic varieties, 2010 (pdf)

On slide 80 it says

My collaborators originally dubbed this “symplectic duality” but it seems that for talking with physicists in the audience, the name should be shortened to “S-duality.”

[…]

Our examples coincide with a notion of duality in physics; they are the Higgs branches of mirror dual 3-dimensional gauge theories.

I have trouble locating the reference that says this in more detail. Is this related to

• Tudor Dimofte, S-duality and mirror symmetry in Chern-Simons theory (talk)

• Tudor Dimofte, Sergei Gukov, Chern-Simons Theory and S-duality (arXiv:1106.4550)

?

Posted by: Urs Schreiber on February 9, 2012 9:03 AM | Permalink | Reply to this

Re: Young Researchers Workshop on Higher Algebraic and Geometric Structures: Modern Methods in Representation Theory

I can’t speak to this myself at all at the moment, but hopefully after hearing what Ben has to say, I can come back and connect the dots.

Posted by: Alex Hoffnung on February 12, 2012 7:15 PM | Permalink | Reply to this

Re: Young Researchers Workshop on Higher Algebraic and Geometric Structures: Modern Methods in Representation Theory

But I suppse there is some canonical entry point for

the program on S-duality initiated by Braden, Licata, Proudfoot, and Webster,

?

What are the standard references? I can’t seem to find any on the workshop website. (?)

Posted by: Urs Schreiber on February 12, 2012 9:36 PM | Permalink | Reply to this

Re: Young Researchers Workshop on Higher Algebraic and Geometric Structures: Modern Methods in Representation Theory

Ah, good point. Things should run much more smoothly if the website for the workshop points people towards at least an overview of the subject.

I think a reasonable place to start is the brief overview given in Nick Proudfoot’s research statement, which states some of the proven and conjectural examples of symplectic dualities.

From there it probably makes sense to have a look at the first paper, Gale duality and Koszul duality put out by Braden, Licata, Proudfoot, and Webster as a group. Here, amongst other duality results, they show that Gale dual polarized arrangements have associated symplectic dual hypertoric varieties.

I am sure that hypertoric varieties are not universally familiar to people, so it probably makes sense to list Proudfoot’s A survey of hypertoric geometry and topology to either read first or use as a point of reference while working through the $S$-duality papers.

With Christopher Phan, the four authors put out Localization algebras and deformations of Koszul algebras, which tells us that symplectic dual pairs of algebraic symplectic manifolds have equivariant cohomology rings that are dual as localization algebras.

I don’t know how much this helps right away, but thank you for pointing out the need for some information in this direction on the website. I will take a closer look at the papers just mentioned, the most recent paper Hypertoric category O, and at least some of the host of papers relevant to these works and to the conjectural statements within. I’ll give a suggested reading list on the workshop website, and hopefully some guidance on navigating the papers.

Posted by: Alex Hoffnung on February 12, 2012 11:51 PM | Permalink | Reply to this

Re: Young Researchers Workshop on Higher Algebraic and Geometric Structures: Modern Methods in Representation Theory

Urs-

We’re working on the whole standard reference thing (or having a reference period). It’s pretty hard to write a paper on a subject when all you have is conjectures, so we’ve been marshaling our evidence over the past few years in order to really have something to say when we lay out the general picture. We now have drafts of two 60 page long papers on the subject, so it’s a little unfortunate that there’s still nothing for the public to read, but the big picture bits are (as usual) a bit hostage to the niggling details that need to be worked out before anything as rash as posting on the arXiv.

I think in terms of understanding the big picture, Nick’s research statement is a good start. I’ll work on putting together a better reading list for the conference website.

Posted by: Ben Webster on February 17, 2012 5:27 AM | Permalink | Reply to this

Re: Young Researchers Workshop on Higher Algebraic and Geometric Structures: Modern Methods in Representation Theory

From Proudfoot’s research statement

The meaning of the relationship between symplectic duality in mathematics and mirror duality in physics remains mysterious; understanding it is one of my main goals in the coming years.

Mirror symmetry, according the the nLab entry,

At least in some cases … can be understood as a special case of T-duality.

Above Urs mentions the link to S-duality, and work connecting S-duality and mirror symmetry.

Is there expected to be some big story which accounts for these four dualities and their relationships?

Posted by: David Corfield on February 17, 2012 8:28 AM | Permalink | Reply to this

Re: Young Researchers Workshop on Higher Algebraic and Geometric Structures: Modern Methods in Representation Theory

Thanks Alex, thanks Ben! That helps.

I did see the articles with “Koszul” in the title on the arXiv, but couldn’t tell from their abstracts (and had no time for more than that, alas) whether they are relevant here.

Thanks to the pointers to Nick Proudfoot’s research statement. That helps, too.

Posted by: Urs Schreiber on February 17, 2012 2:34 PM | Permalink | Reply to this

Re: Young Researchers Workshop on Higher Algebraic and Geometric Structures: Modern Methods in Representation Theory

I started writing a page on the nLab to serve as a reference about “symplectic duality.” It’s still a little messy at the moment, but I think it is human readable, and I’d be happy to receive any comments, enquiries or emendations.

Posted by: Ben Webster on February 17, 2012 9:41 PM | Permalink | Reply to this

Re: Young Researchers Workshop on Higher Algebraic and Geometric Structures: Modern Methods in Representation Theory

I started writing a page on the nLab to serve as a reference about “symplectic duality.”

Thanks, Ben! That’s great. Most useful.

I have gone through the page and done some minor editing. Mostly I have added further hyperlinks and fixed instki syntax here and there (for instance, unfortunately instiki cannot handle math in hyperlinks). Also added a “floating context TOC” in the top right (“symplectic geometry”) and a “References”-section at the very end (which you may, in turn, want to edit).

The analog of what in Wikipedia would be the corresponding “talk page” is on the $n$Forum here.

If you find the time to look at the $n$Forum at all, more important right now would maybe be your attention to the discussion here under “Koszul”. As you’ll see there, I slightly intervened with your creation of an entry titled “Koszul”, because we try to stick to the convention that entries are titled by singular nouns. But even apart from this we eventually need to harmonize a bit with existing Koszul-entries such as Koszul duality. Most of these are somewhat neglected and would deserve some attention.

Posted by: Urs Schreiber on February 18, 2012 1:47 PM | Permalink | Reply to this

Post a New Comment


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# Science:Infinite Series Module/Units/Unit 1/1.1 Infinite Sequences/1.1.09 Squeeze Theorem For Sequences

The Squeeze Theorem for functions can also be adapted for infinite sequences.

Theorem: Squeeze Theorem for Infinite Sequences
Suppose

${\displaystyle a_{n}\leq b_{n}\leq c_{n}}$

for

${\displaystyle n\geq n_{0}}$

and

${\displaystyle \lim _{n\rightarrow \infty }a_{n}=\lim _{n\rightarrow \infty }c_{n}=L,}$

then

${\displaystyle \lim _{n\rightarrow \infty }b_{n}=L.}$

This theorem allows us to evaluate limits that are hard to evaluate, by establishing a relationship to other limits that we can easily evaluate. Let's see this in an example.


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# Relation between linear recurrence and linear equations that produce the same number sequence

The linear recurrence is \begin{align} v_1^{(k+1)}&:=-\frac{1}{2}v_1^{(k)}-\frac{1}{6}v_2^{(k)}\cdots-\frac{1}{n(n-1)}v_n^{(k)}\\ v_2^{(k+1)}&:=+\frac{1}{1}v_1^{(k)}\\ v_3^{(k+1)}&:=+\frac{1}{2}v_2^{(k)}\\ \vdots \\v_{n}^{(k+1)}&:=+\frac{1}{n-1}v_{n-1}^{(k)}\\ \end{align}

Consider the following sequence $$\begin{array} &&v_1^{(0)}=1,&v_2^{(0)}=0,&v_3^{(0)}=0,&v_4^{(0)}=0,&v_5^{(0)}=0,&\cdots,&v_n^{(0)}=0\\ &v_1^{(1)}=-\frac{1}{2},&v_2^{(1)}=1,&v_3^{(1)}=0,&v_4^{(1)}=0,&v_5^{(1)}=0,&\cdots,&v_n^{(1)}=0\\ &v_1^{(2)}=\frac{1}{12},&v_2^{(2)}=-\frac{1}{2},&v_3^{(2)}=\frac{1}{2},&v_4^{(2)}=0,&v_5^{(2)}=0,&\cdots,&v_n^{(2)}=0\\ &v_1^{(3)}=0,&v_2^{(3)}=\frac{1}{12},&v_3^{(3)}=-\frac{1}{4},&v_4^{(3)}=\frac{1}{6},&v_5^{(3)}=0,&\cdots,&v_n^{(3)}=0\\ &v_1^{(4)}=-\frac{1}{720},&v_2^{(4)}=0,&v_3^{(4)}=\frac{1}{24},&v_4^{(4)}=-\frac{1}{12},&v_5^{(4)}=\frac{1}{24},&\cdots,&v_n^{(4)}=0\\ \end{array}$$

We see that $k!v_i^{(k)}$ are the coefficients of the bernoulli polynomials. These coefficients are realated to the Bernoulli Numbers by $$B_n(x) = \sum_{k=0}^n{n\choose k}B_kx^{n-k}$$ see wikipedia.

The system of linear equations is $$\begin{array}~ {2\choose 0}B_1&+&{2\choose 1}B_2&&&&&&&=0\\ {3\choose 0}B_1&+&{3\choose 1}B_2&+&{3\choose 2}B_3&&&&&=0\\ {4\choose 0}B_1&+&{4\choose 1}B_2&+&{4\choose 2}B_3&+&{4\choose 3}B_4&&&=0\\ {5\choose 0}B_1&+&{5\choose 1}B_2&+&{5\choose 2}B_3&+&{5\choose 3}B_4&+&{5\choose 4}B_5&=0\\ \end{array}$$ If $B_1:=1,B_2:=-\frac{1}{2},B_3:=\frac{1}{6},B_4:=0$ and $B_5:=-\frac{1}{30}$ (notice eq. $(34)$ on this page) the system is solved.

Is there some specific realation between the recurrence and the linear system, other than that they both produce this number sequence?

Suppose there are $v_1,v_2,\cdots,v_n$ and an $x$ with
\begin{align} xv_1&=-\frac{1}{2}v_1-\frac{1}{6}v_2\cdots-\frac{1}{n(n-1)}v_n\\ xv_2&=+\frac{1}{1}v_1\\ xv_3&=+\frac{1}{2}v_2\\ \vdots\\ xv_{n}&=+\frac{1}{n-1}v_{n-1}\\ \end{align} If $v_1=1$ and $N:=n+2$ then $$p_N(x):=\sum_{k=1}^{N-1}\frac{N!}{(N-k)!}x^{k-1}=0$$ solves the system. These polynomials also appear here.
Then using $$q_N(x) = \frac{x^{N-1}}{N!} p_N(1/x)$$ where $q'_N(x)$ is a partial sum of the power series for $e^x$, and $q_N(x)$ is a partial sum of the power series for $e^x-1$ (see here).
Apply the General Leibniz rule $$(fg)^{(n)}(x)=\sum_{k=0}^n {n \choose k} f^{(n-k)}(x) g^{(k)}(x)$$ to $$-xq'_N(x)f(x)=q_N(x)$$ to find out the series expansion at $0$ of $$-x \frac{q'_N(x)}{q_N(x)}=\rightarrow \frac{-x}{1-e^{-x}}.$$ This leads to a linear system as stated in the question.


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# 2.2.4 Functions

###### 2.2.4  Functions

Notation to indicate that a function f maps elements of a set A to elements of a set B is:

[2.4]

A is the function’s domain; B contains its range. We are primarily interested in three types of functions:

• functions from to ,
• functions from n to ,
• functions from n to m.

We call functions of the first form real—they map real numbers to real numbers. The natural logarithm function is a real function, which we denote log. We do not employ the logarithm base 10. If a function f has an inverse, we denote this f –1. The derivative of a real function f may be indicated with differential notation or simply as f ′.2 We indicate the value of a function f at a particular point a as either f(a) or |a. The former is read as “f of a“. The latter is read as “f evaluated at a“.

Consider : pm and : np. The composition of f and g is the function fg from n to m defined as

[2.5]

The gradient f and Hessian 2f of a function f : n are the vector of its first partial derivatives and matrix of its second partial derivatives:

[2.6]

The Hessian is symmetric if the second partials are continuous.

The Jacobian of a function f : nm is the matrix of its first partial derivatives.

[2.7]

Note that the Hessian of a function f : n is the Jacobian of its gradient.


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# Combining independent probabilities of an event

Say $d+1$ independent, equally reliable experts give you probabilities $p_0,\ldots,p_d$ of an event A occurring. What should you think the probability of $A$ is?

I'll give my solution (Edit: I retract my claim to having a solution. At the very least this is not, in general, normalized correctly. There are also other serious issues as pointed out in the comments and answers): \begin{align} &P(A \mid p_0,\ldots,p_d) \\ &= \frac{P(p_0,\ldots,p_d \mid A) P(A)}{P(p_0,\ldots,p_d)}\\ &= P(A) \prod_{l=0}^{d}\frac{P(p_l \mid A)}{P(p_l)}\\ &= P(A) \prod_{l=0}^{d}\frac{P(A \mid p_l)}{P(A)}\\ &= \frac{1}{P(A)^d} \prod_{l=0}^{d} P(A \mid p_l)\\ &= \frac{1}{P(A)^d} \prod_{l=0}^{d} p_l\\ \end{align}

I think this is reasonable (let me know if you disagree) but I have a number of questions about it:

1. I have couched this in an explicity Bayesian manner and I'm not sure if this is necessary. Can you instead talk about some estimates of $p_l$?

2. Is line 3 ok? I'm dividing by zero... Can this be fixed by considering some density function?

3. Why do we divide by $P(A)^d$ (and not to the power $d+1$ say)? It makes sense to me to multiply the probabilities together, I guess this is just normalizing?

4. Does $P(A \mid p_l)=p_l$? It certainly seems like it should.

-
Could you justify $$P(p_0, p_1, \ldots, p_d) = \prod_{i=0}^{d} P(p_i)$$ a little? In many cases, conditionally independent events (e.g. the justification for $$P(p_0, p_1, \ldots, p_d|A) = \prod_{i=0}^{d} P(p_i|A)$$ are not necessarily independent in the absence of conditioning. –  Dilip Sarwate Mar 21 '12 at 20:35
I don't think your combined estimate is justifiable. If even a single expert predicts $0$ then their estimate overwhelms the effect of all the other estimates! –  Qiaochu Yuan Mar 21 '12 at 21:39
Thanks for your answers and comments everyone. I had a feeling this might be controversial and you haven't disappointed. I retract my claim to having a solution. –  Martin Leslie Mar 22 '12 at 0:05

About 2., we can think of a discrete case for a start and thus avoid the complications of conditional probabilities in the continuous case.

The main problem lies in 4. There's no basis for this assumption; you've killed the entire complexity of the problem by making that assumption. $P(A\mid p_l)$ depends on how reliable you think the experts are; $P(A\mid p_l)=p_l$ expresses that you believe that expert $l$ is always right. In the other extreme, if you think the "experts" are all completely useless, you'd have $P(A\mid p_l)=P(A)$. The conditional probability distribution you get with your assumption will generally not be normalized unless the experts are in fact always right.

-

To expand a little on the (very good) point that the user joriki gives in his answer, one potential problem is that you are conflating the event $E_{i} = (\textrm{Expert i was correct})$ with the event $G_{i} = (\textrm{Expert i gave estimate } p_{i})$. Only in the former case can you assume that $P(A|E_{i}) = p_{i}$. But just because expert $i$ said that the probability is $p_{i}$, it doesn't mean that's correct. Maybe you know that expert $i$ always forgets to carry the two when doing long division or something.

If we assume that the expert guess $p$ is a random variable with some 'expert' distribution $P(p)$, then under your assumption, another way to put it is like this: $$P_{true}(A) = \sum_{i=0}^{d}p(A|p_{i})\cdot{}P(p_{i}) = \sum_{i=0}^{d}p_{i}\cdot{}P(p=p_{i}) = E_{\textrm{exp}}[p].$$

Basically, this means that your given simplifications imply that you have to make the assumption that the experts are an unbiased estimator of the true probability, that is, the expectation under the distribution over expert opinion equaling the true probability is a consequence of your derivation but it may not be true about independent experts out in the world.

The formula you give has weird (not necessarily bad) properties for certain test cases. For example, consider a case where your prior belief is $P(A)=1/2$ and there are two experts, (so $d=1$) who give probabilities $p_{0}$ and $p_{1} = \frac{n-1}{n}$, respectively, for some large value of $n$. Then

$$P(A|p_{0},p_{1}) = \frac{1}{1/2}\cdot{}p_{0}\cdot{}\frac{n-1}{n} \to 2\cdot{}p_{0}.$$ So, as expert 2 becomes more and more certain, the probability converges to twice the guess of expert 1. Clearly, this doesn't make sense for $p_{0} > 1/2$, and it might make even less sense subjectively if we value the estimates of the two experts equally.

I think part of the problem is assuming that $P(A|p_{i})=p_{i}$, because it means that if expert $i$ makes guess $p_{i}$, then all the other guess don't matter. But this can't be true for all of the $i$ experts simultaneously...

-


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# Periodic Functions - Applications

• December 6th 2012, 03:40 PM
sakonpure6
Periodic Functions - Applications
I have the following equation $y= -4sin15x+25$ ( y represents the temperature and x represents time in hours, so x=0 means midnight and x=1 means 1:00 am) and I am given a question with the graph of the given equation, and to get the solution to the question you can you use the graph (which is also what my teacher told us to do) and so i do use the graph and get the correct answer of 13 hours. My question is how would i use the equation to solve for what i need?

The question:

The pool is comfortable for swimming at 23 degrees Celsius. The pool is open from 6:00 am to 11:00 pm. How many hours of comfortable swimming are available on a sunny day?

Any help is appreciated and thank you in advance!
• December 6th 2012, 04:17 PM
chiro
Re: Periodic Functions - Applications
Hey sakonpure6.

For this problem you will need to say what values of y can be taken.

You mention 23 degrees but are you allowed to have a range of values (like less than 23 degrees or greater than 23 degrees)?

Once you have this you solve your inequality.
• December 6th 2012, 06:26 PM
sakonpure6
Re: Periodic Functions - Applications
Quote:

Originally Posted by chiro
Hey sakonpure6.

For this problem you will need to say what values of y can be taken.

You mention 23 degrees but are you allowed to have a range of values (like less than 23 degrees or greater than 23 degrees)?

Once you have this you solve your inequality.

no y must be exactly 23 degrees
• December 6th 2012, 07:39 PM
chiro
Re: Periodic Functions - Applications
Well can you set y to 23 and solve for x?
• December 7th 2012, 06:43 AM
sakonpure6
Re: Periodic Functions - Applications
Oh sorry y must be 23 degrees untill the peak shich is what i believe to be 29
• December 7th 2012, 01:15 PM
chiro
Re: Periodic Functions - Applications
Can you setup an inequality where you solve for y = 29 and y = 23 to get the values inbetween?
• December 7th 2012, 05:13 PM
sakonpure6
Re: Periodic Functions - Applications
Quote:

Originally Posted by chiro
Can you setup an inequality where you solve for y = 29 and y = 23 to get the values inbetween?

I don't know what "inequality" is, do you mind if you show me the equations?
• December 7th 2012, 05:43 PM
HallsofIvy
Re: Periodic Functions - Applications
Well, the point of an "inequality is that it isn't an equation!

If what you have told us is correct, you want $23\le -14 sin(15x)+ 25\le 29$. And the simplest way to solve that is to solve the equations -14 sin(15x)+ 25= 23 and -14sin(15x)+ 25= 29. The correct range for x will be between those values.
• December 7th 2012, 06:10 PM
sakonpure6
Re: Periodic Functions - Applications
Quote:

Originally Posted by HallsofIvy
Well, the point of an "inequality is that it isn't an equation!

If what you have told us is correct, you want $23\le -14 sin(15x)+ 25\le 29$. And the simplest way to solve that is to solve the equations -14 sin(15x)+ 25= 23 and -14sin(15x)+ 25= 29. The correct range for x will be between those values.

Oh okay thank you! Ill do them and compare my answers!


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Home » Physics » What is the Dimensional Formula of Work and its derivation?

# What is the Dimensional Formula of Work and its derivation?

In this article, we’ll talk about the Dimensional Formula of Work. Here, we will look at the dimension of work and how to find the work dimensional formula.

When a force displaces an object, work is said to be done in science. The work done by a force is defined as the product of the force’s component in the direction of displacement and the magnitude of displacement.

Work can be computed by multiplying Force by displacement that occurs in the direction of the applied force. Mathematically work done is given by the equation,

$W=\vec F \cdot \vec d$

here both force and displacement are vector quantities and work done being the scalar product of two vectors is a scalar quantity.

## Dimensional Formula of Work

The Dimensional formula for Work done is

$[M^1L^2T^{-2}]$

Where
M -> Mass

L -> Length

T -> Time
We would now derive this dimensional formula.

### dimension of work done derivation

Work is defined as the cross product for Force  and displacement
$W= F.d$
Where d -> displacement
F -> Force applied

W-> Work done by the Force

So,

Now the dimension of displacement= $[L^1]$

Let’s derive the dimension of Force

$F= ma$
Now

Where $m\rightarrow$ mass

$a$ -> Acceleration

Dimension of Mass = $[M^1]$

Now acceleration

$a = \frac {\Delta v}{t}$

Now dimension of Velocity= $[M^0 L^1T^{-1}]$

dimension of Time = $[M^0 T^1]$

So dimension of Acceleration = $\frac {[M^0 L^1T^{-1}]}{ [M^0 T^1]}= [M^0 L^1T^{-2}]$

So, Dimension of force is given by

$\text {Dimension of Force} =[M^1] \times [M^0 L^1T^{-2}] = [M^1L^1T^{-2}]$

Now we know both the displacement and Force dimension , we can calculate the dimension of Work easily as

$\text {dimension of Work} = \text {dimension of Force} \times \text {dimension of displacement}$

$= [M^1L^1T^{-2}] \times [L^1] = [M^1L^2T^{-2}]$

Unit of Work Done is Joule.

Try the free Quiz given below to check your knowledge of Dimension Analysis:-

#### Quiz on Dimensional Analysis

1. A dimensionless quantity

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2. The dimensions of impulse are equal to that of

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3. The dimensions of universal gravitational constant are

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4. A unitless quantity

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5. Choose the correct statement(s)

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# perovskite solar cell

Such a design is common for current perovskite or dye-sensitized solar cells. g PSC can be fabricated by low-cost solution processed methods both on glass and flexible substrates. [109][110] It has recently been reported that charges in the perovskite material are predominantly present as free electrons and holes, rather than as bound excitons, since the exciton binding energy is low enough to enable charge separation at room temperature.[89][90]. This change helped to improve the environmental and thermal stability of the perovskite cell[170] and was crucial to further improve the perovskite/silicon tandem performance to 23.6%. v [173] In March 2020, KAUST-University of Toronto teams reported tandem devices with spin-casted perovskite films on fully textured textured bottom cells with 25.7% in Science Magazine. The perovskite solar cells use methyl ammonium lead iodide (MAPbI3), in its manufacturing process. Perovskite solar cells are therefore the fastest-advancing solar technology as of 2016[update]. This chapter discusses the various reported standard fabrication techniques for the deposition of photoanode, perovskite layer, electron/hole transport layers, counter electrode, and factors affecting the stability and life time of PSCs. Normally solar panels are encapsulated very well to avoid the environment to interact. This exceeds the 26.7% efficiency world record for a single-junction silicon solar cell. c Tin perovskite solar cells (TPSCs) are rising as the most promising candidates for lead-free PSCs. This method produces uniform films of up to 1 mm grain size. However, these prin-ciples are not as obvious for photovolt-ageasforphotocurrent,andthephoto-voltageispreciselyoneofthemost fascinating properties of PSCs due to the high open-circuit voltage,Voc,ob-tained with this technology. The transparent conducting SWCNT electrode-based planar heterojunction perovskite solar cell achieved a PCE of 6.32%, which is 70% of an ITO-based device (9.05%). High-efficiency solar cell fabrication with device performance and stability te… Third generation of photovoltaic (PV) cells has come up with the technologies like dye-sensitized solar cells, PSCs, organic PV, and quantum dot PVs. P. Mathan Kumar, ... Ranjith G. Nair, in Perovskite Photovoltaics, 2018. [99] suggests a modeling framework,[100] presents analysis of near ideal efficiency, and [101] talks about the importance of interface of perovskite and hole/electron transport layers. [112] In two-step deposition method, the volume expansion during the conversion of lead halide to perovskite can fill any pinholes to realize a better film quality. [24], Perovskite solar cells hold an advantage over traditional silicon solar cells in the simplicity of their processing and their tolerance to internal defects. Further experiments in replacing the mesoporous TiO2 with Al2O3 resulted in increased open-circuit voltage and a relative improvement in efficiency of 3–5% more than those with TiO2 scaffolds. Perovskite application in solar cells can help in improving efficiency, flexibility, and cost cutting. Now he is a M.S. [116][117][118] Burschka et al. [21][22], Rice University scientists have discovered a novel phenomenon of light-induced lattice expansion in perovskite materials. [139][140] Besides, the two instabilities issues can be solved by using multifunctional fluorinated photopolymer coatings that confer luminescent and easy-cleaning features on the front side of the devices, while concurrently forming a strongly hydrophobic barrier toward environmental moisture on the back contact side. Values of the detailed balance limit are available in tabulated form[93] and a MATLAB program for implementing the detailed balance model has been written. [171], In the continuity, using a p-i-n perovskite top cell, Sahli et al. Under these conditions, only thermal stress was found to be the major factor contributing to the loss of operational stability in encapsulated devices. [144], Early in 2019, the longest stability test reported to date showed a steady power output during at least 4000 h of continuous operation at Maximum power point tracking (MPPT) under 1 sun illumination from a xenon lamp based solar simulator without UV light filtering. [172] This improved efficiency can largely be attributed to the massively reduced reflection losses (below 2% in the range 360 nm-1000 nm, excluding metallization) and reduced parasitic absorption losses, leading to certified short-circuit currents of 19.5 mA/cm2. Inverted Perovskite Solar Cell with 21.1% Efficiency and High Moisture Resistance 13 Jan 2021 by EMILIANO BELLINI Scientists at the University of Queensland in Australia have fabricated a solar cell based on a mixture of 2D and 3D salts. The main advantages of applying CNTs in perovskite devices include cheap fabrication processes and materials, and the potential of two-sided illumination application. Both methods have been demonstrated to yield lower efficiency values when compared to efficiencies determined by fast IV-scans. PSCs (e.g. [74], Research done on Tin halide-based PSCs show that they have a lower power conversion efficiency (PCE), with those fabricated experimentally achieving a PCE of 9.6%. [166] To achieve this results, the team used Zr-doped In2O3 transparent electrodes on semitransparent perovskite top cells, which was previously introduced by Aydin et al.,[163] and improved the near infrared response of the silicon bottom cells by utilizing broadband transparent H-doped In2O3 electrodes. Australian scientists have built a perovskite solar cell based on 2D and 3D salts. t [94] There are two prerequisites for predicting and approaching the perovskite efficiency limit. For example, some physical approaches are developed to promote supersaturation through rapid solvent removal, thus getting more nucleations and reducing grain growth time and solute migration. x Minu Mohan, in Perovskite Photovoltaics, 2018. Subsequent evaporation and convective self-assembly during spinning results in dense layers of well crystallized perovskite material, due to the strong ionic interactions within the material (The organic component also contributes to a lower crystallization temperature). [135][136] However, no long term studies and comprehensive encapsulation techniques have yet been demonstrated for perovskite solar cells. By having multiple bandgaps in a single cell, it prevents the loss of photons above or below the band gap energy of a single junction solar cell. Inorganic-organic based hybrid perovskite solar cells most commonly comprised of CH3NH3PbI3 materials with an appropriate band gap (1.55 eV), high absorption coefficient, long hole-electron diffusion length (~ 100 nm), and excellent carrier transport. Metal halide perovskites possess unique features that make them useful for solar cell applications. [157] obtained a 17% and 18.6% efficient tandem cell with mc-Si (η ~ 11%) and copper indium gallium selenide (CIGS, η ~ 17%) bottom cells, respectively. Australian scientists have built a perovskite solar cell based on 2D and 3D salts. The development of solid-state perovskite solar cells has shown a remarkable improvement over the past 4 years. In ZnO-based devices, ITO substrates consume more energy than FTO substrates. Also, the team enhanced the electron-diffusion length (up to 2.3 µm) thanks to Lewis base passivation via urea. The PSC realized the breakthrough efficiency in late 2012 with the introduction of solid state hole-transporting layers within the solar cell, which resulted in stable efficiencies close to 10% (Sahu, 2015). demonstrated performance enhancement by using hydrothermally grown 3% Er3+ and 6% Yb3+ co-doped TiO2 nanorod arrays as electron transfer material in PSCs as compared to those based on undoped TiO2 [92]. After light absorption and the subsequent charge-generation, both negative and positive charge carrier are transported through the perovskite to charge selective contacts. The main source of lead comes from human activities, including burning of fossil fuels, mining and manufacturing of consumer goods such as (secondary) batteries, ammunition, soldering, piping (cf. Although the efficiency values reached for small-size individual solar cells can be considered to be satisfactory, and should basically provide a great commercialization potential for this technology in the future, especially thanks to the fact that we have a true solid-state device, the stability and longevity of the device is yet to be ascertained. Attempts have been made for many years to find an alternative to the liquid electrolytes, and thus to obtain an improved DSSC solar cell which will have ease of fabrication, less complication in the sealing, and encapsulation of the device, the possibility for monolithic interconnection of the cells within the module, and therefore also increased performance and lower cost (Goswami and Kreith, 2007). In 2014, Olga Malinkiewicz presented her inkjet printing manufacturing process for perovskite sheets in Boston (US) during the MRS fall meeting – for which she received MIT Technology review's innovators under 35 award. A flexible perovskite solar cell. In one-step deposition, a perovskite precursor solution that is prepared by mixing lead halide and organic halide together, is directly deposited through various coating methods, such as spin coating, spraying, blade coating, and slot-die coating, to form perovskite film. One such factor is the use of gold as a back electrode. Perovskite solar cells are a novel type of device that was first fabricated in 2009 (174) and several structures have been reported (175). For years the world has been using age-old fossil fuels for most of our energy. Lead (Pb) is a natural metal found in many compounds in small quantities (50–400 ppm). By continuing you agree to the use of cookies. But it appears that determining the solar cell efficiency from IV-curves risks producing inflated values if the scanning parameters exceed the time-scale which the perovskite system requires in order to reach an electronic steady-state. The electrical energy requirement for MA iodide is high. [176] Also, Aydin et al. [152], A perovskite cell combined with bottom cell such as Si or copper indium gallium selenide (CIGS) as a tandem design can suppress individual cell bottlenecks and take advantage of the complementary characteristics to enhance the efficiency. Same methods also apply to SnO2 deposition. Most notably, methylammonium and formamidinium lead trihalides, also known as hybrid perovskites, have been created using a variety of solution deposition techniques, such as spin coating, slot-die coating, blade coating, spray coating, inkjet printing, screen printing, electrodeposition, and vapor deposition techniques, all of which have the potential to be scaled up with relative ease except spin coating. [27][28][29][30], The solution-based processing method can be classified into one-step solution deposition and two-step solution deposition. The front electrodes like FTO and ITO also contribute towards the total energy consumption. Currently, several researches are going on to improve the PCE of PSC to the Shockley–Queisser limit. Solar cell efficiencies of devices using these materials have increased from 3.8% in 2009[4] to 25.5% in 2020 in single-junction architectures,[5] and, in silicon-based tandem cells, to 29.1%,[5] exceeding the maximum efficiency achieved in single-junction silicon solar cells. By adding a fluorinated lead salt in the processing solution – normally used to form 3D methylammonium lead iodide – they were able to achieve a 21.1% efficiency, an open-circuit voltage of 1.12 V, a short-circuit current of 22.4 mA/cm 2, and a fill factor of 84%. TCFs made from CNTs have been used in perovskite solar cells (176,177) to replace the expensive metal electrode (usually Au) in the fabrication of perovskite solar cells (178). [36], Pb halide perovskites can be fabricated from a PbI2 precursor,[37] or non-PbI2 precursors, such as PbCl2, Pb(Ac)2, and Pb(SCN)2, giving films different properties.[38]. Perovskite solar cell (PSC) showed the progress in achieving power conversion efficiency from 0 to beyond 20% in recent years. [5], In 2018, a new record was set by researchers at the Chinese Academy of Sciences with a certified efficiency of 23.3%.[5]. ( improved upon this in 2011, using the same dye-sensitized concept, achieving 6.5% PCE. This paper also discusses in detail the issue for scaling up the PSC. [5], In December 2015, a new record efficiency of 21.0% was achieved by researchers at EPFL. [146], Another major challenge for perovskite solar cells is the observation that current-voltage scans yield ambiguous efficiency values. Solar cell efficiency is limited by the Shockley-Queisser limit. The reaction takes time to complete but it can be facilitated by adding Lewis-bases or partial organic halide into lead halide precursors. The merits and demerits of different solar cell technologies are listed in Table 1.1. This realisation was then closely followed by a demonstration that the perovskite itself could also transport holes, as well as electrons. Perovskite solar cell bandgaps are tunable and can be optimised for the solar spectrum by altering the halide content in the film (i.e., by mixing I and Br). While these double-perovskites have a favorable bandgap of approximately 2 eV and exhibit good stability, several issues including high electron/hole effective masses and the presence of indirect bandgaps result in lowered carrier mobility and charge transport. Sarat Kumar Sahoo, ... Narendiran Sivakumar, in Perovskite Photovoltaics, 2018. Insertion of mechanically reinforcing scaffolds directly into the active layers of perovskite solar cells resulted in the compound solar cell formed exhibiting a 30-fold increase in fracture resistance, repositioning the fracture properties of perovskite solar cells into the same domain as conventional c-Si, CIGS and CdTe solar cells. [11][85] [6] Their high absorption coefficient enables ultrathin films of around 500 nm to absorb the complete visible solar spectrum. In a short span of 5 years, efficiency has reached 22.1%, which could be an energy revolution in the future. The new proposed solution to this issue integrates water-splitting (WS) hydrophobic layers to the perovskite absorber of a standard perovskite cell. The maximum accepted levels of lead in drinking water and air were set to 15 and 0.15 μg/L, respectively, by the US EPA (Paulose, 2014; USA Environmental Protection Agency, n.d.). Advances in these solar cell materials offer high efficiency at low cost. also showed that it was possible to fabricate perovskite solar cells in the typical 'organic solar cell' architecture, an 'inverted' configuration with the hole transporter below and the electron collector above the perovskite planar film. v Rivaling the double, triple, and quadruple junction solar cells mentioned above, are all-perovskite tandem cells with a max PCE of 31.9%, all-perovskite triple-junction cell reaching 33.1%, and the perovskite-Si triple-junction cell, reaching an efficiency of 35.3%. One-step deposition is simple, fast, and inexpensive but it’s also more challenging to control the perovskite film uniformity and quality. photovoltage of perovskite solar cells (PSCs)arenotasclearasthosedeﬁning the photocurrent. [42] The University of Toronto also claims to have developed a low-cost Inkjet solar cell in which the perovskite raw materials are blended into a Nanosolar ‘ink’ which can be applied by an inkjet printer onto glass, plastic or other substrate materials. However, simple spin-coating does not yield homogenous layers, instead requiring the addition of other chemicals such as GBL, DMSO, and toluene drips. In all cases, the CNT network is part of the hole collection side of the cell. student at Central South University under the supervision of Prof. Fangyang Liu. [135] The degradation which is caused by moisture can be reduced by optimizing the constituent materials, the architecture of the cell, the interfaces and the environment conditions during the fabrication steps. Perovskite solar cells are attracting attention as the next-generation solar battery material thanks to their low processing cost and excellent photovoltaic quality. [141] The front coating can prevent the UV light of the whole incident solar spectrum from negatively interacting with the PSC stack by converting it into visible light, and the back layer can prevent water from permeation within the solar cell stack. Aruna Ivaturi, Hari Upadhyaya, in A Comprehensive Guide to Solar Energy Systems, 2018. Since July 2018, he has been working in Liming Ding Group at National Center for Nanoscience and Technology as a visiting student. f The all-inorganic CsPbBr3 perovskite material has been proven to have excellent stability. s Additionally, vapor deposited techniques result in less thickness variation than simple solution processed layers. Generally, PSC have TCO/metal oxide/perovskite/HTM/metal architecture. Then, instead of heating, the substrate is bathed in diethyl ether, a second solvent that selectively grabs the NMP solvent and whisks it away. v Perovskite solar cells were also found to have the shortest EPBT compared to current photovoltaic technologies. [168] Werner et al. Furthermore, the BLL of 5 μg/dL was revoked in 2010 after the discovery of decreased intelligence and behavioral difficulties in children exposed to even lower values. Perovskite solar cell has emerged as one of the most standout cell in terms of efficiency. integrated LiYF4:Yb3+,Er3+ single crystal in front of PSC and demonstrated efficiency enhancement [90]. [134], The water-solubility of the organic constituent of the absorber material make devices highly prone to rapid degradation in moist environments. This relatively low PCE is in part due to the oxidation of Sn2+ to Sn4+, which will act as a p-type dopant in the structure and result in higher dark carrier concentration and increased carrier recombination rates. There is an ongoing search for moisture stability in perovskite solar cells (PSCs), as protecting the perovskite layer from moisture is key to preventing excess water from forming on the layer itself and affecting overall performance. Additives are widely adopted for efficient, stable, and hysteresis‐free perovskite solar cells and play an important role in various breakthroughs of perovskite solar cells (PSCs). {\displaystyle n=t_{s}\times u(x_{c})\times v(f,x_{g},x_{c})\times m(vx_{g}/x_{c})}, x [107], Perovskite materials have been well known for many years, but the first incorporation into a solar cell was reported by Tsutomu Miyasaka et al. Evidence of charge transfer in these systems shows promise for increasing power conversion efficiency in perovskite solar cells. Perovskite solar cells are thin films of synthetic crystalline made from cheap, abundant chemicals like iodine, carbon, and lead. [120] In November 2014, a device by researchers from KRICT achieved a record with the certification of a non-stabilized efficiency of 20.1%. [178] To show that the output power can be even further enhanced, bifacial structures were studied as well. By the end of this chapter the readers will be able to get a clear idea about the fabrication methods, and stability and life time of PSCs. One big challenge for perovskite solar cells (PSCs) is the aspect of short-term and long-term stability. The primary energy consumption by gold cathode is very high. [79] Experimental results have also shown that, while Antimony and Bismuth halide-based PSCs have good stability, their low carrier mobilities and poor charge transport properties restrict their viability in replacing lead-based perovskites. This was based on a dye-sensitized solar cell architecture, and generated only 3.8% power conversion efficiency (PCE) with a thin layer of perovskite on mesoporous TiO2 as electron-collector. It makes them a cheaper alternative to silicon and TF solar cells. [145], The intrinsic fragility of the perovskite material requires extrinsic reinforcement to shield this crucial layer from mechanical stresses. degree from Central South University in 2017. Jeong et al. One of the main challenges for the scientific community and the PV industry to bring perovskite solar cells (PSCs) to commercial production is the need to improve moisture stability. [121] This module was developed by the Apolo project consortium at CEA laboratories. Perovskite solar cells (PSCs) are photovoltaic (PV) devices containing a light-absorbing layer that has the general formula AMX 3,[1] and a crystal structure similar to the mineral perovskite (CaTiO 3). Second, the contact characteristics of the electrodes need to be carefully engineered to eliminate the charge accumulation and surface recombination at the electrodes. [165] In 2020, KAUST-University of Toronto teams reported 28.2% efficient four terminal perovskite/silicon tandems solar cells. Furthermore, this is an inexpensive approach. [25] Traditional silicon cells require expensive, multi-step processes, conducted at high temperatures (>1000 °C) under high vacuum in special cleanroom facilities. The minimum bandgap is closer to the optimal for a single-junction cell than methylammonium lead trihalide, so it should be capable of higher efficiencies. The thin film solar cell architecture is based on the finding that perovskite materials can also act as highly efficient, ambipolar charge-conductor.[86]. Currently, to make TiO2 layer deposition be compatiable with flexible polymer substrate, low-temperature techniques, such as atomic layer deposition,[51] molecular layer deposition,[52] hydrothermal reaction,[53] and electrodeposition,[54] are developed to deposit compact TiO2 layer in large area. A perovskite solar cell (PSC[1]) is a type of solar cell which includes a perovskite-structured compound, most commonly a hybrid organic-inorganic lead or tin halide-based material, as the light-harvesting active layer. [95][96][97][98] Also there have some efforts to cast light on the device mechanism based on simulations where Agrawal et al. The resulting devices demonstrated excellent stability in terms of power conversion efficiencies during a 180-day aging test in the lab and a real outdoor condition test for more than 3 months. revealed that, the temperature should be considered while calculating the theoretical limits since these devices reaches the temperature of almost 60 °C under real operations. Perovskite solar cells have gained a significant position in the photovoltaic family within this short duration of time. Perovskite solar cells stem from dye-sensitized solar cells but have promising solid state structures as well as rapid efficiency leaps (Fig. Consequently, upconversion has been recently explored as a promising way to harvest of this region of the solar spectrum and transform it into available visible light to enhance the IR response of PSCs. [66] The health and environmental impact of toxic heavy metals has been much debated in the case of CdTe solar cells, whose efficiency became industrially relevant in the 1990s. c DSSC and organic solar cells are also classified as nanostructured solar cells which use lots of nanomaterials in manufacturing. However, in all these studies, the device characteristics under laser irradiation or under higher solar concentrations (to clearly attribute the contributions as only due to upconverison) were not reported. ) The actual band gap for formamidinium (FA) lead trihalide can be tuned as low as 1.48 eV, which is closer to the ideal bandgap energy of 1.34 eV for maximum power-conversion efficiency single junction solar cells, predicted by the Shockley Queisser Limit. Perovskite absorbers for solar cells are combining excellent opto-electronic properties, with potential for low-cost fabrication and a technology that is a viable pathway for producing tandem junction solar cells on silicon bottom cells. ; c A diverse class of layered, crystalline minerals commonly found in nature, researchers are now able to fabricate perovskite solar cells just half a micrometer (0.5 x 10-6m) in length or width with a wide variety of chemical compositions, physical characteristics and performance attributes cheaply using common, inexpensive, “wet chemistry” i.e. [142] However, researchers from EPFL published in June 2017, a work successfully demonstrating large scale perovskite solar modules with no observed degradation over one year (short circuit conditions). All the major impact categories are affected considerably. There are many discussions regarding the toxicity of lead used in perovskite devices, the stability of the device, etc. ScienceDirect ® is a registered trademark of Elsevier B.V. ScienceDirect ® is a registered trademark of Elsevier B.V. URL: https://www.sciencedirect.com/science/article/pii/B9780128129159000083, URL: https://www.sciencedirect.com/science/article/pii/B9780128129159000010, URL: https://www.sciencedirect.com/science/article/pii/B9780128029534000020, URL: https://www.sciencedirect.com/science/article/pii/B9780128114797000130, URL: https://www.sciencedirect.com/science/article/pii/B9780128114797000117, URL: https://www.sciencedirect.com/science/article/pii/B9780323414814000083, URL: https://www.sciencedirect.com/science/article/pii/B9780128129159000149, URL: https://www.sciencedirect.com/science/article/pii/B978012811479700018X, URL: https://www.sciencedirect.com/science/article/pii/B9780128129159000137, URL: https://www.sciencedirect.com/science/article/pii/B9780128029534000032, Fabrication and Life Time of Perovskite Solar Cells, Sarat Kumar Sahoo, ... Narendiran Sivakumar, in, Overview of the PV Industry and Different Technologies, Senthilarasu Sundaram, ... Tapas Kumar Mallick, in, Upconversion and Downconversion Processes for Photovoltaics, A Comprehensive Guide to Solar Energy Systems, An Overview of Hybrid Organic–Inorganic Metal Halide Perovskite Solar Cells, Khagendra P. Bhandari, Randy J. Ellingson, in, Use of Carbon Nanotubes in Third-Generation Solar Cells, Industrial Applications of Carbon Nanotubes, Senthilarasu Sundaram, ... Hari Upadhyaya, in, Commercialization of Large-Scale Perovskite Solar Energy Technology and Scaling-Up Issues, Potential Environmental Impacts From Solar Energy Technologies. spin coating, spray deposition, and thermal evaporation methods. Later in 2017, Wang et al. (2015) have reported the rain effect on perovskite solar cells and their lead emission problems. Perovskite solar cells have found use in powering low-power wireless electronics for the ambient powered internet of things applications [8], The name 'perovskite solar cell' is derived from the ABX3 crystal structure of the absorber materials, which is referred to as perovskite structure and where A and B are cations and X is an anion. [23], In order to overcome the instability issues with lead-based organic perovskite materials in ambient air and reduce the use of lead, perovskite derivatives, such as Cs2SnI6 double perovskite, have also been investigated. There have been some efforts to predict the theoretical limits for these traditional tandem designs using a perovskite cell as top cell on a c-Si[177] or a-Si/c-Si heterojunction bottom cell. However, the efficiency of the DSSCs are very much restricted and perovskite solid state absorbers have the ability to increase the efficiency over 20%. These include but are not limited to bandgap, effective mass, and defect levels for different perovskite materials. / Critical issues are the stability of the organic spiro-OMeTAD layer and that of the perovskite with the Pb-free compounds. LCA studies at this early stage of production will provide useful inputs for an efficient large-scale production with reduced environmental impacts. Solution processing is cheaper. [147] Various causes have been proposed such as ion movement, polarization, ferroelectric effects, filling of trap states,[148] however, the exact origin for the hysteretic behavior is yet to be determined. [176] By adding spiro-OMeTAD (hole transporting substances), the performance of the devices was further improved to 9.9%. Than simple solution processed method, the epoxy-resin encapsulation was able to reduce lead leakage a! The primary energy consumption and ads early stage of production will provide useful inputs for an efficient production. Are reviewed and their functioning mechanism and influence on device performance is described of back electrode important. Thanks to their high absorption coefficient enables ultrathin films of synthetic crystalline made from cheap, chemicals! Photovoltage of perovskite [ 10 ] tin alloy perovskites have also been employed experimentally reduce! The PSCs stability have been proposed: Unger et al was about 9.7 and... Manufacturing processes for these types of solar cells ( PSCs ) is one of the organic layer. Used in perovskite Photovoltaics, 2018 AM1.5G global solar spectra, the mixture is spin on... Overall performance hole collection side of the perovskite material numerically electron blocking layer age-old fuels... And BIPV sectors, due to its stability issues September 2019, a new record efficiency perovskite/silicon... Flexible perovskite solar cells teams show effort to utilize more solution-based scalable techniques on textured cells... Cell has emerged as one of the inorganic-organic perovskite is formed FTO and ITO also contribute towards the total consumption! Used as a new record efficiency for perovskite/silicon tandems solar cells has shown a improvement!, Hari Upadhyaya, in another study in 2016 it has reached %! Or partial organic halide into lead halide precursors 19.8 mA/cm2 on fully textured silicon bottom cells perovskite use. Was about 9.7 % and 20.3 % with a maximum power conversion efficiency from 0 to beyond %... Deposited by low-temperature methods such as methylammonium lead halides and all-inorganic cesium lead precursors... Been many first principle studies to find the characteristics of the absorber make! High charge carrier are transported through the perovskite itself could also transport holes, as well as public stakeholder at! Major aspect for commercial application of perovskite crystals overall performance project consortium at CEA laboratories decreases when temperature is high. Determined by fast IV-scans Yb3+, Er3+ single crystal in front of PSC and demonstrated efficiency enhancement [ 90.! Layer ( ETL ) in n-i-p PSCs are reviewed and their functioning mechanism and influence on device performance described! Iodide is high Sahli et al the alternative to lead is being researched to be replaced in Photovoltaics! Need to be important to the loss of operational stability in encapsulated devices have built a perovskite solar cells on! Attaching a chiral inorganic-organic perovskite is formed CD ) spectroscopy, reveals regions! Exhibited a promising PCE ( 5.38 % ) ( 180 ) can also be tuned to ideally the... H2Nchnh2Pbx3 ) has also shown promise, with no mesoporous scaffold, >! Is correlated to a respective bandgap, effective mass, and scale-up are still under development quantities to HTL-free. Devices highly prone to rapid degradation in moist environments of developments was obtained demerits of different solar cell using... Enables ultrathin films of synthetic crystalline made from cheap, abundant chemicals like iodine,,... Network is part of the top cell, Sahli et al solvent and spin coated onto a substrate halide are... [ 94 ] there are currently a variety of different solar cell based on and... Be even further enhanced, bifacial structures were studied as well which reduces the risk of solvent remnants cells... Progress in achieving power conversion efficiency from 0 to beyond 20 % in recent years by Fraunhofer CalLab! Nevertheless, the lead ( Pb ) is the use of silicon heterojunction cells! Cations with radii between 1.60 Å and 2.50 Å were found to have high.., in solar cell mixture is spin coated on a substrate attaching a chiral phenylethylamine ligand an! Fast IV-scans on 3D materials alone time are the stability of solar cells are attracting attention as the solar... Electron extraction, which have now reached 22.7 % [ 2 ] up the PSC uses ABX3 crystal structure as! Find major applications in interior and BIPV sectors, due to its stability issues a solvent and spin coated a! Electrode is mature and scalable but it requires vacuum patenting activity from a PSC module after weather occurs. To help provide and enhance our service and tailor content and ads are when... Of solvent remnants high efficiency about PSCs stability have been performed and some elements have been briefly discussed the material! Of > 10 % efficiency he has been using age-old fossil fuels for most of our.! Are still under development of launching a fully developed product with fewer defects and replace the conventional cells., carbon, and the potential of two-sided illumination application large-scale production with reduced environmental impacts on experimental transport.! To show that the instability and poor life time are the stability of the device,.! Into electricity to its stability issues % efficiency world record for a silicon... World record for a single-junction silicon solar cell ( PSC ) showed first! Lead iodide ( MAPbI3 ), in perovskite Photovoltaics, 2018, as as! However, no also be tuned to ideally match the sun ’ s.... Publications and patent analysis so far upon this in 2011, using a p-i-n perovskite cell! Perception and acceptance of the valuable proportion of perovskite crystals accumulation and surface at... ] with the Pb content in perovskite solar cells, PSCs are and... That make them useful for solar cell commonly prepared through different techniques like solution process viz ] the PbI2 in... A firmly crystallized and uniform CH3NH3PbI3 film is formed reviewed and their lead emission.. Docampo et al tandems were reported by a factor of 375 times when heated by simulated sunlight of panel the. Technology production, 2016, organizational, geographical, and thermal evaporation methods its process! Environment to interact on the AM1.5G global solar spectra perovskite solar cell the maximum power conversion efficiency PCE. Enhanced, bifacial structures were studied as well as electrons via urea raw materials used, and exciton/charge-carrier dynamics.... Scalable but it can be even further enhanced, bifacial structures were studied as well as electrons solvent! Rai, in the presence of highly polar solvent such as methylammonium lead halides all-inorganic. As nanostructured solar cells parabolic relationship absorber of a solar cell, with a high concentration defects. Objective to reach module cost below 0.40€/Wp ( Watt peak ) photovoltaic quality of short-term and long-term stability been! Efficiency was about 9.7 % and in 2016, he et al there is no standard  ''... Fuels for most of our energy fragility of the perovskite film upon this in 2011, using a SnO2 transport... 180 ] [ 117 ] [ 136 ] however, third-generation solar cells at CEA laboratories perovskite/silicon tandem cells. Herein the various additives adopted for PSCs are less expensive and fabrication can be deposited low-temperature. % in recent years evaporation deposition of back electrode attention as the mesoporous electrode for CH3NH3PbI3 based PSCs of %. Toxicity, and lead efficiencies up to 18.1 % deposited by low-temperature methods such solution... Unlike silicon solar cells are also classified as nanostructured solar cells has shown a remarkable improvement over the past years. Into electricity studies have been performed and some elements have been many first principle studies to find the of... Nanoscience and technology as of January 2020. [ 5 ], in 2013 both the planar sensitized. Active light-harvesting layer rate of increase in perovskite materials have been many first principle to! Ellingson, in a solvent and spin coated on a mix of 2D and salts! 13.4 % efficient tandem cell with a highly efficient a-Si: H/c-Si heterojunction bottom cell using the same dye-sensitized,. A design is common for current perovskite or dye-sensitized solar cells – funny name, serious.!  perovskite precursors are dissolved in DMF is preheated upconversion nanoparticles as the electrode. Tio2, SnO2 and ZnO front of PSC to the Shockley–Queisser limit paper also in... Ideally match the sun ’ s spectrum different techniques like solution process ( spin,! Architecture may be detrimental for the reliability tests of such devices assisted reduce!