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7fb3aaf2c1c65d2790bc8b0c934e4fd0710c5638 | subsection | 38 | 60 | Private and Verifiable Smart Contracts | In order to merge the Private smart contracts (see Section )
and the Verifiable smarts contracts (see Section ),
the core of the present work has been re-developed and this will be
the version that is going to be open-sourced:private smart contracts are developed in Obliv-Java, a modified Open-JDK
for secure computatio... | {
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8e6c1958056951de9f9d9ec93c68a270c27041ca | subsection | 39 | 60 | Detailed Description | In this section we present our secure protocol for private and verifiable
smart contracts, specified in the Functionality below:Functionality 6.1: Private and Verifiable Smart Contracts
Parties: E_{1},...,E_{N}, set of nodes N_{i} of a blockchain
B
Inputs: smart contract SC, private inputs from parties
E_{1}:\overrig... | {
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5f8693940fd6f2f21206ebae32a833c88367af5a | subsection | 40 | 60 | Detailed Description | The parties invoke Functionality 4.1 (Secure computation of smart
contracts), implemented using Protocol 4.2 where party E_{1} inputs
\overrightarrow{x}, party E_{2} inputs \overrightarrow{y}
and the rest of parties E_{N} their corresponding inputs. Parties
receive results E_{1}:\overrightarrow{r_{1}}, E_{2}:\overright... | {
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3e2627843c5e2943d506f01d967bb8f601142460 | subsection | 41 | 60 | Security Analysis | The security analysis follows the standard definition of static semi-honest
security in the standalone setting and assumes
semi-honest security of basic building blocks: Yao's protocol
and oblivious transfer accelerated with OT-extension, ,
implying the security for the Protocol 4.2 realising Functionality
4.1. Additio... | {
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d55e8953f56a75debc711baef9727f91c2d1f45e | subsection | 42 | 60 | Extended verification of smart contracts | Trusted third parties (e.g., governments, central banks, regulating
bodies) may provide specifications against which proofs must be generated.
Thus, Gyges attacks can be prevented if
parties and executing nodes require that all the executed smart contracts
must adhere to said specifications and be digitally signed by t... | {
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1dc54b76d68acf8e13eae3342f8719a9911aefc2 | subsection | 43 | 60 | Extended verification of smart contracts | Output: true if the verification was correct, false
otherwise.Finally, the general Protocol 6.4 is reviewed with the new Protocol
6.6 for the extended verification of smart contracts:Protocol 6.7: Realising Functionality 6.1 (Private and Extended
Verifiable Smart Contracts)
Parties: E_{1},E_{2},...,E_{N}, set of nodes... | {
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851a22257ac500bc541dcf865aa0bffc28327de4 | subsection | 44 | 60 | Malicious security | Smart contracts are compiled to efficient Boolean circuits that could
be executed on multiple secure computation frameworks with the added
benefit of obtaining malicious security, , , , :
protocols and theorems above can trivially be updated with their security
definitions and proofs. In case of using ORAMs, security a... | {
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c81721322cf9b01c2f618d78d00d210735b58816 | subsection | 45 | 60 | Private Function Evaluation | In this setting, the smart contract itself is kept secret: that is,
only one of the parties knows the function f\left(x\right) computed
by the smart contract, whereas other parties provide the input to
the private function without learning about f besides the size
of the circuit defining the function and the number of ... | {
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62fdeb06a62f3cb6afbb7dadd6c864db68911284 | subsection | 46 | 60 | Economic Impact and Legal Analysis | Previous legal analysis has considered how secure
computation fits current legal frameworks, although it's much more
interesting to evaluate how it could alter legal frameworks and its
economic impact. | {
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976e8c099323c2729311f39827fc82416f99a5f4 | subsection | 47 | 60 | Data Privacy as Quasi-Property | Quasi-property interests refer to situations
in which the law seeks to simulate the functioning of property’s exclusionary
apparatus, through a relational entitlement mechanism, by focusing
on the nature and circumstances of the interaction in question, which
is thought to merit a highly circumscribed form of exclusion... | {
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36764a04483e789c7610aada13210b1e71082bfc | subsection | 48 | 60 | A Solution to Arrow's Paradox | Arrow's Paradox states that “there
is a fundamental paradox in the determination of demand for information;
its value for the purchaser is not known until he knows the information,
but then he has in effect acquired it without cost”. Ex-ante,
the purchaser cannot value the original information since it can only
be know... | {
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cb40fc980a41222b0753ddbb73335706e48e911d | subsection | 49 | 60 | Expansion of Trade Secrecy | While software copyright and patentability protection have been weakened,
trade secrecy stands firm. As defined in the United States' Uniform
Trade Secrets Act: “Trade secret means information,
including a formula, pattern, compilation, program, device, method,
technique, or process, that: (i) derives independent econo... | {
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4182ee6db4ef85a4584a105aa208249c0600c654 | subsection | 50 | 60 | Verifiability and Self-Enforcement | The term Lex Cryptographia designates
a new body/subset of law, containing rules administered through smart
contracts and decentralized autonomous organizations: this concept
is inspired on Lex Mercatoria, an old subset of customs that
became recognized as a customary body of law for international commerce.
Smart contr... | {
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6929ecb1114d514aefdc8b17b3853a588faee121 | subsection | 51 | 60 | Markets for Smart Contracts | Paradoxically, smart contracts are hardly bought/sold: the inexistence
of strong property rights hampers the development of markets on smart
contracts; actually, not even companies developing and operating them
are being acquired/merged based on the value of their smart contracts
since they can easily be reverse engine... | {
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e5b09970fd3dc98c310dd04c50c365d5353e7527 | subsection | 52 | 60 | Effects on Currency Competition | Latest analysis on the monetary policy of cryptocurrencies
provide insights on the effects of currency competition:Monetary equilibrium between private cryptocurrencies will not deliver
price stability: profit-maximizing entrepreneurs issuing cryptocurrencies
do not have real incentives to provide stable currencies, on... | {
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9cbf9dcb1582fc1426319f5bfcaa3884f22980a6 | subsection | 53 | 60 | Token vs. Account-based Cryptocurrencies | Most cryptocurrencies are based on the model of issuing and transacting
tokens, using some form of distributed ledger to keep track of the
ownership of said tokens (e.g., Bitcoin, Ethereum and Zcash): they're
the digital equivalents of cash.Another unexplored model of cryptocurrency is that of holding funds
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d6e30e3c18029ad469f3e7bc76fed5e3e98bff23 | subsection | 54 | 60 | Impact on Market Structures | Although disintermediation will be one of the first consequences of
the application of cryptographic smart contracts to financial markets,
as happened in the past with the introduction of the Internet,
bank and fund concentrations may also increase: information asymmetry
is associated with more concentration and inform... | {
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bffe941f1bf6aa54b1b8ddeffe020b81fee3981d | subsection | 55 | 60 | The Valuation of Secrecy and the Privacy Multiplier | The use of private smart contracts (see Section )
has inmediate economic benefits for the secured information, because
when information is withholden to prevent data leakages then its valuation
is discounted. This secrecy discount is given byS_{T}=e^{-yT}\left(2\left(\frac{\sigma \sqrt{T}}{2}\right)-1\right)with T deno... | {
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abdb460751eb40407f8ce554a4d51e73de7ab0ce | subsection | 56 | 60 | Related work | A number of related works using cryptographic techniques are as follows:Enigma, is coded in Python, a language
lacking verification libraries/toolkits and formal semantics so it
can't be used for proof-carrying code; its cryptographic protocols
are not constant-round, thus settings with some latency will be it
excrucia... | {
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"raw": "Guy Zyskind, Oz Nathan, and Alex Pentland. Enigma: Decentralized Computation Platform with Guaranteed Privacy, 2015. http://www.enigma.... | 1807.09484 | Raziel: Private and Verifiable Smart Contracts on Blockchains | [
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1a9017f6cf4f4da87de74806f964377075a908b5 | subsection | 57 | 60 | Conclusions, subsequent and future work | The present paper has tackled and successfully solved the problem
of improving the privacy, correctness and verifiability of smart contracts,
resolving the DAO and Gyges attacks. Examples have been shown to demonstrate
its practical viability. | {
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} | 1807.09484 | Raziel: Private and Verifiable Smart Contracts on Blockchains | [
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6eaf5229882ebd68a9d93d70f679ef157da0c950 | subsection | 58 | 60 | Subsequent work | The ability to save the state of garbled circuits and restore them
at later times is a major improvement, bringing them closer to secret
sharing techniques:partial garbled circuits: for each wire value,
the generator sends two values to the evaluator, transforming the
wire labels of the evaluator to another garbled cir... | {
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e1bd93efce84ceccb1d7718ee31cdfb2acc3e97d | subsection | 59 | 60 | Future work | Some encrypted smart contracts will be perpetual (e.g., consols):
if they ever store any kind of encrypted secure computation (e.g.,
secret shares, garbled circuits, homomorphic encryptions, IO) eventually
there will be the need to update their encrypted contents to upgrade
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87d70a5ba2b92653387092b29d8209169d798f0d | abstract | 0 | 32 | Abstract | We show how increasing returns to scale in urban scaling can artificially
emerge, systematically and predictably, without any sorting or positive
externalities. We employ a model where individual productivities are
independent and identically distributed lognormal random variables across all
cities. We use extreme valu... | {
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} | 10.1177/2399808320942366 | 1807.09424 | Artificial Increasing Returns to Scale and the Problem of Sampling from
Lognormals | [
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9146d272be3f85eaab65848b770a418b2c7c0542 | subsection | 1 | 32 | Introduction | There is a substantial body of theoretical and empirical research on the origins of “Increasing Returns to Scale” in cities (, , , , ). This urban productivity premium, whereby workers tend to be more productive in larger cities, can be estimated using wages , and is thus referred to as the urban wage premium (from her... | {
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67cc522d092b9b4d182c05c394020696e84dcd97 | subsection | 2 | 32 | Introduction | Our analytic results are validated on simulated data, as well as on an administrative dataset of all formal workers in Colombia.To develop the intuition for why individual productivities that are lognormally distributed can give rise to UWP not caused by any sorting or agglomeration effects, it is illustrative to think... | {
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} | 10.1177/2399808320942366 | 1807.09424 | Artificial Increasing Returns to Scale and the Problem of Sampling from
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57c897b22943c02d4fb088758bd3cf6ff64bd5e4 | subsection | 3 | 32 | Introduction | Assuming independent and identically distributed productivities, denoted in general with the letter X, and assuming the difference in average productivities between the groups is \bar{x}_2 - \bar{x}_1>\epsilon , the likelihood of the conjunction is \Pr (X>\bar{x}_1 + \epsilon )^{n_2} while the likelihood of the disjunc... | {
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} | 10.1177/2399808320942366 | 1807.09424 | Artificial Increasing Returns to Scale and the Problem of Sampling from
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3a37a54827c65a9091a0bf028df50473cd0f8fa1 | subsection | 4 | 32 | Introduction | Even more, the conjunction and the disjunction examples provide a mathematically correct explanation of large deviations probabilities which describe highly unlikely events (e.g., that the average productivity attains an atypically large value above the expected one).Conjunctions correspond to light-tailed cases, where... | {
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} | 10.1177/2399808320942366 | 1807.09424 | Artificial Increasing Returns to Scale and the Problem of Sampling from
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0d95d5c873ef020e7322c0b69aca385a26f3484a | subsection | 5 | 32 | Background | Few studies have addressed the relationship between probability distributions and production functions exhibiting increasing returns to scale (IRS). One of the earliest attempts at connecting a probability distribution with a production function was made by , who showed that a Cobb-Douglas production function arises wh... | {
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fa667e8ae2961593013638ddf4e41fadf4e41d43 | subsection | 6 | 32 | Background | In contrast, we will derive our results from a model in which all individuals have productivities that are independently and identically sampled from the same distribution whose moments are all fixed and finite. Under this model, we show that total output shows increasing returns for a wide range of scales, a result th... | {
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fc11f8221d2cab8f72f40cc5d2c03676dcd54cad | subsection | 7 | 32 | Analytic Results | Our analytic results are based on a very simple model whereby individuals, regardless of the city they live in, have productivities independently and identically distributed (i.i.d.), sampled from a lognormal distribution \mathcal {LN}(x_0,\sigma ^2), whose probability density function isp_X(x;x_0,\sigma ^2)=\frac{1}{x... | {
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6c95daa002e2c81bf78bdb8e49a82d7c04f96cd9 | subsection | 8 | 32 | Elasticity for a single city | Let us proceed by calculating first the change in the expected value of urban output if population size is increased by \lambda >1 according to this simple model:{\mathrm {E}}\left[ Y(\lambda n) \right] &= {\mathrm {E}}\left[ \sum _{i=1}^{\lambda n} X_i \right], \\
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} | 10.1177/2399808320942366 | 1807.09424 | Artificial Increasing Returns to Scale and the Problem of Sampling from
Lognormals | [
"Andres Gomez-Lievano",
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bf6c5c2ee1a112e023d629c5a802ca5d2412c852 | subsection | 9 | 32 | Elasticity for a single city | Thus, for sizes n\rightarrow \infty , the elasticity of total output with respect to size is\beta &=\frac{\mathrm {d}\:\ln (Y(n))}{\mathrm {d}\:\ln (n)} \\
&\approx \frac{\mathrm {d}\:\ln (\mu n)}{\mathrm {d}\:\ln (n)} \\
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c81f37c47c0459731d32b8e081365bd9912ab974 | subsection | 10 | 32 | The maximum of lognormal random variables | The random variables representing the productivity of individuals in our model can be conveniently represented as X_i = \text{e}^{\sigma Z_i + \ln x_0}, where Z_1, Z_2, \ldots are i.i.d. random variables sampled from the standard normal distribution \mathcal {N} (0,1). We will consider lognormal distributions with vari... | {
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} | 10.1177/2399808320942366 | 1807.09424 | Artificial Increasing Returns to Scale and the Problem of Sampling from
Lognormals | [
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b97c2ce5471526e716857a75a9e2d8a412e310a0 | subsection | 11 | 32 | The maximum of lognormal random variables | ThenY(n) &= \sum _{i=1}^n X_i = \sum _{i=1}^n \text{e}^{\sigma Z_i - \sigma ^2/2} = \text{e}^{\sigma L(n) - \sigma ^2/2} \sum _{i=1}^n \text{e}^{\sigma (Z_i - L(n))}, \\
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36367b296687b17c93414da199207a3f9a57b94f | subsection | 12 | 32 | The maximum of lognormal random variables | The remaining terms \text{e}^{ \sigma (L_i(n) - L(n))} for i \ge 3 decay to 0 much faster since so do exponents with larger negative powers.Thus, we have \Delta _n \approx 1 when \sigma \gg \sqrt{\ln (n)}. Moreover, our simulations (not shown) reveal that a similar conclusion applies even when \sigma is larger than, bu... | {
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6eb9cbcb433e1266dd0d0378c5f89a4fbc907a6a | subsection | 13 | 32 | The maximum of lognormal random variables | For \sigma =4 one needs to increase size to n=10^7 (a ten-thousand-fold increase) in order to decrease the dominance of the maximum to about 10\% (see lightest blue line). The fact that as population size increases (i.e., as the color of the lines become lighter) the dominance of the maximum value over the sum decrease... | {
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cb7b7dc56e425da40ea2f06fb39fb4307ad6fa1b | subsection | 14 | 32 | Elasticity from a cross-sectional regression of many cities | The elasticity of the urban productivity premium effect is a relative rate of change of total output with size. This rate may be different for different sizes. In regression analysis, however, we often estimate empirically an average elasticity across many sizes. Assuming the artificial UWP described in the previous se... | {
"cite_spans": []
} | 10.1177/2399808320942366 | 1807.09424 | Artificial Increasing Returns to Scale and the Problem of Sampling from
Lognormals | [
"Andres Gomez-Lievano",
"Vladislav Vysotsky",
"Jose Lobo"
] | [
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e8fb9d24624871db5375ad8d54a47744728c281e | subsection | 15 | 32 | Elasticity from a cross-sectional regression of many cities | The parameter n_{\min } determines the minimum value above which sizes follow a Pareto distribution.The regression coefficient can be re-written as\beta _{ave}(n_{\min }, \sigma , \alpha ) &= \frac{{\mathrm {E}}\left[ \ln (N)\ln (Y(N)) \right]-{\mathrm {E}}\left[ \ln (N) \right]{\mathrm {E}}\left[ \ln (Y(N)) \right]}{\... | {
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} | 10.1177/2399808320942366 | 1807.09424 | Artificial Increasing Returns to Scale and the Problem of Sampling from
Lognormals | [
"Andres Gomez-Lievano",
"Vladislav Vysotsky",
"Jose Lobo"
] | [
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33a62cc58da36fd1e167cfec7c0ac8978bc0d789 | subsection | 16 | 32 | Elasticity from a cross-sectional regression of many cities | Hence, researchers often estimate average productivites across cities using samples of the city populations. To model this situation we only need to introduce a new parameter f, a number between 0 and 1, premultiplying n_{\min }, using the convenient property that taking a fixed percentage f of all city sizes does not ... | {
"cite_spans": []
} | 10.1177/2399808320942366 | 1807.09424 | Artificial Increasing Returns to Scale and the Problem of Sampling from
Lognormals | [
"Andres Gomez-Lievano",
"Vladislav Vysotsky",
"Jose Lobo"
] | [
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986ca23e2d961a08be5b80d945c796d516d39589 | subsection | 17 | 32 | Simulations | We simulate m=900 synthetic cities using our model. Each city has a population n_k taken from a Pareto distribution with parameters n_{\min }=10,000 and \alpha =1, and for each of them, we generate n_k productivities sampled i.i.d. from eq:lognormalpdf. We will generate simulations for different values of the model par... | {
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} | 10.1177/2399808320942366 | 1807.09424 | Artificial Increasing Returns to Scale and the Problem of Sampling from
Lognormals | [
"Andres Gomez-Lievano",
"Vladislav Vysotsky",
"Jose Lobo"
] | [
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0b3df0d58e21f4ad113ec98047132d2331f115fc | subsection | 18 | 32 | Simulations | For each value of \sigma we generated 100 simulations. The values of \widehat{\beta }, \widehat{\ln (Y_0)} and R^2 change systematically with the value of the \sigma -parameters of the individuals' lognormal. As \sigma increases, the scaling exponent also increases, the intercept decreases, and the goodness of the line... | {
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} | 10.1177/2399808320942366 | 1807.09424 | Artificial Increasing Returns to Scale and the Problem of Sampling from
Lognormals | [
"Andres Gomez-Lievano",
"Vladislav Vysotsky",
"Jose Lobo"
] | [
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156844e97a2073cc0f75aae3f2450b7b3218bd31 | subsection | 19 | 32 | An Application | In sec:analytic we derived eq:predictedbeta, which highlights two important effects. On the one hand, that the elasticity \beta of total output in a city with respect to its population size will increase if the standard deviation of log-productivity, \sigma , increases. On the other, it tells us that \beta will also in... | {
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Lognormals | [
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ffc838b71731b3f2bf77ddebf4cea8c0c17a1cbb | subsection | 20 | 32 | Data, Descriptives and Distributions | The data used here is the 2014 administrative records of the social security system in Colombia (the Spanish acronym is PILA, for Integrated Report of Social Security Contributions). We refer the reader to the Appendix A for the source, details and preparation of the data, as the dataset has been cleaned and prepared f... | {
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Lognormals | [
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ce0887397209ac845a9ca587c751dad734de0e8f | subsection | 21 | 32 | Data, Descriptives and Distributions | Specifically, wages were best fit by the former while sizes were best fit by the latter, as quantified by three criteria: the largest likelihood, the minimum Akaike Information Criterion (AIC), and the minimum Bayesian Information Criterion (BIC). fig:fitdiagnosticsSizes,fig:fitdiagnosticsWages in Appendix C show the d... | {
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} | 10.1177/2399808320942366 | 1807.09424 | Artificial Increasing Returns to Scale and the Problem of Sampling from
Lognormals | [
"Andres Gomez-Lievano",
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3f42956cb6c4433541a2304c7338c24494b4ad1a | subsection | 22 | 32 | Telling Apart Real Versus Artificial UWP | fig:WagesVSSizeScatter plots the cross-section of the average monthly wage per municipality with respect to municipality size. There is clearly a positive and significant elasticity \widehat{\delta }\approx 0.06.
[Figure: Larger municipalities have, on average, workers with higher monthly wages.]The strategy to estimat... | {
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} | 10.1177/2399808320942366 | 1807.09424 | Artificial Increasing Returns to Scale and the Problem of Sampling from
Lognormals | [
"Andres Gomez-Lievano",
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f586db56804fad6a91cc992b7e2898e23406cc2f | subsection | 23 | 32 | Telling Apart Real Versus Artificial UWP | To summarize the procedure, first, we will sample without replacement a fraction f of all workers, second, we will estimate the real unconditional elasticity, and third, we will randomize several times the locations of individuals by applying random permutations of the location of individuals in the sample, estimating ... | {
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} | 10.1177/2399808320942366 | 1807.09424 | Artificial Increasing Returns to Scale and the Problem of Sampling from
Lognormals | [
"Andres Gomez-Lievano",
"Vladislav Vysotsky",
"Jose Lobo"
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39b96517936fe350ba2dea7cc2a8a3ae83f8b9c4 | subsection | 24 | 32 | Telling Apart Real Versus Artificial UWP | The dotted blue line is the elasticity \delta = \beta _{ave}(f\times n_{\min }, \sigma , \alpha )-1 predicted by eq:predictedbetaave, with values n_{\min }=287, \sigma =2.0, \alpha =0.67, and f=0.001, 0.005, 0.01, 0.05, 0.10 and 1.00. Panel B plots the values of the z-score statistic for each elasticity from the OLS re... | {
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Lognormals | [
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72fdcf12e9b93f277da4c84e35e0849c5f3e7c96 | subsection | 25 | 32 | Telling Apart Real Versus Artificial UWP | Under the null hypothesis that these two estimated elasticities are equal, and assuming the number of municipalities large, we can construct the following z-score:z_{\text{stat}} = \frac{\widehat{\beta }^{(\text{real})} - \widehat{\beta _k}^{(\text{rand})}}{\sqrt{(se^{(\text{real})})^2 + (se_k^{(\text{rand})})^2}}which... | {
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} | 10.1177/2399808320942366 | 1807.09424 | Artificial Increasing Returns to Scale and the Problem of Sampling from
Lognormals | [
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9bb022b82866cf76b026c1c785f01138b3c5002c | subsection | 26 | 32 | Discussion and Conclusions | The argument presented here provides evidence for there being a mechanism, previously unaccounted in the urban economics literature, that can give rise to a statistical effect which can be mistaken for a city-size productivity premium. Our goal is to persuade the reader that increasing returns to scale exhibited by dat... | {
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Lognormals | [
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46a129e10a84c5008bd1d8bc7756fd7e3909dcb1 | subsection | 27 | 32 | Discussion and Conclusions | We derived a precise formula to compute the null model elasticity for both a single city and a cross section of cities, the latter being solely a function of the distributional parameters of productivities (\sigma ) and city sizes (n_{\min } and \alpha ). Our approach shifts attention away from the study of averages to... | {
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211dd29ad4b14e312c888d4f8b3be20f20a0d1f6 | subsection | 28 | 32 | Discussion and Conclusions | We regard the effect we have studied here as a bias, but we distinguish it from other types of statistical biases in that the sampling effect is a real tangible property of a statistical distribution which has consequences on measures of central tendency such as the sample mean. As mentioned in the Background section, ... | {
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8f48fe1996433a521cbd653b45c3aa18b21be129 | subsection | 29 | 32 | Appendix A. Data | In sec:application we use data of the formal workforce in Colombia to analyze the unconditional elasticity of nominal wages on municipality population size. These come from the administrative records of the social security system in Colombia (abbreviated as PILA in Spanish, meaning the Integrated Report of Social Secur... | {
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} | 10.1177/2399808320942366 | 1807.09424 | Artificial Increasing Returns to Scale and the Problem of Sampling from
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5fa891457a493d523eeee9a3e9f7c979ccf7b5a8 | subsection | 30 | 32 | Appendix B. The null model | The model of lognormally distributed productivities is built on arguments given by about the distribution of the earnings of individuals. Roy's central assumptions are that (i) earnings are proportional to the output of workers, and (ii) that output is the product of many internal characteristics of the individuals, in... | {
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c6fe6f2de11a9b9f62a84f2029d0ac387ecbbbec | subsection | 31 | 32 | Appendix B. The null model | Hence, we can restate Assumptions 2 and 3 into a single statement: individuals have a productivity X_i i.i.d. sampled from a lognormal distribution \mathcal {LN}(x_0,\sigma ^2) with probability density functionp_X(x;x_0,\sigma ^2)=\frac{1}{x\sqrt{2\pi \sigma ^2}}\text{e}^{-\frac{(\ln x - \ln x_0)^2}{2\sigma ^2}},where ... | {
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97061131d2f8197bc0f64ab3cba1aadf21047b23 | abstract | 0 | 24 | Abstract | Organic molecules such as steroids or amino acids form crystals that can
facilitate the formation of ice -- arguably the most important phase transition
on earth. However, the origin of the ice nucleating ability of organic crystals
is still largely unknown. Here, we combine experiments and simulations to
unravel the m... | {
"cite_spans": []
} | 10.1039/C8SC02753F | 1809.01232 | Unravelling the Origins of Ice Nucleation on Organic Crystals | [
"Gabriele C. Sosso",
"Thomas F. Whale",
"Mark A. Holden",
"Philipp Pedevilla",
"Benjamin J. Murray",
"Angelos Michaelides"
] | [
"cond-mat.soft"
] | 2,018 | en | Physics | [
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54f8a59275ac67eee111fb5848c084cdcb2e5f28 | subsection | 1 | 24 | Introduction | The freezing of liquid water into crystalline ice is a ubiquitous phenomenon which is part of our everyday
experience and has countless reverberations in fields as diverse as
cryobiology , , and atmospheric
science , . Strikingly, the overwhelming majority of ice on earth forms
heterogeneously, i.e. thanks to the pres... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1038/494027a",
"end": 220,
"openalex_id": "https://openalex.org/W1972222459",
"raw": "Thorsten Bartels-Rausch. Chemistry: Ten things we need to know about ice and snow. Nature, 494(7435):27–29, February 2013. ISSN 0028-0836. doi: 10.1038... | 10.1039/C8SC02753F | 1809.01232 | Unravelling the Origins of Ice Nucleation on Organic Crystals | [
"Gabriele C. Sosso",
"Thomas F. Whale",
"Mark A. Holden",
"Philipp Pedevilla",
"Benjamin J. Murray",
"Angelos Michaelides"
] | [
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b2d8219c1683895f360b67d16e08e7a24da7992e | subsection | 2 | 24 | Introduction | We find via micro-litre droplet nucleation measurements (\mu l-NIPI) that CHL crystals
display an outstanding ice nucleation ability (stronger than most inorganic ice nucleating agents), with freezing events
initiating at very mild supercooling \Delta T_S = T_{\text{Melt}}-T = 4 K down to \Delta T_S = 20 K. Scanning
el... | {
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{
"arxiv_id": "",
"doi": "10.1021/ja411507a",
"end": 1293,
"openalex_id": "https://openalex.org/W2327383586",
"raw": "Laura Lupi, Arpa Hudait, and Valeria Molinero. Heterogeneous Nucleation of Ice on Carbon Surfaces. J. Am. Chem. Soc., 136(8):3156–3164, February 201... | 10.1039/C8SC02753F | 1809.01232 | Unravelling the Origins of Ice Nucleation on Organic Crystals | [
"Gabriele C. Sosso",
"Thomas F. Whale",
"Mark A. Holden",
"Philipp Pedevilla",
"Benjamin J. Murray",
"Angelos Michaelides"
] | [
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6c6212f0c82313f291a2f9772c8d181c9098c75f | subsection | 3 | 24 | Body | The ice nucleation efficiency of CHLM was evaluated using an adapted version of the \mu L Nucleation by
Immersed Particles Instrument (\mu l-NIPI) described in detail in Ref. . To make the flat
plates we dissolved 2g of pure CHL (Sigma Aldrich) in approximately 30 ml of hot (\sim 343 K) 95% ethanol
(Sigma-Aldrich). The... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.5194/amt-8-2437-2015",
"end": 176,
"openalex_id": "https://openalex.org/W2170544731",
"raw": "T. F. Whale, B. J. Murray, D. O'Sullivan, T. W. Wilson, N. S. Umo, K. J. Baustian, J. D. Atkinson, D. A. Workneh, and G. J. Morris. A technique... | 10.1039/C8SC02753F | 1809.01232 | Unravelling the Origins of Ice Nucleation on Organic Crystals | [
"Gabriele C. Sosso",
"Thomas F. Whale",
"Mark A. Holden",
"Philipp Pedevilla",
"Benjamin J. Murray",
"Angelos Michaelides"
] | [
"cond-mat.soft"
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a269aaf5832a784f054b48e95e36d1daad3a8d9a | subsection | 4 | 24 | Body | SEM was performed with an FEI Nova NanoSEM 450 in high vacuum mode, using an Everhart-Thornley Detector
(ETD). | {
"cite_spans": []
} | 10.1039/C8SC02753F | 1809.01232 | Unravelling the Origins of Ice Nucleation on Organic Crystals | [
"Gabriele C. Sosso",
"Thomas F. Whale",
"Mark A. Holden",
"Philipp Pedevilla",
"Benjamin J. Murray",
"Angelos Michaelides"
] | [
"cond-mat.soft"
] | 2,018 | en | Physics | [
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d74fceaa9cb2ce9739240b2944a97213b25647a3 | subsection | 5 | 24 | Molecular dynamics simulations | The computational setup we have used is depicted in Fig. REF a. A single layer of CHL molecules,
cleaved along the (001) plane (perpendicular to the normal to the slab) was prepared by starting from the experimental
cell parameters and lattice positions . Specifically, a CHLM crystal system made of two
mirroring slabs ... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1038/260727a0",
"end": 255,
"openalex_id": "https://openalex.org/W2036313288",
"raw": "Bryan M. Craven. Crystal structure of cholesterol monohydrate. Nature, 260(5553):727–729, April 1976. doi: 10.1038/260727a0. URL http://www.nature.com... | 10.1039/C8SC02753F | 1809.01232 | Unravelling the Origins of Ice Nucleation on Organic Crystals | [
"Gabriele C. Sosso",
"Thomas F. Whale",
"Mark A. Holden",
"Philipp Pedevilla",
"Benjamin J. Murray",
"Angelos Michaelides"
] | [
"cond-mat.soft"
] | 2,018 | en | Physics | [
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9eb806408024346f4cf97fa5fde8deac67948f3b | subsection | 6 | 24 | Molecular dynamics simulations | The Forward Flux Sampling (FFS) simulations reported in this work rely on a massive collection of unbiased Molecular
Dynamics (MD) runs, all of which have been performed using the GROMACS package, version 4.6.7. The code was compiled in
single-precision, in order to alleviate the huge computational workload needed to c... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1063/1.2408420",
"end": 978,
"openalex_id": "https://openalex.org/W2057477511",
"raw": "Giovanni Bussi, Davide Donadio, and Michele Parrinello. Canonical sampling through velocity rescaling. J. Chem. Phys., 126(1):014101, January 2007. I... | 10.1039/C8SC02753F | 1809.01232 | Unravelling the Origins of Ice Nucleation on Organic Crystals | [
"Gabriele C. Sosso",
"Thomas F. Whale",
"Mark A. Holden",
"Philipp Pedevilla",
"Benjamin J. Murray",
"Angelos Michaelides"
] | [
"cond-mat.soft"
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... |
9e92184906aefbd00067f8bb74e0299ca148a3b6 | subsection | 7 | 24 | Cholesterol promotes the formation of ice across a wide range of temperatures | We start by experimentally investigating the ice nucleating ability of cholesterol crystals - as a function of
supercooling. CHL can crystallise into two different polymorphs, namely anhydrous (CHLA) and
CHLM . The latter is the most relevant to ice formation, as it spontaneously forms in aqueous
environments , , , , ... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1038/267287a0",
"end": 211,
"openalex_id": "https://openalex.org/W2023593626",
"raw": "H. S. Shieh, L. G. Hoard, and C. E. Nordman. Crystal structure of anhydrous cholesterol. Nature, 267(5608):287–289, May 1977. doi: 10.1038/267287a0. U... | 10.1039/C8SC02753F | 1809.01232 | Unravelling the Origins of Ice Nucleation on Organic Crystals | [
"Gabriele C. Sosso",
"Thomas F. Whale",
"Mark A. Holden",
"Philipp Pedevilla",
"Benjamin J. Murray",
"Angelos Michaelides"
] | [
"cond-mat.soft"
] | 2,018 | en | Physics | [
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5d39f9aef17eca3b8b503282c5f6efa14142c050 | subsection | 8 | 24 | Cholesterol promotes the formation of ice across a wide range of temperatures | It can be seen that CHLM
nucleates ice far more efficiently than kaolinite and more efficiently even than the feldspar at warm temperatures.
Thus, CHLM has the potential to be a highly efficient ice nucleating agent in immersion mode across a wide range of
temperatures - which is the scenario typically encountered when... | {
"cite_spans": []
} | 10.1039/C8SC02753F | 1809.01232 | Unravelling the Origins of Ice Nucleation on Organic Crystals | [
"Gabriele C. Sosso",
"Thomas F. Whale",
"Mark A. Holden",
"Philipp Pedevilla",
"Benjamin J. Murray",
"Angelos Michaelides"
] | [
"cond-mat.soft"
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610b09177285742d07e7eb3c836b372d88f2ec70 | subsection | 9 | 24 | The role of surface topography | As noted above the spread of freezing temperatures we report for CHLM is very broad. This behaviour suggests that the
nucleation behaviour of CHLM is spatially heterogeneous, i.e. different parts of the surface nucleate ice with differing
efficiency. This is commonly known as site specific nucleation
behaviour , , , , ... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.5194/acp-14-5271-2014",
"end": 650,
"openalex_id": "https://openalex.org/W2109631880",
"raw": "G. Vali. Interpretation of freezing nucleation experiments: singular and stochastic; sites and surfaces. Atmos. Chem. Phys., 14(11):5271–5294,... | 10.1039/C8SC02753F | 1809.01232 | Unravelling the Origins of Ice Nucleation on Organic Crystals | [
"Gabriele C. Sosso",
"Thomas F. Whale",
"Mark A. Holden",
"Philipp Pedevilla",
"Benjamin J. Murray",
"Angelos Michaelides"
] | [
"cond-mat.soft"
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892e9c6182e03edbe96091e67b477eced1a5827a | subsection | 10 | 24 | The cholesterol-water interface | In order to investigate the molecular-level details of the CHLM-water interface, we have performed unbiased
molecular dynamics simulations at strong supercooling (\Delta T_S = 42 K) employing the
CHARMM36 , and the TIP4P/Ice force
fields for CHL and water molecules respectively. Computational details and results conce... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1021/ct900549r",
"end": 281,
"openalex_id": "https://openalex.org/W2052038479",
"raw": "Pär Bjelkmar, Per Larsson, Michel A. Cuendet, Berk Hess, and Erik Lindahl. Implementation of the CHARMM Force Field in GROMACS: Analysis of Protein S... | 10.1039/C8SC02753F | 1809.01232 | Unravelling the Origins of Ice Nucleation on Organic Crystals | [
"Gabriele C. Sosso",
"Thomas F. Whale",
"Mark A. Holden",
"Philipp Pedevilla",
"Benjamin J. Murray",
"Angelos Michaelides"
] | [
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df781a7a0a3640bc782d80fef60a835530ebec4b | subsection | 11 | 24 | The cholesterol-water interface | Note thatthese cages are made of both 6-membered (panel d, side view) and 5-membered (panel e, side view) hydrogen bonded rings,involving water molecules as well as a hydroxyl group provided by the CHLM surface.]Because of this flexibility of the CHL molecules and the low-density of hydroxyl groups at the water-CHLM in... | {
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{
"arxiv_id": "",
"doi": "10.1021/jacs.5b08748",
"end": 584,
"openalex_id": "https://openalex.org/W3102140724",
"raw": "Martin Fitzner, Gabriele C. Sosso, Stephen J. Cox, and Angelos Michaelides. The Many Faces of Heterogeneous Ice Nucleation: Interplay Between Surf... | 10.1039/C8SC02753F | 1809.01232 | Unravelling the Origins of Ice Nucleation on Organic Crystals | [
"Gabriele C. Sosso",
"Thomas F. Whale",
"Mark A. Holden",
"Philipp Pedevilla",
"Benjamin J. Murray",
"Angelos Michaelides"
] | [
"cond-mat.soft"
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dbf5a1dfbddba2ba242f5542e3face3c3c9f4bce | subsection | 12 | 24 | The cholesterol-water interface | REF b), we have found that the rather generous cutoff of 5.0 Å\ is sufficient to accommodate
the substantial extent of the hydrogen bonded cages depicted in Fig. REF d and Fig. REF e. We have also verified that
by choosing the second minimum of the density profile (\sim 7Å) our results, including the trends within the ... | {
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{
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"doi": "10.1021/jacs.5b08748",
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"raw": "Martin Fitzner, Gabriele C. Sosso, Stephen J. Cox, and Angelos Michaelides. The Many Faces of Heterogeneous Ice Nucleation: Interplay Between Surf... | 10.1039/C8SC02753F | 1809.01232 | Unravelling the Origins of Ice Nucleation on Organic Crystals | [
"Gabriele C. Sosso",
"Thomas F. Whale",
"Mark A. Holden",
"Philipp Pedevilla",
"Benjamin J. Murray",
"Angelos Michaelides"
] | [
"cond-mat.soft"
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45ef660a9601db632788fe2ae2498c74344c3e45 | subsection | 13 | 24 | Ice nucleation mechanism and kinetics | In order to characterise the mechanism as well as the kinetics of ice nucleation on CHLM^{-OH}_{001} we have
performed forward flux sampling (FFS)
simulations , , , , , , , .
While other enhanced sampling techniques are in principle available, such as metadynamics ,
transition path sampling , and seeded molecular
dynam... | {
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{
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"doi": "10.1088/0953-8984/21/46/463102",
"end": 174,
"openalex_id": "https://openalex.org/W3122702518",
"raw": "Rosalind J Allen, Chantal Valeriani, and Pieter Rein ten Wolde. Forward flux sampling for rare event simulations. J. Phys. Cond. Matt., ... | 10.1039/C8SC02753F | 1809.01232 | Unravelling the Origins of Ice Nucleation on Organic Crystals | [
"Gabriele C. Sosso",
"Thomas F. Whale",
"Mark A. Holden",
"Philipp Pedevilla",
"Benjamin J. Murray",
"Angelos Michaelides"
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aca16edad0176b7ad004242b3d4d2c887486a5a0 | subsection | 14 | 24 | Ice nucleation mechanism and kinetics | The calculated growth probability P(\lambda |\lambda _{0}) as a function of lambda,
together with the fraction of ice nuclei that can be found at the water-CHLM^{-OH}_{001} interface are reported in the
SI (Fig. S2b). By the time the FFS algorithm has reached \lambda =125, no nuclei within the bulk of the water slab
su... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1021/acs.jpclett.6b01013",
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"raw": "Gabriele C. Sosso, Tianshu Li, Davide Donadio, Gareth A. Tribello, and Angelos Michaelides. Microscopic Mechanism and Kinetics of Ice Forma... | 10.1039/C8SC02753F | 1809.01232 | Unravelling the Origins of Ice Nucleation on Organic Crystals | [
"Gabriele C. Sosso",
"Thomas F. Whale",
"Mark A. Holden",
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"Benjamin J. Murray",
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35d87e352a98264c3dfd0b3e0782b22ca9231376 | subsection | 15 | 24 | Ice nucleation mechanism and kinetics | REF in the SI), which
are likely to be characterised by a much smaller interfacial energy if compared to the hemispherical shape predicated by CNT in the
case of perfectly flat, featureless substrates. We note however that in order to probe this aspect of CNT
quantitatively, it would be desirable to improve the current... | {
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"Gabriele C. Sosso",
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f1a37c2fd4bac92ff1162fa2ba5f52d1d47ed484 | subsection | 16 | 24 | Ice nucleation mechanism and kinetics | Asphericity parameter \alpha and spatial extent of the ice nuclei along the directionnormal to the CHLM slab \Delta Z as a function of \lambda for ice nuclei at the water-CHLM^{-OH}_{001} interface.The insets correspond to top and side views of typical ice nuclei forming at the water-CHLM^{-OH}_{001} interface,containi... | {
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12d9c77dacd0769d30ec75fdab1857ea3a5c3635 | subsection | 17 | 24 | Competition between cubic and hexagonal ice | At the strong supercooling considered here (\Delta T_S = 42 K), homogeneous ice nucleation results in a
mixture of ice I_c and ice I_h known as stacking disordered ice
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df1b5878aa5b9ac0a76efab57d43a3d85cd11992 | subsection | 18 | 24 | Competition between cubic and hexagonal ice | Moreover, we have shown in this work that some of these functional groups -
such as the hydroxyl groups characterising the water-CHLM^{-OH}_{001} interface - can even promote a different ice
polytype at the same time, possibly according to different supercooling.Finally, we note that, in agreement with previous simulat... | {
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7796addea4d9dce2f65e95530608ce9cc50203f0 | subsection | 19 | 24 | Conclusions | By means of a blend of experiments and simulations, we have unravelled the the origins of ice nucleation on
cholesterol (CHL), a prototypical organic crystal of relevance to cryopreservation. Our results suggest that its
exceptional ice nucleating activity stems from the ability of its flexible hydrophilic surface to f... | {
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512101dde7b5beb41190bc9f83aea76351106d77 | subsection | 20 | 24 | Conclusions | In fact, the n_s data reported in Fig. REF b suggest that two populations of potentially different ice nucleating sites may
coexist on the CHLM surface. The change in the slope of the CHLM data is reminiscent of that observed for freezing
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cryoprotectants, the design of which, at the mome... | {
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e4c321ac03a2c6c2f57714fb957846095f35a248 | subsection | 22 | 24 | Conflicts of interest | There are no conflicts to declare. | {
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19d551467324fa7f0e4d33d175c5403f5e566f5d | subsection | 23 | 24 | Supplementary Information | Supplementary Information (SI) can be found at: http://www.rsc.org/suppdata/c8/sc/c8sc02753f/c8sc02753f1.pdf | {
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cf216e4df9e63ce783604960d2c71758a69e6253 | abstract | 0 | 61 | Abstract | Persistence modules are a central algebraic object arising in topological
data analysis. The notion of interleaving provides a natural way to measure
distances between persistence modules. We consider various classes of
persistence modules, including many of those that have been previously studied,
and describe the rel... | {
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312b853e3f772760517a37a953a58ac08166d393 | subsection | 1 | 61 | Introduction | A standard tool in topological data analysis is persistent homology , , , , . It is often applied as follows. One starts with some data, constructs an increasing family of complexes or spaces, and applies homology with coefficients in some fixed field to obtain a persistence module. Next, one computes a summary (e.g. b... | {
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72c41a05ba69c492c46d2f60d04bc6d57ef529f9 | subsection | 2 | 61 | Introduction | We identify which of the inclusions in Figure REF are inclusions of open sets (Proposition REF ).We show that these topological spaces are large and poorly behaved in the following ways. They do not have the T_0 or Kolmogorov property (Corollary REF ), they are not locally compact (Corollary REF ), and their topologica... | {
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9fd9a76b8e4efcfa7fe494641cdb4a731c0c11b0 | subsection | 3 | 61 | Persistence modules and persistence diagrams | Topological data analysis tends to focus on persistence diagrams rather than persistence modules. Readers more familiar with persistence diagrams may wonder why we work with persistence modules and what our results imply for persistence diagrams.Let us present three responses.
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fa10849b585f652e0becbf281ea1cf0c79e2bd09 | subsection | 4 | 61 | For the data scientist | For the reader primarily interested in topological data analysis, we would summarize our results by stating that the extended metric space of persistence diagrams with the bottleneck distance is “big”. Say we fix c<d and restrict ourselves to persistence diagrams with finitely many points (a_i,b_i) each of which satisf... | {
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1c009ea965ea6a00d8c3b7dd0755ec8e7e5c19f6 | subsection | 5 | 61 | Extended pseudometric spaces | The results presented here for extended pseudometric spaces are straight-forward extensions of the standard results for metric spaces
(Lemmas REF , REF , REF , and REF and Theorem REF ).
However, in order to keep the material accessible to applied mathematicians without a background in point-set topology, we include th... | {
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ecc339a56940bff3ac48274c939b0488ee42d31d | subsection | 6 | 61 | Related work | Mileyko, Mukherjee, and Harer consider the set of persistence diagrams with countably many points in \mathbb {R}^2 together with the topology induced by the p-Wasserstein distance for 1 \le p < \infty . They show that the subspace consisting of persistence diagrams with finite distance to the empty persistence diagram... | {
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eb23c9cb951b2a4a7687c3b095459345defe74f3 | subsection | 7 | 61 | Organization of the paper | In Section , we provide background on persistence modules, indecomposable modules, interleaving distance, and pseudometric spaces. In Section , we define the classes of persistence modules that we consider, study the relationships between them, and identify which of them are sets. In Section , we study the basic topolo... | {
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0c460fb9bbc19a9817a22dd2f03cd18340c1529d | subsection | 8 | 61 | Background | In this section we define persistence modules and interleaving distance, giving examples and basic properties. We also define extended pseudometric spaces and their induced topological spaces. | {
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fe55735143c34d079af4e04f234db357c6152fd4 | subsection | 9 | 61 | Persistence modules | Let \mathbf {k} be a fixed field.
A persistence module M is a set of
\mathbf {k}-vector spaces \lbrace M(a) \ | \ a \in \mathbb {R}\rbrace together with \mathbf {k}-linear maps \lbrace v_a^b: M(a) \rightarrow M(b) \ | \ a\le b \rbrace such thati)
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6eaba9f4072c2555128c508e716c71e1778afb84 | subsection | 10 | 61 | Persistence modules | For simplicity, we will abuse notation and denote this persistence module by [0,2).Example 2.3 Replacing [0,2) in the above with an arbitrary interval J \subset \mathbb {R} we obtain a persistence module that we call an interval module and we will also denote by J.Example 2.4 A trivial but important example is the zero... | {
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adc5e241ce97355b3a6d9637cf206392aa070de4 | subsection | 11 | 61 | Indecomposables | Given two persistence modules M and N, their direct sum is the persistence module M \oplus N given by (M \oplus N)(a) = M(a) \oplus N(a) and (M \oplus N)(a \le b) = M(a \le b) \oplus N(a \le b). In the same way we can define the direct sum of a collection of persistence modules indexed by an arbitrary set.A persistence... | {
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"raw": "Frédéric Chazal, Vin de Silva, Marc Glisse, and Steve Oudot. The structure and stability of persistence modules. SpringerBriefs in Mathematic... | 10.1007/s41468-018-0022-4 | 1802.08117 | Topological spaces of persistence modules and their properties | [
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27c0524b0f5f506e20920ea70b643a6ee48b30e7 | subsection | 12 | 61 | Interleaving distance | Interleaving distance was introduced in and further studied in the context of multiparameter persistence in . Here we also adopt the categorical point of view from .Definition 2.8
Let \varepsilon \ge 0.
An \varepsilon -interleaving between persistence modules M and N consists of morphisms
\varphi _a: M(a) \rightarrow... | {
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ab217da353148f8deffd36dd6467defe029356b8 | subsection | 13 | 61 | Interleaving distance | Then the interleaving distance d_I(M,N) between M and N is defined asd_I(M,N):= \inf \big (\varepsilon \in [0,\infty ) \ | \ M \ \text{and} \ N \ \text{are} \ \ \varepsilon \text{-interleaved} \ \big )If no such \varepsilon exists, then d_I(M,N)=\infty .Example 2.13
The interval modules [0,2] and (0,2) are not 0-inter... | {
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3997e17686899601116ee4537a135ed7fffc5586 | subsection | 14 | 61 | Pseudometric spaces | Definition 2.16 A pseudometric on a set X is a map d: X\times X \rightarrow [0,\infty ) that satisfiesM1)
d(x,x)=0,
M2)
d(x,y)=d(y,x), and
M3)
d(x,y)\le d(x,z)+d(z,y)for all x,y,z \in X.
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ab4508d56040dab92a81677bffb60ffb6345e19d | subsection | 15 | 61 | Pseudometric spaces | Note that in an extended pseudometric space we no longer have unique limits, but we do have that if x and x^{\prime } are limits, then by the triangle inequality d(x,x^{\prime }) = 0.A sequence (x_n)_{n \ge 1} in an extended pseudometric space is a
Cauchy sequence if for all \varepsilon >0 there exists an N>0 such that... | {
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43f7534ef73f254b55e7ebb08b0c016e4f484abb | subsection | 16 | 61 | Sets and classes of persistence modules | In this section we define classes of persistence modules that contain many of the persistence modules considered in the literature. We study the relationships between these classes and determine which of them are in fact sets.For the remainder of the paper, we will only consider isomorphism classes of persistence modul... | {
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} | 10.1007/s41468-018-0022-4 | 1802.08117 | Topological spaces of persistence modules and their properties | [
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5896452d1afbe4c2b5460b5dac0fcdc8647710a1 | subsection | 17 | 61 | Classes of persistence modules | In this section, we consider the classes of persistence modules in Figure REF , which we now describe.
[Figure: Hasse diagram of sets and classes of persistence modules.]\mathrm {(pm)} is the class of persistence modules.
\mathrm {(id)} is the class of interval-decomposable persistence modules: those isomorphic to \bi... | {
"cite_spans": []
} | 10.1007/s41468-018-0022-4 | 1802.08117 | Topological spaces of persistence modules and their properties | [
"Peter Bubenik",
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756b0324761c4b272e1025e44c038fbd6825c972 | subsection | 18 | 61 | Classes of persistence modules | A persistence module M is said to be constructible if there exists a finite subset A=\lbrace a_1,\ldots ,a_n\rbrace of \mathbb {R} such thatfor t< a_1, M(t)=0,
for a_i \le s \le t < a_{i+1}, M(s\le t) is an isomorphism
where i\in \lbrace 1,\ldots ,n-1\rbrace , and
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d13a0a8ff075c5a5e4ce3e9462754c282b434c28 | subsection | 19 | 61 | Inclusions | Lemma 3.2
Let M be an ephemeral module. Then M \cong \bigoplus _{\alpha \in A} M_{\alpha }, where each M_{\alpha } \cong [r,r] for some r \in \mathbb {R}.Let M \in \mathrm {(eph)}.
For r \in \mathbb {R}, let M_r be the persistence module with M_r(x) = M(r) if x=r and otherwise M_r(x)=0.
Then M \cong \oplus _{r \in \ma... | {
"cite_spans": []
} | 10.1007/s41468-018-0022-4 | 1802.08117 | Topological spaces of persistence modules and their properties | [
"Peter Bubenik",
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5a9214056fd1fd652038f2bb727490fafca06985 | subsection | 20 | 61 | Inclusions | \mathrm {(qtame)}\lnot \subset \mathrm {(id)}: \prod _{k=1}^{\infty }[0,\frac{1}{k}) is in \mathrm {(qtame)} but is not in \mathrm {(id)} . | {
"cite_spans": []
} | 10.1007/s41468-018-0022-4 | 1802.08117 | Topological spaces of persistence modules and their properties | [
"Peter Bubenik",
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bdc1fe4b45e3ff4fe5cd68ef57b55ea7cdd722a1 | subsection | 21 | 61 | Almost inclusions | Definition 3.4 Say that a class of persistence modules \mathcal {A} almost includes in a class of persistence modules \mathcal {B} if for each A \in \mathcal {A} there exists an element B \in \mathcal {B} such that d_I(A,B) = 0.Lemma 3.5
A finite sequence of inclusions and almost inclusions is an almost inclusion.This... | {
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"raw": "Frédéric Chazal, William Crawley-Boevey, and Vin de Silva. The observable structure of persistence modules. Homology Homotopy Appl., 18(2)... | 10.1007/s41468-018-0022-4 | 1802.08117 | Topological spaces of persistence modules and their properties | [
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