text:
Introduction

In Smart Cities, Wireless networks have progressed quickly in the past years. Considering the natural frequency spectrum’s constraints, it’s clear that the present stagnant frequency allocation techniques won’t be able to accommodate a wide range of new wireless services. Cognitive radio emerges as a solution to this issue of spectral overload by allowing secondary users to utilize the unused portions of licensed spectrum bands1. A cognitive radio, as a smart city wireless communication system, is cognizant of the radio frequency environment. It optimizes spectrum utilization by selecting communication characteristics (namely waveform, throughput, and power consumption). Spectrum sensing is among the most important aspects of cognitive radio technology. Attributed to the influence of geographic location on spectrum sensing, only one node’s sensing efficiency cannot be well attained, whereas multi-node CSS can resolve the sensing weaknesses of one node and make preparations by integrating information from various geographic locations2. As a result, in cognitive wireless networks, multi-node CSS is a widely utilized sensing technique that can successfully enhance perception reliability, but as the number of sensor nodes grows, so does the energy utilization of the cognitive wireless network, and because the front end of the cognitive network is a rechargeable battery, the energy utilization grows as well. Many enhanced ways are used to increase the lifetime of cognitive wireless networks3. The technique of flipping on and off time is being used to prolong the lifetime of the cognitive network and hence reduces the perception efficiency which is ineffective for the spectrum access. Therefore, to lessen the cognitive network workload this technique uses integration between the nodes and the primary users.

In4 for achieving the effectiveness, every secondary user doesn’t have to collaborate in the network, and secondary users with the strongest primary user’s signal-to-noise ratio (SNR) engage in spectrum sensing. In reduced SNR cognitive radio (CR) networks, the detecting efficiency for spectrum sensing is examined. However, this publication does not provide analytical formulations for their issues. The process in which the CR nodes choose the appropriate frequency band based on various parameters including the amount of time the spectrum is vacant or the available bandwidth is known as spectrum selection5. The process in which CR nodes negotiate communication link with some other CR users is known as spectrum sharing. Blockchain can help to make this spectrum utilization efficient, thus helping directly to support B5G and 6G applications and services. In the previous works, the main problem is that the sensor nodes used for spectrum sensing have restricted energy budgets. They are usually powered by batteries, that must be replenished or refilled (e.g., using solar power) when they run out. Both options are not feasible for some nodes, which means they will be eliminated after their energy supply is exhausted. In6 the nodes with low sensing capability would not aid cooperative sensing, but will have an impact on the fusion center’s overall judgment. As a result, nodes with inadequate sensing efficiency should be eliminated from cognitive wireless networks, and only the most consistent nodes should be chosen to engage in cooperative sensing. Nevertheless, the node’s efficiency is not static; it will fluctuate due to internal causes as well as changes in the external environment7. As a result, the role of technological changes must be included in the real cognitive wireless radio which thereby enhance the sensing efficient.

Problem definition

In this article, a security-based CSS using blockchain for smart cities is developed for a cognitive wireless network consisting of a FC and cognitive sensors for spectrum sensing in a defined time period, thereby, we explore the challenge of reducing energy utilization in CSS. The developed technique can adapt to changes in the environment and adjust the number of sensor nodes engaging in CSS in real time, as well as analyze the consistency of sensing nodes in real time and calculate the node’s trust value using an node evaluation algorithm. This technique not only recalls each node’s energy utilization and sensing efficiency, but also its trust value. The trust value is maintained in the blockchain’s consistency list, which is encrypted by the blockchain’s management center to verify that each node conforms to its own trust value without ambiguity. Later, a fusion rule is used by FC to make the ultimate decision on the channel’s occupancy. We employ the energy detector for channel sensing since it is easy to install and does not require existing knowledge of the primary user signal. The energy received on a licensed band is evaluated using this technique. When the level rises beyond a predetermined threshold, it is considered as recognizing primary user transmissions; else, a spectrum hole is verified. The SNR and distance between each node and the FC are assumed to be known.

Main contributions

The main contributions of the paper are explained as follows:

1. 1.

We have developed the secure spectrum sensing method for smart cities based on the performance of minimizing the energy utilization and accuracy of the blockchain-enabled cognitive wireless networks. This method will detect the probability of sensing all the nodes ensuring SNR which is computed by calculating the distance between the FC and each node.

2. 2.

The optimum result is based on cooperative sensing which changes the number of nodes participating and the method may adapt to changes in the environment and alter the number of sensor nodes engaged in cooperative sensing in real-time, as well as evaluate the actual dependability of sensing nodes, and the node’s trust value is determined using the node selection algorithm.

3. 3.

The optimum conditions acquired are based on the first-order necessary conditions, the algorithm that is developed to find the optimum nodes selection strategy.

4. 4.

To gain more energy efficiency, the method not only preserves each node’s energy utilization and sensing efficiency, but also its trust value. The trust value is maintained in the blockchain’s consistency list, which is encrypted by the blockchain’s management center to verify that each node conforms to its trust value without ambiguity.

5. 5.

The obtained simulations were utilized to evaluate the suggested algorithm’s performance in terms of finding the solution for the sensing nodes, energy utilization, and average number of sensing nodes.

The paper is represented as follows: section 2 demonstrates the background and motivation of the blockchain and CWN in blockchain; section 3 explains the design analysis of the proposed work; section 4 describes the flowchart for the evaluation and interpretation of nodes; section 5 shows the simulation results and evaluation of the performance; and lastly section 6 concludes the paper.

Background and motivation

Blockchain

Digital Ledger Technology (DLT) uses Peer-2-Peer (P2P) communication technologies to preserve records and transactions in a dispersed, decentralized way8. Data management and organizing in DLT can be accomplished in a variety of methods. Data can be maintained in a number of ways, including as a linear linked list of blocks or as a Directed Acyclic Graph (DAG)9. The term “blockchain” refers to the management of data is in the form of a linear linked list of blocks. One of the most distinguishing characteristics of blockchain is that it fully excludes the need of a trusted third party in the network’s maintenance10.

A “block” is a building block of a blockchain network where transactions are constructed. Figure 1 depicts a blockchain network with multiple blocks interconnected (each block contains numerous transactions). To prevent tampering, these transactions are put together in a block utilizing cryptographic methods11. The blocks are then combined together to form a blockchain12. This linkage can be done in a variety of ways. One simple method is to connect these blocks in a straight line. However, there could be concerns with sustainability, quick access to these blocks, and privacy.

Blockchain is also used from the context of spectrum licensing:

• Unlicensed Spectrum Band: Users can use the spectrum without paying any fees or obtaining permission in the unlicensed spectrum concept, enabling anybody to easily utilize any piece of the unlicensed spectrum. Users in the unlicensed spectrum band seek to work together to ensure some quality of service, limit harmful interference, and minimize channel contention. From a licensing standpoint, blockchain can provide a critical part in these spectrum-sharing concepts.

Cognitive wireless networks in blockchain for smart cities

Constraining spectrum availability only to licensed users is a very ineffective use of resources, as actual dimensions show that such resources are left unused for extended periods13. This observation, combined with the advancement of sophisticated nodes aware of exploring licensed spectrum and adapting transmission parameters correspondingly, inspired the concept of cognitive radio, in which the spectrum is made accessible to both licensed (also known as primary)users and unlicensed (secondary/cognitive) users who proactively obtain the sensed spectrum.

A cluster of neighboring nodes is formed in blockchain-enabled cognitive radio networks (CRN). These node groups are in charge of allocating radio resources within the group. Well within CRN, the nodes inside each group exchange information they expect to use14. This set of resources will be vetted among group members first, but once resources have been allocated without dispute, they will be disseminated and added to the blockchain. The CRN will be able to allocate resources efficiently in this manner. All nodes in the network will be notified of which resources are assigned to which nodes, lowering the likelihood of a CRN clash. Secondly, because blockchain offers a replica of the ledger to every node (in this case, the copy of the ledger will constitute available resources to devices) as well as every CR node would have a broad perspective of assigned network resources, the entire network nodes would be informed of assigned resources amongst these CR nodes15.

In this article, we have designed the architecture of blockchain-enabled CWN for Smart cities. The blockchain’s management center, licensed users, FC and cognitive users (various clusters or nodes) are all included in the technique. Some clusters or nodes are in the spectrum detecting mode, while others are in the sleep mode. A centralized CSS mechanism is used in this technique16. Only the FC and each node can share resources. Each node sensing data is delivered to the FC for evaluation, and the FC sends information back to each node. Since there is no immediate contact among nodes, a centrally controlled design aids in the efficient processing of data. The FC is accountable for not only communicating with the nodes but also with the blockchain management center, delivering node information to the management center for storing and having the blockchain management center notify the FC with node information.

The blockchain management center and the FC play a significant role in determining which nodes are reliable. Figure 2 shows that not all nodes engage in cooperative sensing at the same time while some of the nodes are nodes tampering17. Since the nodes are clustered and the node with the greatest trust value is chosen to engage in cooperative sensing, nodes’ tampering is excluded. The goal is to increase energy efficiency and extend the lifetime of the cognitive network. The energy utilization is lowered based on fulfilling the sensing efficiency.

Design analysis

In Fig. 3, we examine a network with C cognitive sensors and an FC. The frames time is denoted by F. We consider that almost all cognitive sensors have a similar spectrum sensing time, thus we employ $$\varphi$$, $$\varphi<0<F$$ to represent the CSS sensing time. For every user u, $$\forall$$ u $$\in$$ C, we presume that fn is the Nyquist frequency of the acquired signal from the licensed user18. $$\mu$$ is considered as the manifold of $$\frac{1}{fn}$$, therefore the total samples are taken as $$\varphi \frac{1}{fn}$$. Based on the observed scenario, as decided on the status every sensed node has $$A_u[m]$$ where $$m=1,2,3,\dots , \varphi fn$$.

In this, every observed scenario comprises two hypotheses. $$H_1$$, hypotheses signifies that licensed users are in the spectrum detecting mode and $$H_0$$ hypotheses signify licensed users are in the sleep mode.

\begin{aligned} H_1 \rightarrow A_u[m]= & {} n_u[m] + p_u[m] \end{aligned}
(1)
\begin{aligned} H_0 \rightarrow A_u[m]= & {} p_u[m] \end{aligned}
(2)

The signal of licensed users at the uth sensor is denoted as $$n_u [m]$$ and is considered to be the process done randomly with the 0th mean and variance $$\sigma ^2_{nu}$$. Therefore, the noise $$p_u[m]$$ is a gaussian, independent and similar dispersed random process with the 0th mean and the variance $$\sigma ^2_u$$. It is presumed that the independent processes are $$n_u[m]$$ and $$p_u[m]$$. Using hypothesis $$H_1$$, we represent u as the licensed user’s acquired SNR determined at the uth sensor. The possibility of identifying Id and possibility of triggering Tr are the possibilities of recognizing a licensed user under $$H_1$$ and $$H_0$$, respectively, in node sensing. A cognitive sensor can thus detect whether a node is active or sleep based on these possibilities19. An increased Id safeguards a licensed user broadcast from interruption with an unlicensed user broadcast, whereas a reduced Tr allows unlicensed users to access sleep nodes. As a result, a network with an increased Id and a reduced Tr is desired. The following is the decision rule used by the energy detection for the sensor u:

If $$H_0$$, then $$R_u = 0$$;

\begin{aligned} \in D_u = \frac{1}{\varphi fn} \sum ^{\varphi fn}_{m=1} A_{um}^2 < H_0 \end{aligned}
(3)

If $$H_1$$, then $$R_u = 1$$;

\begin{aligned} \in D_u = \frac{1}{\varphi fn} \sum ^{\varphi fn}_{m=1} A_{um}^2 > H_1 \end{aligned}
(4)

where $$\in$$ signifies all the cognitive sensors for detection of threshold and $$R_u$$ signifies the decision made by the node u. As a result, each user transmits a single bit to notify the FC of whether the licensed user signal is 1 or 0. The test constant $$D_u$$ is a random variable with a chi-square distribution with $$2\varphi fn$$ degrees of freedom during $$H_0$$, and a non-central chi-square probability with $$2\varphi fn$$ degrees of freedom and a non-centrality parameter $$2_{yu}$$ under $$H_1$$.

We are assuming the OR rule to be applied for the final decision. In other words, if either of the sensors detects an active licensed user, the final decision proclaims the channel to be congested. We suppose that every CR in the identical channel makes its own decisions. The final decision for Id and Tr are shown in Eqs. (5) and (6) as below:

\begin{aligned} Tr (\varphi ) = 1 - \Pi ^C_{u=1} (1- Tr_u (\varphi )) \end{aligned}
(5)
\begin{aligned} Id (\varphi ) = 1 - \Pi ^C_{u=1} (1- Td_u (\varphi )) \end{aligned}
(6)

It is demonstrated that spectrum sensing does not necessitate the participation of all the nodes2. For spectrum sensing, our method selects nodes with greater detection probability and shorter distances from the FC. As a result, we can change (5) and (6) with the Eqs. (7) and (8) below. In this $$\beta u \in {0,1}$$ which indicates the assignment index, 1 and 0 are used for sensing and not sensing the spectrum by the CSS respectively:

\begin{aligned} Tr (\varphi ) = 1 - \Pi ^C_{u=1} (1- \beta u Tr_u (\varphi )) \end{aligned}
(7)
\begin{aligned} Id (\varphi ) = 1 - \Pi ^C_{u=1} (1- \beta u Td_u (\varphi )) \end{aligned}
(8)

Proposed Method for Evaluation and Interpretation of nodes

It is crucial to use the node trust value as an essential measure to cooperate in CSS to enhance the safety of CWN. As a result, integrating the node trust value with the basic system design can improve sensing accuracy while reducing energy utilization. To avoid data ambiguity, the blockchain management center can be more efficient20. The proposed flowchart enhances the accuracy of sensing and performance of the CWN so, this method begins by estimating the system’s consistency. This estimate is based on accessible statistical information. When an accused node is identified, it generates an instantaneous decision to isolate the node’s sensing data. The method achieves the system’s resiliency but boosts energy utilization, and the effects of global variations on the node are not taken into account. The licensed user’s living conditions have an impact on node sensing. For example, when the licensed user’s location changes, nodes with strong sensing may turn mischievous in the next instant, whereas nodes with low performance become a trusted nodes. As a result, to detect modifications in node status, a real-time evaluation system for nodes must be established. When a node’s efficiency worsens, it can cease detecting work in real-time, and when it improves, it can be moved to work in real-time.

This article establishes an interpretation of nodes and an evaluation of the nodes method to determine and identify nodes more effectively. Before executing spectrum sensing procedures, the CWN determines the consistency of each node, which is based on scientific data. The original aim will continue working whenever the global environment is stable, but when the global environment changes, the node’s consistency must be re-evaluated. To prevent issues, the node’s trustworthiness level is computed using Eq. (9), and the FC creates a nodes list and transmits node data to the blockchain’s management center. The management center effectively delivers node data and is in charge of scheduling nodes to engage in cooperative sensing based on the fusion center’s needs.

\begin{aligned} y_u = \frac{\sum ^m_{a=1} |L_{u,a}| * l_{u,a}}{\sum ^m_{a=1} |L_{u,a}|} \end{aligned}
(9)

$$y_u$$ represents the starting trust value for the $$u_{th}$$ node, $$|L_{u,a}|$$ signifies the CSS in the ath cycle of sensing of the uth node, $$l_{u,a}$$ denotes the worth value acquired in the ath cycle of sensing of the uth node. When the $$l_{u,a} = 1$$; signifies the uth node in the ath cycle of sensing is reliable with the FC, and $$l_{u,a} = 0$$; signifies the uth node in the ath cycle of sensing is not reliable with the FC. The evaluation and interpretation of nodes achieved by Eq. (9) are used to store the value in the blockchain management center. The steps to evaluate and interpretation of nodes is explained as:

• Firstly, check whether the global environment has been modified, if yes, then re-evaluate the trust value of the nodes, otherwise, sensing nodes need not be modified.

• Then, FC will establish the list of nodes’ trust values.

• Later, the blockchain management center is accountable for managing and scheduling nodes.

• Further, adjust the number of sensing nodes and then call the nodes whose trust value is greater than the threshold value to engage in CSS.

Efficiency return value er, energy utilization return value eu, overall return value or, co-efficient of efficiency correction $$\rho$$, coefficient of energy utilization correction ec, and overall correction coefficient oc are the three return values and three correction coefficients set. These are computed in the given Eq. (10) for the Efficiency return value er:

\begin{aligned} er = \frac{1}{m} \sum ^m_{a=1} [(1-\beta _a)(\alpha _a * W_C + (1-\alpha _F)P_C)] + \beta _a(\gamma _a * W_C + (1-\gamma _a) * P_C) \end{aligned}
(10)

In the above Eq. (10), the value of $$\beta a$$ is either 1 or 0 which indicates that if its 1 means, the licensed user is in the sleep mode and 0 indicates the licensed user is in the active mode, u represents the same as given above, $$W_C$$ signifies the worth coefficient and $$P_C$$ signifies the unlawful coefficient. In this equation, $$\alpha _a$$ and $$\gamma _a$$ are the weighted coefficients which are represented in Eq. (11).

In $$H_0$$ $$\rightarrow$$ $$\alpha _a$$ $$=$$ 1, $$\beta _a$$ $$=$$ 0, $$H_1$$ $$\rightarrow$$ $$\alpha _a$$ $$=$$ 0, $$\beta _a$$ $$=$$ 0

\begin{aligned} In H_0 \rightarrow \gamma _a = 0, \beta _a = 1 \quad and \quad In H_1 \rightarrow \gamma _a = 1, \beta _a = 1 \end{aligned}
(11)

The representation for computation of energy utilization eu is shown in Eq. (12)

\begin{aligned} eu = \frac{1}{m} \sum ^m_{a=1} [E_WZ_a + E_P(1-Z_a)] \end{aligned}
(12)

where $$E_W$$ represents the worth energy utilization which says that the node used for energy utilization is lower than the threshold value; $$E_P$$ represents the punishable energy utilization which says that the node used for energy utilization is higher than the threshold value. $$Z_a$$ is the energy return value for the weighted coefficient and its value is denoted in Eq. (13) as: $$Z_a = 1$$, $$\tau _0$$, $$\sum ^{I_v}_{a,u}$$ = 0

\begin{aligned} Z_a = 0, \tau _0 - \sum ^{I_v}_{a,u} < 0 \end{aligned}
(13)

where $$\tau _0$$ represents the threshold of energy utilization in a sensing duration. The overall return value for energy utilization is computed in Eq. (14) as:

\begin{aligned} or = 0.3er + 0.7eu \end{aligned}
(14)

This equation describes that 30% of the weight is assigned for the energy utilization return value and the rest 70% of the weight is assigned for the efficiency return value. Thereby, authors have focused on the sensing efficiency while taking into consideration the minimizing of energy utilization. The equation for calculating coefficient of coefficient correction $$\rho$$ is:

\begin{aligned} \rho _u = \sum _a (\mu _{a,u} - \beta _a) \end{aligned}
(15)

The total number of repetitions the uth node communicates incorrect information to the FC is represented by the correction coefficient $$\rho _u$$; $$\mu _a$$,u shows that in the ath sensing cycle, the outcome provided by the uth node to the fusion center; $$\beta _a$$ reflects the outcome of the ath sensing cycle’s decision. The energy utilization correction coefficient is denoted by ec is computed in the Eq. (16).

\begin{aligned} ec_u = \sum _a S_{a,u} * J_{a,u} \end{aligned}
(16)

The total number of repetitions the uth node increases the value of threshold is represented by the energy utilization; the Eq. (17) shows the significance of $$J_{a,u}$$. In case $$J_{a,u} = 1$$ means $$v_{a,u} - \tau _0 >= 0$$ and $$J_{a,u} = 0$$ means $$v_{a,u} - \tau _0 < 0$$. $$\tau _0$$ means the threshold value has been raised for energy utilization and its value is computed where I is the total number of nodes present in the CWN; and $$v_{a,u}$$ tells that the energy utilized for the uth node in the ath sensing cycle is shown in Eq. (17).

\begin{aligned} \tau _0 = \frac{\sum ^I_{u=1} v_{a,u}}{I} \end{aligned}
(17)

The predefined value for the overall correction coefficient oc is shown in Eq. (18).

\begin{aligned} oc_u = 0.3\rho _u + 0.7ec_u \end{aligned}
(18)

$$oc_u$$ is the overall correction coefficient for the uth node which has been acquired by the total weighted count of the efficient and energy utilization correction coefficient. The efficiency has been evaluated by 30% and 70% for the efficient and energy utilization correction coefficient respectively. The trust value of the nodes are computed as shown in Eq. (19).

\begin{aligned} y_u^{r+1} = y_u^r + (\omega oc - (1-\varphi ) oc_u^r) y_u^r \end{aligned}
(19)

In the above Eq. (19), $$y_u^r$$ shows the nodes trust value in the ath sensing cycle for the uth node; $$y_u^{r+1}$$ shows the present nodes trust value for the $$u_{th}$$ node; $$oc_u^r$$ is the overall return value of the sensing cycle; the $$\varphi$$ denotes the value either 1 or 0. More the value of $$\varphi$$ gives better efficiency for energy utilization. The flowchart for the evaluation and interpretation of nodes is shown in Fig. 4.

The complexity of the proposed flowchart is O(I!) where O is denoted Big O Notation, I is the total number of nodes in the trust value. In the design flowchart, the main difficulty of blockchain-enabled CWN among the IoT devices shows that this article consists of the blockchain system, CWN’s and IoT devices. The FC is where users interact with the blockchain system. The IoT device provides node data to the FC, which searches the blockchain system for node data. The node then transmits verified by the private key to the FC, which validates if the sensing node has a matching private key pair. If that’s the case, send the node’s request to the blockchain system, and the blockchain system’s confirmation to the sensing node. The data verified by the sensing node can verify the identity of those taking part in CSS and guarantee that their message has not been tampered with. The steps to follow for the designing of CWN are:

• Firstly, check the sensing nodes in the CWN. It transmits the information of nodes and then requests for the identification of encryption to the FC.

• Secondly, examine the verification request to the blockchain management center.

• Thirdly, the blockchain management center returns the verification information to the FC and then returns the encrypted data to the sensing nodes of CWN.

Discussion

Let us assume that the CWN changes from quantity ten to sixty whose main objective is to select the best location of the sensor nodes and choose the sensing sensors regarding the limitations on the possibility of identifying Id and possibility of triggering Tr. Simulation outcome displays the $$\alpha = 1.5$$ and $$\gamma = 2.5$$ and acquired $$\alpha = 0.8$$ and $$\gamma = 1.2$$ in which all the nodes simulations outcome comes out to be over 2000 iterations. This experiment has been implemented in the solidity tool for the blockchain and the tests were This study involves a workspace is shown in Table 1 with two Intel Xeon Gold6 128 3.7GHz CPUs, 512 GB of memory, and a Panasonic S16GB SDHC Memory Card to run VMware ESXi 6.7.0 bare metal virtualized system. There are 8 auxiliary nodes and 30 sensor nodes in the modeled CWN. In this simulation environment, the total number of tampered users (TU) are described in two different cases: the first case, in case there are 5 TU’s then the SNR is − 16d B for 1 TU working under SNR; the second case, in case there are 10 TU’s then the SNR is taken as − 18d B for 3 TU’s working under SNR.

Authors have utilized the 5.2 MHz IEEE 802.15.4/ZigBee for the CSS communication which interacts with each other to sense the spectrum. For the performance evaluation, authors have assumed that the sensors that are located at the sensing field are 200m in length where the sensing nodes are uniformly dispersed and in the middle of CSS, FC is situated. In this article, while taking the evaluation and interpretation of nodes flowchart into account, the total number of TU’s are taken as 10. Therefore, to enhance the sensing efficiency of the nodes the comparison for various sensing has been discussed in this article.

The licensed users are based on the location which is placed on the edge of the circular area; the total number of nodes required is assumed as 30 (10 nodes having SNR = − 16 dB, − 18 dB and − 12 dB) respectively for all the nodes. The total TU’s are 3 when the SNR is − 18 dB. In the situation of four TU’s in CWN, the flowchart in this article’s identification Id is greater than the classical method under the same possibility of triggering Tr. That’s because the flowchart in this article takes into account the environment and other variables, and the CWN is more resilient and anti-attack competent. The node information may be monitored in real-time in any location, and blockchain technology encrypts the node information. The FC can request that high-performing nodes engage in CSS, ensuring that sensing performance is always improved.

In this article, various algorithms have been compared with the proposed work. In the Reduced energy Algorithm (REA), Sensors are ordered in increasing order as per their distances from the FC in this approach, and nodes with the shortest distances are chosen to sense the channel such that Id $$> \gamma$$ is fulfilled. There are fewer sensing nodes than I. If REA can identify nodes that meet the Id $$> \gamma$$ criterion, then its solution is close to optimum since it selects nodes with the shortest distances from the FC. In the Increased possibility of Identification Algorithm (IPIDA), Sensors are ordered in decreasing order as per their $$Id_u$$ in this flowchart. As a result, the nodes with the highest $$Id_u$$ are chosen, and the selection nodes are less than I, satisfying the $$Id_u > \gamma$$ condition. If there is a response for the algorithm, IIPDA discovers it. The randomized sensor choosing algorithm (RSCA) selects sensors for CSS at random, with the number of choosing nodes being less than I. This algorithm is the simplest to use to identify a solution for the problems.

The REA algorithm when the number of nodes rises, the energy consumed in the network is minimized because the nodes comes closer to one another and their distance will be decreased while transferring the data as shown in Fig. 6. But on the other side IIDPA uses the less number of nodes. Our proposed algorithm and REA are closer to the IIDPA which displays that our proposed algorithm consumes less energy while maximising the number of nodes.

Figure 5 shows the success rate of finding a solution for several methods that meet $$Id > \gamma$$. This statistic for each method demonstrates the algorithms’ capacity to locate the answer when the problem conditions are met. It is shown that the REA method is compatible with the IIPDA algorithm, implying that if the problem has a solution, our proposed scheme can locate it. On the other hand, probably, REA and RSCA will not discover a solution to the issue while it has one.

The average number of sensing nodes for various methods is displayed in Fig. 6. IIPDA employs the smallest number of nodes possible. IIPDA and our algorithm and REA are extremely similar. This shows that our approach uses the least amount of energy while keeping $$Id > \gamma$$ constant. The RSCA algorithm uses the most sensing nodes.

Figure 7 shows that when the number of sensing cycles surpasses 60, the energy consumption of the method in this paper is much lower than the other three main techniques. The more sensing cycles there are, and the longer the sensing duration is, the more energy is saved. CWN will be greatly aided in extending the operating life. That’s because the methodology in this work can always choose the node with the best performance to participate in CSS in real-time, saving more energy than the conventional method when sensing efficiency is preferable.

Conclusion

This paper examines the concept of CWN for smart cities in detail. CSS sensing collaboration is critical for limiting the effects of shadowing and fading, as well as accurate sensing. Nevertheless, as the number of users grows, we spend lots of time observing, analyzing, and transmitting data. Furthermore, expanding the amount of spectrum sensing users does not enhance the system’s detection accuracy exponentially. As a result, it is not required to use all users at all times for spectrum sensing. So, when nodes sense data in practical implementation situations of CWN, there are generally major mistakes that cause the sensing values to depart from the normal range, or some nodes intentionally relay the erroneous data to the FC. As a result, this study offers the Evaluation and interpretation of nodes flowchart to address the security challenge of TU’s attack in CWN for smart cities. In this article, the authors have made a comparison with various existing algorithms based on the percentage of finding the solution of sensing nodes, the average number of sensing nodes and energy utilization. It has been found that the proposed methodology achieves 10% sensing efficiency in finding the solution for identification and triggering possibilities with the value of $$\alpha = 1.5$$ and $$\gamma = 2.5$$ with the varying number of nodes.


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## Geometry & Topology RTG

Previously this week (week of August 3rd) I was able to attend the Geometry & Topology RTG workshop at the University of Notre Dame. The workshop was a week long event consisting of two parts, I and II, the first being an introduction into geometry & topology, and the latter being lectures on more advanced topics including student presentations. I’ve included their website link here. I attended part II and thought I’d speak about my experiences.

#### Closed Geodesics

This talk was given by post-doc G. Szekelyhidi. One starts with a closed Riemann surface $M \subset \mathbb{R}^{3}$. We call a smooth non-trivial loop $r:[0,1] \to M$ such that $r'(0) = r'(1)$ a closed geodesic if it is a map of minimal length or equivalently a critical point of the length functional; think a non-squiggly curve at small scales. The goal is to deduce if there are closed geodesics in $M$. Szekelyhidi broke the argument into two cases, either $M$ is simply-connected or it is not. If not, then we have the following well-known theorem:

Theorem: If $\pi_{1}(M) \ne \{1\}$, then every non-trivial homotopy class contains a closed geodesic.

The proof is essentially finding the shortest loop in each homotopy class and we know such a loop exists because on $M$ there always exists some length $L$ such that if the length of a loop $r$ is less than $L$, written $\ell(r) < L$, then $r$ must be homotopic to the identity. For example, any loop less than length $2\pi$ in the torus cannot go around length-wise or width-wise and so must be trivial). In the picture below $C_{2}$ is homotopic to the identity while $C_{1}$ is not:

To find the shortest loop we apply curve shortening flow, which is a process that minimizes the curvature everywhere on the loop in a continuous manner without affecting the homotopy class of the loop. For example, curve shortening flow on the loop

produces the loop

On the other hand, if $M$ is simply-connected then all loops are homotopic to the identity and we need to work a little harder. Instead of considering classes of loops we consider loops of loops commonly called sweepouts. Formally, a sweepout is a loop $\Phi:[0,1] \to \Omega M$ into the space of all loops on $M$ (not homotopy classes of loops) such that the induced map $S^{2} \to M$ viewing $S^{2} \cong [0,1] \times S^{1}/\sim$ is not trivial. We can think of a sweepout as taking a non-trivial loop in $M$ and pulling it over $M$ so it traces out $M$ in terms of the loop. For example, a few loops in a sweepount on the sphere may look like:

We then define $W(\Phi) = \max_{s \in [0,1]}\ell(\Phi(s))$ and try to find the sweepout minimizing width because this sweepout will contain the geodesic. In fact, Birkhoff proved this in 1917.

Theorem (Birkhoff 1917): There exists closed geodesics in $M$.

Current research aims to generalize this idea to higher dimensional manifolds where we are now looks for surfaces of minimal area, $3$-manifolds of minimal volume, etc. In fact, in 2016 and again in 2018 it was shown (by Sang and Marques-Neves respectively) that if $M$ is a closed Riemann $3$-manifold, then it has infinitely many minimal surfaces.

#### Relating Topology and Geometry of Manifolds

This talk was given by Notre Dame professor S. Stolz. He started with a short introduction about the differences between topology and geometry. His view, was that topologists are interested in qualitative aspects of spaces such as the number of connected components, holes, punctures, and twists. While on the other hand, geometers are more interested in quantitative aspects of spaces such as the curvature, area, length, and area. This viewpoint is quite nice because it also alludes to the fact that most homeomorphisms (mapping that preserve topological structure) are often very far off from being Riemannian isometries (a smooth homeomorphism which respects the Riemannian metric between manifolds), that is qualitative aspects being preserved doesn’t imply the quantitative ones are as well. However in this viewpoint the converse is true because every Riemannian isometry is in particular a homeomorphism.

The next piece of his talk introduced the Euler characteristic of a surface $M$. Its obtained, in a crude manner, by putting a pattern of of polygons $\Gamma$ on $M$, defining $v(\Gamma)$, $e(\Gamma)$, and $f(\Gamma)$ to be the number of vertices, edges, and faces in $\Gamma$ respectively, and then defining the Euler characteristic $\chi(M)$ by the alternating sum $\chi(M) = v(\Gamma)-e(\Gamma)+f(\Gamma)$. It turns out that $\chi(M)$ is independent of the type of patterning and that it is topologically invariant, so homeomorphic manifolds have the same Euler characteristic. This gives a nice method for determining if two surfaces are not homeomorphic, and if we assume the surfaces $M$ and $N$ are compact orientable then they are homeomorphic if and only if they have the same Euler characteristic.

Stolz then shifted gears to talk a little about Riemannian manifolds: smooth manifolds $M$ with an inner product $g_{p}$ on the tangent space $T_{p}M$ such that the inner product varies smoothly with respect to $p$. With this inner product we have a way to measure lengths of curves, distance between points, and areas in $M$ analogous to how such quantities are measured in calculus. More importantly, we have a notation of scalar curvature at $x \in M$ (up to normalization) which is defined to be $-3(n+2)$ times the second derivative with respect to $r$ evaluated at $r = 0$ of $\text{vol\,}B_{r}(x,M)/\text{vol\,}B_{r}(0,\mathbb{R}^{2})$ (this is a measure of how fast the manifold is curving away or towards $x \in M$). Varying $x$ over all of $M$ we get a function $sc:M \to \mathbb{R}$ called the scalar curvature function. Incredibly, this relates to the Euler characteristic of $M$ in the following manner:

Theorem (Gauss-Bonnet): If $M$ is a compact orientable surface, then $\int_{M}s(x)\,dx = 4\pi\chi(M)$.

As an interesting corollary, it can be shown that the sphere is the only compact orientable surface admitting a positive Euler characteristic (namely $2$). Hence by Gauss-Bonnet it is the only compact orientable surface that can possibly admit an everywhere positive scalar curvature function (and in fact it does).

#### Some Soft Comments

Overall I really enjoyed the level of mathematics the workshop was at. I understood a majority of the content, and for the the things I didn’t quite understand I either had a good sense of intuition for them or I had someone else give me more detail after the talk. There were quite a few more talks then the two presented here by the way (seven professor/post-doc talks and twelve student presentations). Between talks there was a $30$ minute break intended to be a time to discuss the talk in greater detail, ask questions, and chat with other members of the conference. However, there was really no external indication that a workshop was taking place (no direction signs, banner, etc.) and that gave a feeling of “if I was in the local area and didn’t get invited I could of just shown up and no one would of minded”. I would have also preferred if there was some type of lunch session where invitees could go grab food with some of the speakers and talk about their research and interests in a less formal setting. Perhaps these things occurred in Part I and not in Part II, but I would have preferred them nonetheless. Overall, it was a good experience and I’ll be applying again next summer.

By the way, this is a neat shot my roommate got of me after the conference when I was playing around with some Hagoromo chalk. If you’re wondering what the math on the blackboard is, it’s diagrams for operad multiplication of strings.

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# Detect triangles within triangulation in 3D

I have a triangulated polygon(lies approximately on a single plane). I know the vertices and the edges(polygon edges not the triangulation edges), but i dont know the exact order of them(but it probably can be found).

Inside the polygon additional smaller polygons will be added and re-triangulated. I need a way to detect the colored triangles.

I made an error here, the red areas are segmented into quads, but they are supposed to be segmented into triangles.

I know that one solution would be to do a sweepline across the triangulation and then use the edges to see if its inside or outside. But is there any easier methods? Since the polygons are in 3D space its hard to do a sweepline.

This is the polygonal area that is know to me. And the colored area needs to be found.

So given the knowledge of the edges(in no particular order), vertices and vertices that make up the inner added polygonal shapes, like in the last image, is there a way to detect what triangles are within the added polygons?

• If you are in 3D, can't you change the coordinates and the the plane of the figure to become the $x-y$ plane ? – G Cab Mar 26 '18 at 18:07
• @GCab yes i could do the change of basis vectors. But is there a way to detect the areas with out a sweepline? – Marko Taht Mar 26 '18 at 18:13
• In your third pic, one of the red regions is a quad, not a triangle. – John Hughes Mar 26 '18 at 19:52
• When you say "given the knowledge of the edges, ...", can you say in what form we "know" them? Are we given the three (or more) black polylines as an ordered list, where each edge shares a vertex with the previous edge, etc.? Are these lists oriented (i.e., is the outer black cycle oriented clockwise, while the inner hexagon is oriented counterclockwise, and the inner triangle is oriented...in some direction? And are we given a list of all mesh triangles as unordered triples of vertices? – John Hughes Mar 26 '18 at 19:55
• Triangles are given as separate structures that have 3 vertices and 3 edges. Edges re shared between triangles, so i can ask for neighbors of a triangle. – Marko Taht Mar 27 '18 at 6:23

In $n-$D a point $P$ is internal to a segment $AB$ iff $$P = t\,A + s\,B\quad \left| {\;\left\{ \matrix{ \;0 \le t,s \le 1 \hfill \cr \;t + s = 1 \hfill \cr} \right.} \right.$$

in $n-$D a point $P$ is internal to a triangle $ABC$ iff $$P = t\,A + s\,B + r\,C\quad \left| {\;\left\{ \matrix{ \;0 \le t,s,r \le 1 \hfill \cr \;t + s + r = 1 \hfill \cr} \right.} \right.$$ i.e. if it is the weighted average of the vertices of the triangle, with the weights non-negative and summing to $1$.

IIn $2-$D we can write the above in matrix notation as $$P = \left( {\matrix{ x \cr y \cr } } \right) = \left( {\matrix{ {x_{\,a} } & {x_{\,b} } & {x_{\,c} } \cr {y_{\,a} } & {y_{\,b} } & {y_{\,c} } \cr } } \right)\left( {\matrix{ t \cr s \cr r \cr } } \right)$$ and using the fact that $r+s+t=1$ we can rewrite the above using a square matrix $$\left( {\matrix{ x \cr y \cr 1 \cr } } \right) = \left( {\matrix{ {x_{\,a} } & {x_{\,b} } & {x_{\,c} } \cr {y_{\,a} } & {y_{\,b} } & {y_{\,c} } \cr 1 & 1 & 1 \cr } } \right)\left( {\matrix{ t \cr s \cr r \cr } } \right)\quad \Rightarrow \quad \left( {\matrix{ t \cr s \cr r \cr } } \right) = \left( {\matrix{ {x_{\,a} } & {x_{\,b} } & {x_{\,c} } \cr {y_{\,a} } & {y_{\,b} } & {y_{\,c} } \cr 1 & 1 & 1 \cr } } \right)^{\, - \,1} \left( {\matrix{ x \cr y \cr 1 \cr } } \right)$$

We recognize in the above that it implies the use of homogeneous coordinates, and that the determinant of the matrix equals the area of the triangle, and therefore it is non-null and the matrix is invertible, unless the triangle is degenerated.

We are employing a Barycentric system.

So, to check if a point is internal to $\triangle {ABC}$, we multiply its homogeneous coordinates by the inverted matrix as above and verify that the resulting vector of parameters have non-negative components.

Concerning the arrangement of the points that you have into a polygon, let's consider first three points $A, B, C$ out of which we construct a triangle as per above. Then let's take a fourth point $P$.

With the above method we can determine if $P$ is external or internal to $\triangle {ABC}$. If it is external, one or two of its barycentric coordinates are negative.

When only one of them is negative, $s$ in the example shown in the drawing, then we know that $P$ is opposite to $B$ wrt $AC$ and we can (supposedly) add the triangle $\triangle {PAC}$.

But, if two coordinates are negative, then we have a double choice as for the triangle to add. Suppose e.g. that also $t$ be negative and $P=(10,3)$ in the $x-y$ reference. Then the possible additional triangles would be $\triangle {PBC}$ and $\triangle {PAC}$, and it is as well possible that $\triangle {ABC}$ be actually excluded.

Then, when $P$ is internal to $\triangle {ABC}$, the choice is triple.

In case of multiple choices, we cannot decide unless we have additional information.
If I understood properly, you have the list of the edgeswhich are part or not of the polygon to build. Then the choices become obvious, except for the case of a triangular hole in the polygon.
In this last case I do not know what additional information you have, that might help to decide whether the triangle is internal to the polygon or is to be excluded.

• This tests if the point is within the triangle. But how would you use it to determine if the triangle is within a polygon? – Marko Taht Mar 27 '18 at 6:28
• @MarkoTaht: I added a 2nd part to my answer. But it is possible that I did not catch clearly which is your problem. – G Cab Mar 27 '18 at 13:25
• I think i kinda understand where you are trying to go with this. But to clearify. My goal is to remove the colored triangles from the overall triangulation. – Marko Taht Mar 27 '18 at 17:10

Here's a strategy using just a little bit of graph theory.

TL;DR: from the dual graph of the triangulation, delete all edges corresponding to the "(boundary) edges of the inner polygonal shapes"; then compute connected components.

Details:

Step 1: Build a graph structure, $G$, with one vertex for each triangle of your triangulation, and an edge between $v_1$ and $v_2$ exactly when the corresponding triangles share an edge. Thus if the vertex $v_1$ corresponds to triangle $(1, 4, 5)$ and $v_2$ corresponds to triangle $(4, 7, 5)$, then we include an edge between $v_1$ and $v_2$ in the graph $G$.

(Intuitively: put a dot in the middle of each triangle in your triangulation; draw an edge between corresponding dots if the triangles share an edge.)

Step 2: when the edge between two triangles is one of your boldface black edges (a divider between red and white, one of the things you described as "edges, vertices, and vertices that make up the inner polygonal shapes", or more intuitively, "the edges that you drew in your last picture"), mark the corresponding edge in $G$. Each edge of $G$ is now either marked or unmarked.

Step 3: starting at any node of G, perform a depth-first traversal, attaching a boolean label to each vertex you encounter.

a. Label the first vertex "t" (for "true").

b. As you traverse edges, if the edge you traverse is unmarked, use the same label for the newly-encountered vertex as for the one from which you're traversing; if the edge you traverse is marked, and the starting vertex has label $b$, then label the new vertex with "not $b$".

When you're done, all the vertices with one boolean label will correspond to triangles in the original mesh with one color; all those with the other label will correspond to triangles in the original mesh with the other color.

Which one is which? Who knows!? If your input mesh consisted of a single triangle, there's no way to know whether it should be white or red, from the data you've given us. So I leave that final determination -- the color of one particular triangle -- to you.

Just to make that last claim clear, I'm going to start with your last example, and make step by step edits that don't alter the nature of the problem in any essential way, and arrive at a situation where the solution is self-evidently ambiguous:

The upper left is a region with two interior regions, very similar to your final example; I simplify by removing one of them; I simplify again by removing an inner region from the other. Then (upper right) I move some of the vertices on the bottom edge, and insert a bottom vertex. Then I continue simply moving vertices until several line up, and delete those. I arrive at a mesh consisting of two triangles, with the edge between them being one where we change from white to red. But the same input could correspond to the opposite coloring, where the bottom triangle is red and the upper is white, by symmetry. That's why my algorithm can't tell you which is which: your data doesn't contain that information. But I can tell you all the triangles in the same color group.

• uuh.. i like this. im gona try this. THe color i think can actually be determinedrather easily, if the triangle is connected to the outer perimeter(marked edge) but on the other side there is no triangle. That should mean that this triangle must remain. Using that i think this approach can work rather well :). If this works il mark this as an asnwer. – Marko Taht Mar 28 '18 at 7:08


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# nLab F4

Contents

## Philosophy

group theory

### Cohomology and Extensions

#### Lie theory

Background

Smooth structure

Higher groupoids

Lie theory

∞-Lie groupoids

∞-Lie algebroids

Formal Lie groupoids

Cohomology

Homotopy

Related topics

Examples

$\infty$-Lie groupoids

$\infty$-Lie groups

$\infty$-Lie algebroids

$\infty$-Lie algebras

# Contents

## Idea

One of the exceptional Lie groups.

## Definition

###### Definition/Proposition

(Jordan algebra automorphism group of octonionic Albert algebra is F4)

The group of automorphism with respect to the Jordan algebra structure $\circ$ on the octonionic Albert algebra is the exceptional Lie group $F_4$:

$Aut\left( Mat_{3\times 3}^{herm}(\mathbb{O}), \circ \right) \;\simeq\; F_4 \,.$

(e.g. Yokota 09, section 2.2)

• G2, F4,

E6, E7, E8, E9, E10, E11, $\cdots$

## References

### General

Cohomological properties:

• Y. Cchoi, S. Yoon, Homology of the triple loop space of the exceptional Lie group $F_4$, J. Korean Math. Soc. 35 (1998), No. 1, pp. 149–164 (pdf)

### In string theory

That the group $F_4$ controls the massless degrees of freedom of 11-dimensional supergravity was observed and explored in

Last revised on May 14, 2019 at 04:51:19. See the history of this page for a list of all contributions to it.


text:
#### Howdy, Stranger!

It looks like you're new here. If you want to get involved, click one of these buttons!

Options

# This week's progress

I've been writing weekly progress reports to my grad students. It may make sense to copy them here. Let me give it a try.

• Options
1.

17 November 2016:

Hi -

Daniel Cicala points out the math conference December 3rd and 4th at Union College in upstate NY. There will be talks on category theory organized by Susan Niefeld - she does this every year, and I've spoken there once. If you can go, do it!

(If you read the category theory mailing list you can stay up to date on such conferences.)

Here is this week's progress:

1) Blake and I went to San Diego and learned about Metron's "ExAMS" software for designing complex systems. Then John Foley came up and, with help from Joseph Moeller, we figured out a bunch of stuff.

This software raises lots of interesting questions. I believe it's based on "timed hierarchical colored Petri nets with guards". I would like to make sure this is true, and understand this kind of network category-theoretically. In case anyone wants to help me, here's an intro:

When we met, Tom Mifflin at Metron seemed pretty eager for our work to go in this direction.

2) I went to the Mathematical Association of America conference and gave a talk on The answer to the ultimate question of life, the universe and everything. Brandon and Daniel also went there.

3) I finally blogged about Brendan's thesis:

As you can see, it's a lazy blog article - yet it still manages to give a detailed introduction to his work! If you haven't yet learned everything that Brendan is doing, this is a good place to start.

Comment Source:17 November 2016: Hi - Daniel Cicala points out the [math conference December 3rd and 4th at Union College](http://www.math.union.edu/%7Etoddg/ucc/) in upstate NY. There will be talks on category theory organized by Susan Niefeld - she does this every year, and I've spoken there once. If you can go, do it! (If you read the category theory mailing list you can stay up to date on such conferences.) Here is this week's progress: 1) Blake and I went to San Diego and learned about Metron's "ExAMS" software for designing complex systems. Then John Foley came up and, with help from Joseph Moeller, we figured out a bunch of stuff. I blogged about ExAMS here: * [Complex Adaptive System Design (Part 2)](https://johncarlosbaez.wordpress.com/2016/10/18/complex-adaptive-system-design-part-2/) This software raises lots of interesting questions. I believe it's based on "timed hierarchical colored Petri nets with guards". I would like to make sure this is true, and understand this kind of network category-theoretically. In case anyone wants to help me, here's an intro: * Wil M. P. van der Aalst, Christian Stahl, and Michael Westergaard, [Strategies for modeling complex processes using colored Petri nets](http://wwwis.win.tue.nl/%7Ewvdaalst/publications/p710.pdf). When we met, Tom Mifflin at Metron seemed pretty eager for our work to go in this direction. 2) I went to the Mathematical Association of America conference and gave a talk on [The answer to the ultimate question of life, the universe and everything](http://math.ucr.edu/home/baez/42/). Brandon and Daniel also went there. 3) I finally blogged about Brendan's thesis: * [Open and interconnected systems](https://johncarlosbaez.wordpress.com/2016/10/23/open-and-interconnected-systems/). As you can see, it's a lazy blog article - yet it still manages to give a detailed introduction to his work! If you haven't yet learned everything that Brendan is doing, this is a good place to start.
• Options
2.

23 November 2016:

Lots of progress this week:

1) Jason uploaded his thesis to the arXiv! That's great, because it establishes priority - now he can prove he did this stuff before anyone else, even if it takes a while for him to publish a paper.

2) Blake gave a talk about open chemical reaction networks at the Santa Fe Institute workshop on Statistical Mechanics, Information Processing and Biology:

It was a real hit. Susanne Still said it was "great" - she works on Markov processes and the information bottleneck method for predictive inference. Jim Crutchfield liked it so much he invited Blake to give a talk up at U. C. Davis! He's a real bigshot: the Wikipedia article on him says

Over the last three decades Prof. Crutchfield has worked in the areas of nonlinear dynamics, solid-state physics, astrophysics, fluid mechanics, critical phenomena and phase transitions, chaos, and pattern formation. His current research interests center on computational mechanics, the physics of complexity, statistical inference for nonlinear processes, genetic algorithms, evolutionary theory, machine learning, quantum dynamics, and distributed intelligence. He has published over 100 papers in these areas.

If Blake can strike up a relationship with Crutchfield and maybe work on a project, that'll be excellent.

3) Joshua Tan, a grad student at Oxford (and friend of Brendan), invited me to join a bunch of people in writing a grant proposal.

It's for an NSF grant called "Smart & Connected Communities", and part of the plan would be to model cities as composable, open systems using category theory. Here are the other people involved in writing the proposal:

• Dennis Frenchman is a professor at MIT and an expert in building digital tools for cities. He is the likely PI.

• Sokwoo Rhee is a director at NIST managing 100+ smart cities projects and will be collaborating directly with us, but he is a silent partner due to federal rules.

• Stephen Walter is a program director at the City of Boston, Mayor's Office of New Urban Mechanics.

• Matthew Claudel is a student of Dennis' who has been working with me to write the proposal. His research is in urban innovation.

• Possibly: Eric Gordon, a professor at the Engagement Lab at Emerson College, who works on civic participation.

• Other personnel attached to the project include Elizabeth Christoforetti (Harvard, MIT urban planner) and Nissia Sabri (startup, hardware specialist).

This would be a great step toward my ultimate goal: using network theory for studying complex systems like biological systems and designing systems to deal with climate change.

Comment Source:23 November 2016: Lots of progress this week: 1) Jason [uploaded his thesis to the arXiv](https://arxiv.org/abs/1611.07591)! That's great, because it establishes priority - now he can prove he did this stuff before anyone else, even if it takes a while for him to publish a paper. 2) Blake gave a talk about open chemical reaction networks at the Santa Fe Institute workshop on [Statistical Mechanics, Information Processing and Biology](http://www.santafe.edu/gevent/detail/science/2452/): * [Compositional frameworks for open systems](https://johncarlosbaez.wordpress.com/2016/11/27/compositional-frameworks-for-open-systems/). It was a real hit. [Susanne Still](http://www2.hawaii.edu/%7Esstill/) said it was "great" - she works on Markov processes and the information bottleneck method for predictive inference. [Jim Crutchfield](http://csc.ucdavis.edu/%7Echaos/) liked it so much he invited Blake to give a talk up at U. C. Davis! He's a real bigshot: the Wikipedia article on him says > Over the last three decades Prof. Crutchfield has worked in the areas of nonlinear dynamics, solid-state physics, astrophysics, fluid mechanics, critical phenomena and phase transitions, chaos, and pattern formation. His current research interests center on computational mechanics, the physics of complexity, statistical inference for nonlinear processes, genetic algorithms, evolutionary theory, machine learning, quantum dynamics, and distributed intelligence. He has published over 100 papers in these areas. If Blake can strike up a relationship with Crutchfield and maybe work on a project, that'll be excellent. 3) [Joshua Tan](http://www.joshuatan.com/research/), a grad student at Oxford (and friend of Brendan), invited me to join a bunch of people in writing a grant proposal. It's for an NSF grant called "[Smart & Connected Communities](https://www.nsf.gov/pubs/2016/nsf16610/nsf16610.htm#pgm_desc_txt)", and part of the plan would be to model cities as composable, open systems using category theory. Here are the other people involved in writing the proposal: * Dennis Frenchman is a professor at MIT and an expert in building digital tools for cities. He is the likely PI. * Sokwoo Rhee is a director at NIST managing 100+ smart cities projects and will be collaborating directly with us, but he is a silent partner due to federal rules. * Stephen Walter is a program director at the City of Boston, Mayor's Office of New Urban Mechanics. * Matthew Claudel is a student of Dennis' who has been working with me to write the proposal. His research is in urban innovation. * Possibly: Eric Gordon, a professor at the Engagement Lab at Emerson College, who works on civic participation. * Other personnel attached to the project include Elizabeth Christoforetti (Harvard, MIT urban planner) and Nissia Sabri (startup, hardware specialist). This would be a great step toward my ultimate goal: using network theory for studying complex systems like biological systems and designing systems to deal with climate change.
• Options
3.
edited December 2016

28 November 2016:

1) Daniel Cicala passed his oral exam today! He spoke about this paper that he put on the arXiv last week:

Abstract. We introduce the notion of a span of cospans and define, for them, horizontal and vertical composition. These compositions satisfy the interchange law if working in a topos C and if the span legs are monic. A bicategory is then constructed from C-objects, C-cospans, and doubly monic spans of C-cospans. The primary motivation for this construction is an application to graph rewriting.

2) Tobias Fritz is visiting us! He'll be speaking in the network theory seminar tomorrow and also joining our group meeting on Wednesday at 11 am. Here's his talk:

Abstract. The problem of causal inference is to determine if a given probability distribution on observed variables is compatible with some hypothetical Bayesian network structure. In the presence of hidden nodes (unobserved variables), this is a challenging problem for which no exact methods are known. The inflation technique of http://arxiv.org/abs/1609.00672 provides a new practical tool for approaching this problem. It has the potential to be generalized to other kinds of networks, in particular those that live in semicartesian monoidal categories.

3) My former grad student Chris Rogers will be giving a special seminar on symplectic stuff on Thursday 3:40-5:00, either in the Undergraduate Study Room or in some better room like room 284 or 268 - it's not exactly clear, but I'll try to inform you when I find out.

It will be very good for Brandon and Adam to attend this, since they're doing symplectic stuff. However, Chris will blow them out of the water with his erudition.

• From Hamiltonian mechanics to homotopy Lie theory

Abstract. In Hamiltonian mechanics, physicists model the phase space of a physical system using symplectic geometry, and they use Lie algebras to describe the space's infinitesimal symmetries. Given such a Lie algebra of symmetries, the geometry naturally produces a new Lie algebra called a "central extension''. This central extension plays a crucial role, especially in quantum mechanics. The famous Heisenberg algebra, for example, arises precisely in this way.

In this talk, I will explain how the above recipe can be enhanced to geometrically produce examples of "homotopy Lie algebras''. A homotopy Lie algebra is a topologist's version of a Lie algebra: a chain complex equipped with structures which satisfy the axioms of a Lie algebra only up to chain homotopy. They provide important tools for rational homotopy theory and deformation theory. The homotopy Lie algebras produced from our construction turn out to have interesting relationships with the theory of loop groups and what are called "string structures'' in algebraic topology

Comment Source:28 November 2016: 1) Daniel Cicala passed his oral exam today! He spoke about this paper that he put on the arXiv last week: * [Spans of cospans](https://arxiv.org/abs/1611.07886). > **Abstract.** We introduce the notion of a span of cospans and define, for them, horizontal and vertical composition. These compositions satisfy the interchange law if working in a topos C and if the span legs are monic. A bicategory is then constructed from C-objects, C-cospans, and doubly monic spans of C-cospans. The primary motivation for this construction is an application to graph rewriting. 2) Tobias Fritz is visiting us! He'll be speaking in the network theory seminar tomorrow and also joining our group meeting on Wednesday at 11 am. Here's his talk: * [Inferring hidden network structure: the case of causal inference](https://simons.berkeley.edu/talks/tobias-fritz-12-06-2016). > **Abstract.** The problem of causal inference is to determine if a given probability distribution on observed variables is compatible with some hypothetical Bayesian network structure. In the presence of hidden nodes (unobserved variables), this is a challenging problem for which no exact methods are known. The inflation technique of [http://arxiv.org/abs/1609.00672](http://arxiv.org/abs/1609.00672) provides a new practical tool for approaching this problem. It has the potential to be generalized to other kinds of networks, in particular those that live in semicartesian monoidal categories. 3) My former grad student Chris Rogers will be giving a special seminar on symplectic stuff on Thursday 3:40-5:00, either in the Undergraduate Study Room or in some better room like room 284 or 268 - it's not exactly clear, but I'll try to inform you when I find out. It will be very good for Brandon and Adam to attend this, since they're doing symplectic stuff. However, Chris will blow them out of the water with his erudition. * From Hamiltonian mechanics to homotopy Lie theory > **Abstract.** In Hamiltonian mechanics, physicists model the phase space of a physical system using symplectic geometry, and they use Lie algebras to describe the space's infinitesimal symmetries. Given such a Lie algebra of symmetries, the geometry naturally produces a new Lie algebra called a "central extension''. This central extension plays a crucial role, especially in quantum mechanics. The famous Heisenberg algebra, for example, arises precisely in this way. > In this talk, I will explain how the above recipe can be enhanced to geometrically produce examples of "homotopy Lie algebras''. A homotopy Lie algebra is a topologist's version of a Lie algebra: a chain complex equipped with structures which satisfy the axioms of a Lie algebra only up to chain homotopy. They provide important tools for rational homotopy theory and deformation theory. The homotopy Lie algebras produced from our construction turn out to have interesting relationships with the theory of loop groups and what are called "string structures'' in algebraic topology
• Options
4.

10 December 2016:

Some very good news this time:

1) Brendan Fong has accepted a postdoc at MIT working with David Spivak. Having seen them discuss math together, I think we can expect great things!

2) I'm 99% sure that Daniel Cicala has been accepted to the Kan Extension Seminar, a high-powered online course on category theory. This time it'll be about functorial semantics - you can see the papers they'll discuss by clicking the link.

3) I gave a talk on Compositionality in network theory at this week's workshop on Compositionality at the Simons Institute for the Theory of Computing. You can see a video by clicking the link. I explained Brendan's theory of decorated cospans, illustrating it with a paper that Blake and I are writing about Petri nets.

4) Brendan gave a talk on Modelling interconnected systems with decorated corelations at the same workshop. This goes further, introducing decorated corelations, which are a generalization of decorated cospans. Again you can see a video by clicking the link.

These talks seem to have gone over well, along with other closely connected talks by David Spivak, Ross Duncan, Pawel Sobocinski and others. I was invited by Michael Mislove, who edits a column on semantics at the journal Logic, Semantics and Theory of Programming, to contribute a column. The whole lot of us were invited to participate more in various conferences on logic and computer science, since what we're doing seems to fit into that heading.

5) Blake's work on Markov processes was cited in at least two talks, and Prakash Panagaden gave me a draft of his paper on a bicategory of Markov processes, which I append here - Blake, Kenny and Daniel should read it!

I think we can and should do better, but we'll have to avoid stepping on Prakash's toes. For one thing, we can build a symmetric monoidal bicategory. For another thing, they are doing discrete-time Markov processes, with 2-morphisms being maps called 'simulations'. We can do something else. Daniel's work on perfect measure spaces should come into this, as well as what Kenny has been doing on bicategories with coarse-grainings as 2-morphisms.

6) Some negative news: the grant proposal I mentioned recently, engineered by Joshua Tan, has fallen through. I'm not too upset.

Comment Source:10 December 2016: Some very good news this time: 1) Brendan Fong has accepted a postdoc at MIT working with David Spivak. Having seen them discuss math together, I think we can expect great things! 2) I'm 99% sure that Daniel Cicala has been accepted to the [Kan Extension Seminar](https://golem.ph.utexas.edu/category/2016/10/the_kan_extension_seminar_retu.html), a high-powered online course on category theory. This time it'll be about functorial semantics - you can see the papers they'll discuss by clicking the link. 3) I gave a talk on [Compositionality in network theory](https://johncarlosbaez.wordpress.com/2016/11/29/compositionality-in-network-theory/) at this week's workshop on Compositionality at the Simons Institute for the Theory of Computing. You can see a video by clicking the link. I explained Brendan's theory of decorated cospans, illustrating it with a paper that Blake and I are writing about Petri nets. 4) Brendan gave a talk on [Modelling interconnected systems with decorated corelations](https://johncarlosbaez.wordpress.com/2016/12/09/modelling-interconnected-systems-with-decorated-corelations/) at the same workshop. This goes further, introducing decorated corelations, which are a generalization of decorated cospans. Again you can see a video by clicking the link. These talks seem to have gone over well, along with other closely connected talks by David Spivak, Ross Duncan, Pawel Sobocinski and others. I was invited by Michael Mislove, who edits a column on semantics at the journal Logic, Semantics and Theory of Programming, to contribute a column. The whole lot of us were invited to participate more in various conferences on logic and computer science, since what we're doing seems to fit into that heading. 5) Blake's work on Markov processes was cited in at least two talks, and Prakash Panagaden gave me a draft of his paper on a bicategory of Markov processes, which I append here - Blake, Kenny and Daniel should read it! I think we can and should do better, but we'll have to avoid stepping on Prakash's toes. For one thing, we can build a symmetric monoidal bicategory. For another thing, they are doing discrete-time Markov processes, with 2-morphisms being maps called 'simulations'. We can do something else. Daniel's work on perfect measure spaces should come into this, as well as what Kenny has been doing on bicategories with coarse-grainings as 2-morphisms. 6) Some negative news: the grant proposal I mentioned recently, engineered by Joshua Tan, has fallen through. I'm not too upset.
• Options
5.
edited December 2016

Finally catching up to today, 13 December 2016:

1) Kenny Courser's paper A bicategory of decorated cospans was accepted by Theory and Applications of Categories!

The referee wants him to say more about examples. That makes a lot of sense. I'd also like to deal with this issue: in our favorite examples, the 2-morphisms in Kenny's bicategory are a bit too restrictive.

For example, in the bicategory of cospans of finite sets where the apex is decorated by a graph, the 2-morphism do not allow arbitrary graph morphisms, only those that are "cocartesian lifts" of maps between finite sets.

2) On December 14th, Brendan is giving a talk called "All hypergraph categories are decorated corelation categories" at Macquarie University in Australia.

Brendan: make sure to say hi to Ross Street and my old friend James Dolan!

3) At Berkeley, it became clear that the stuff we do fits into "theoretical computer science", which is a very broad subject by now.

All of us were invited to submit papers to CALCO 2017, a conference on algebra and coalgebra in computer science. Daniel reminded me of this, saying:

Actually, a few of the gang could probably submit, since their interests include:

• String Diagrams and Network Theory

- Combinatorial approaches

- Theory of PROPs and operads

- Rewriting problems and higher-dimensional approaches

- Automated reasoning with string diagrams

- Applications of string diagrams

- Connections with Control Theory, Engineering and Concurrency

So, think of submitting papers here! Daniel has a plan to do this.

Comment Source:Finally catching up to today, 13 December 2016: 1) Kenny Courser's paper [A bicategory of decorated cospans](https://arxiv.org/abs/1605.08100) was accepted by _Theory and Applications of Categories_! The referee wants him to say more about examples. That makes a lot of sense. I'd also like to deal with this issue: in our favorite examples, the 2-morphisms in Kenny's bicategory are a bit too restrictive. For example, in the bicategory of cospans of finite sets where the apex is decorated by a graph, the 2-morphism do not allow arbitrary graph morphisms, only those that are "cocartesian lifts" of maps between finite sets. 2) On December 14th, Brendan is giving a talk called "All hypergraph categories are decorated corelation categories" at Macquarie University in Australia. Brendan: make sure to say hi to Ross Street and my old friend James Dolan! 3) At Berkeley, it became clear that the stuff we do fits into "theoretical computer science", which is a very broad subject by now. All of us were invited to submit papers to [CALCO 2017](http://coalg.org/calco17/index.html), a conference on algebra and coalgebra in computer science. Daniel reminded me of this, saying: > Actually, a few of the gang could probably submit, since their interests include: > * String Diagrams and Network Theory > - Combinatorial approaches > - Theory of PROPs and operads > - Rewriting problems and higher-dimensional approaches > - Automated reasoning with string diagrams > - Applications of string diagrams > - Connections with Control Theory, Engineering and Concurrency So, think of submitting papers here! Daniel has a plan to do this.
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6.

For some reason dates aren't showing up in Azimuth Forum entries, so it's good I included the dates in the entries here! I have some new entries...

Comment Source:For some reason dates aren't showing up in Azimuth Forum entries, so it's good I included the dates in the entries here! I have some new entries...
• Options
7.

22 December 2016:

Here are two things that happened this week:

1) Brandon and Brendan's paper Corelations are the prop for extraspecial commutative Frobenius monoids has been accepted for publication by Theory and Applications of Categories subject to making some small corrections.

2) It doesn't really count as mathematics, but I've started the Azimuth Backup Project to help back up climate data before Trump becomes president - because almost all his big hires are people who deny the importance of global warming.

Other teams are doing this too, and you can get the basic idea in this article of mine:

You can see our team's progress here:

We've got a great team, including a guy who used to drive a Mars rover for NASA, and so far we've backed up about a terabyte of data! In a couple of days I'll start a Kickstarter campaign to raise funds to store the data. We'll try to store it at least until larger institutions accept this responsibility.

Comment Source:22 December 2016: Here are two things that happened this week: 1) Brandon and Brendan's paper [Corelations are the prop for extraspecial commutative Frobenius monoids](https://arxiv.org/abs/1601.02307) has been accepted for publication by _Theory and Applications of Categories_ subject to making some small corrections. 2) It doesn't really count as mathematics, but I've started the Azimuth Backup Project to help back up climate data before Trump becomes president - because almost all his big hires are people who deny the importance of global warming. Other teams are doing this too, and you can get the basic idea in this article of mine: * [Saving Climate Data (Part 1)](https://johncarlosbaez.wordpress.com/2016/12/13/saving-climate-data/). You can see our team's progress here: * [Azimuth Backup Project (Part 1)](https://johncarlosbaez.wordpress.com/2016/12/16/azimuth-backup-project/). * [Azimuth Backup Project (Part 2)](https://johncarlosbaez.wordpress.com/2016/12/20/azimuth-backup-project-part-2/). We've got a great team, including a guy who used to drive a Mars rover for NASA, and so far we've backed up about a terabyte of data! In a couple of days I'll start a Kickstarter campaign to raise funds to store the data. We'll try to store it at least until larger institutions accept this responsibility.
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8.
edited January 13

13 January 2017:

We had a hugely productive meeting on Wednesday and I'm really excited about the new ideas:

• The connection between Brandon Coya's work on bond graphs and Ross Street's work on weak bimonoids, noticed by Brendan, is really fascinating - it implies that there's a "quantum groupoid" associated to electrical circuits, and it implies that our conjectured list of axioms characterizing the category of bond graphs was missing some highly nonobvious relations.

• Daniel Cicala is revisiting Franciscus Rebro's work on the bicategory of cobordisms and will prove it's a symmetric monoidal bicategory.

• Kenny Courser had the smart idea of revisiting Jeffrey Morton and Jamie Vicary's work on Khovanov's categorified Heisenberg algebra and making it rigorous using our new ability to get ahold of symmetric monoidal bicategories, and I realized we can actually do this.

• Adam Yassine seems to have proved that there's a bicategory of symplectic manifolds and cospans whose legs are Poisson fibrations — good for the study of open systems in classical mechanics.

And that's not all! In our Metron project,

• Blake Pollard and John Foley are developing a new framework for search and rescue operations (and many other distributed optimization problems).

• Joseph Moeller created a new algebraic structure generalizing the "operad for communication networks", and I think we can prove this new structure has an elegant category-theoretic description.

It's all great stuff. But these weekly reports are supposed to be about things that have been completed, just to focus your attention on getting things finished. So here are two things like that:

• The math department at U.C. Riverside is hosting the Fall Meeting of the AMS Western Section on Saturday and Sunday, Nov. 4 and 5, 2017. My proposal for a special session on Applied Category Theory has been accepted! I hope you submit proposals for talks — if you're able to come despite the fact that, as usual for such meetings, we have no money. I'll say in a while how you can propose a talk: there will be a webpage where you can do this.

• U. C. Riverside has agreed to be a repository of climate data collected by the Azimuth Backup Project. This means we don't have to figure out how to hold this data permanently. We've raised over $10,000 by now, so we're fine in the short term. Comment Source:13 January 2017: We had a hugely productive meeting on Wednesday and I'm really excited about the new ideas: * The connection between Brandon Coya's work on bond graphs and Ross Street's work on weak bimonoids, noticed by Brendan, is really fascinating - it implies that there's a "quantum groupoid" associated to electrical circuits, and it implies that our conjectured list of axioms characterizing the category of bond graphs was missing some highly nonobvious relations. * Daniel Cicala is revisiting Franciscus Rebro's work on the bicategory of cobordisms and will prove it's a symmetric monoidal bicategory. * Kenny Courser had the smart idea of revisiting Jeffrey Morton and Jamie Vicary's work on Khovanov's categorified Heisenberg algebra and making it rigorous using our new ability to get ahold of symmetric monoidal bicategories, and I realized we can actually do this. * Adam Yassine seems to have proved that there's a bicategory of symplectic manifolds and cospans whose legs are Poisson fibrations — good for the study of open systems in classical mechanics. And that's not all! In our Metron project, * Blake Pollard and John Foley are developing a new framework for search and rescue operations (and many other distributed optimization problems). * Joseph Moeller created a new algebraic structure generalizing the "operad for communication networks", and I think we can prove this new structure has an elegant category-theoretic description. It's all great stuff. But these weekly reports are supposed to be about things that have been completed, just to focus your attention on getting things finished. So here are two things like that: * The math department at U.C. Riverside is hosting the Fall Meeting of the AMS Western Section on Saturday and Sunday, Nov. 4 and 5, 2017. My proposal for a special session on **Applied Category Theory** has been accepted! I hope you submit proposals for talks — if you're able to come despite the fact that, as usual for such meetings, we have no money. I'll say in a while how you can propose a talk: there will be a webpage where you can do this. * U. C. Riverside has agreed to be a repository of climate data collected by the Azimuth Backup Project. This means we don't have to figure out how to hold this data permanently. We've raised over$10,000 by now, so we're fine in the short term.
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9.
edited February 28

21 January 2017:

1) Kenny Courser dealt with the referee's comments on his paper A bicategory of decorated cospans, resubmitted it to Theory and Applications of Categories, and put the new improved version on the arXiv.

2) Daniel Cicala has submitted his paper Spans of cospans to Theory and Applications of Categories. Is that true, Daniel? If not, make it so. Also, it probably makes sense to update the version on the arXiv.

(Kenny and Daniel are now teaming up to write a wonderful paper on bicategories of spans and cospans, including the "maps-of-spans" case already dealt with more generally by Mike, but more importantly the "spans-of-spans" case and "spans-of-cospans" case.)

3) Joseph Moeller has written up the statement of a theorem on generalizations of the operad of communication networks, suitable for inclusion in the paper that Blake is working on for DARPA. This paper is due the day after tomorrow!

4) I've been invited to join another project run by the US Defense Department!

I got an email from Michael A. Smith, a microbiologist who's the director of the Defense Biological Product Assurance Office. These are the people who check whether mysterious white powders showing up in people's mail are actually anthrax... and they also keep track of diseases that suddenly spring up, like Ebola or MIRS. They used to be called the Critical Reagents Program.

Smith is interested in whether modifications of the network of hospitals and laboratories in Africa could help them better keep track of new epidemics that show up on this continent. He read my book with Jacob Biamonte on Petri nets, and he thinks I might able to help. I'll start by listening in to a phone conversation he's having with Gary Kobinger, who helped develop a vaccine for Ebola, and who

developed and pioneered use of small mobile laboratories — a lab in a suitcase, essentially — that have changed the way testing is done during Ebola outbreaks.

I'm already way too busy, but this sounds interesting! Michael Smith said that his higher-ups mainly evaluate him on whether he spends money fast enough. I could certainly help with that... but it would also be cool if network ideas could help combat disease.

5) I arranged to visit the Institute for Scientific Interchange in Turin, Italy during the last week of June. This center is very interested in the mathematical foundations of complex systems, including networks.

I ran a workshop there in May 2015 and the director, Mario Rasetti, said I was welcome to visit any time. I decided to see if this offer was real: I'm visiting the category theorist Marco Grandis in nearby Genoa in mid-June, so I emailed Rasetti and asked if I could visit after that.

It turns out he's been reading our papers with interest! He'd actually like me to visit for a sabbatical or maybe even get some long-term affiliation with this center. I'll find out more in June.

Comment Source:21 January 2017: 1) Kenny Courser dealt with the referee's comments on his paper [A bicategory of decorated cospans](https://arxiv.org/abs/1605.08100), resubmitted it to _Theory and Applications of Categories_, and put the new improved version on the arXiv. 2) Daniel Cicala has submitted his paper [Spans of cospans](https://arxiv.org/abs/1611.07886) to _Theory and Applications of Categories_. Is that true, Daniel? If not, make it so. Also, it probably makes sense to update the version on the arXiv. (Kenny and Daniel are now teaming up to write a wonderful paper on bicategories of spans and cospans, including the "maps-of-spans" case already dealt with more generally by Mike, but more importantly the "spans-of-spans" case and "spans-of-cospans" case.) 3) Joseph Moeller has written up the statement of a theorem on generalizations of the operad of communication networks, suitable for inclusion in the paper that Blake is working on for DARPA. This paper is due the day after tomorrow! 4) I've been invited to join another project run by the US Defense Department! I got an email from [Michael A. Smith](https://www.linkedin.com/in/michael-smith-b6468844), a microbiologist who's the director of the [Defense Biological Product Assurance Office](https://globalbiodefense.com/2017/01/02/dod-seeks/). These are the people who check whether mysterious white powders showing up in people's mail are actually anthrax... and they also keep track of diseases that suddenly spring up, like Ebola or MIRS. They used to be called the Critical Reagents Program. Smith is interested in whether modifications of the network of hospitals and laboratories in Africa could help them better keep track of new epidemics that show up on this continent. He read [my book with Jacob Biamonte on Petri nets](http://math.ucr.edu/home/baez/stoch_stable.pdf), and he thinks I might able to help. I'll start by listening in to a phone conversation he's having with Gary Kobinger, who helped develop a vaccine for Ebola, and who > developed and pioneered use of small mobile laboratories — a lab in a suitcase, essentially — that have changed the way testing is done during Ebola outbreaks. I'm already way too busy, but this sounds interesting! Michael Smith said that his higher-ups mainly evaluate him on whether he spends money fast enough. I could certainly help with that... but it would also be cool if network ideas could help combat disease. 5) I arranged to visit the [Institute for Scientific Interchange](https://en.wikipedia.org/wiki/Institute_for_Scientific_Interchange) in Turin, Italy during the last week of June. This center is very interested in the [mathematical foundations of complex systems](http://www.isi.it/en/research/mathematics-foundation-of-complex-systems), including networks. I ran a workshop there in May 2015 and the director, Mario Rasetti, said I was welcome to visit any time. I decided to see if this offer was real: I'm visiting the category theorist Marco Grandis in nearby Genoa in mid-June, so I emailed Rasetti and asked if I could visit after that. It turns out he's been reading our papers with interest! He'd actually like me to visit for a sabbatical or maybe even get some long-term affiliation with this center. I'll find out more in June.
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10.

14 February 2017:

1) I gave a talk called Biology as Information Dynamics at a workshop called Biological Complexity: Can it be Quantified? at the Beyond Center at Arizona State University. The last result in my talk is new: it's an improved version of Fisher's fundamental theorem of natural selection, which is more general than the usual version, and phrased in terms of information geometry.

I talked about gene regulatory networks with Sara Walker, who works there. A gene regulatory network is a very simple thing: a finite set of 'genes' with two kinds of arrows from some genes to some other genes: 'express' and 'repress'. It would be nice to study 'open' gene regulatory networks using the decorated cospan formalism. There seems to be a fairly simple semantics for these networks where time goes in integer steps and at each time each gene is either on (expressed) or off (repressed).

I also had some conversations with Jim Crutchfield, who wants Blake Pollard to visit him up at U.C. Davis.

2) Tom Mifflin of Metron said that DARPA likes our work so much that they're starting two new programs related to these ideas. One will be at the Strategic Technology Office, and it will be big. Tom suggested I could get more money for grad students. The other is a small "seedling" program, just $500,000, for Metron to help develop autonomous vehicles for the Navy. 3) Blake noticed that Jason Erbele's paper Categories in control was cited by Dominique Luzeaux in a paper on the category-theoretic foundations of systems engineering. The main interesting thing about this paper is that Lzueaux is the Deputy Director of the Joint Directorate for Networks, Infrastructure and Information Systems, part of the French defense department. It may seem weird that the military is interested in category theory and operads, but it makes sense. They have a lot of money, they're willing to experiment to stay ahead of other countries, and they have huge organizational/strategic problems that involve complex networked systems. Thus, they've traditionally been at the forefront of "systems of systems engineering". Comment Source:14 February 2017: 1) I gave a talk called [Biology as Information Dynamics](http://math.ucr.edu/home/baez/bio_asu/) at a workshop called [Biological Complexity: Can it be Quantified?](https://beyond.asu.edu/workshop/biological-complexity-can-it-be-quantified) at the [Beyond Center](https://beyond.asu.edu/) at Arizona State University. The last result in my talk is new: it's an improved version of Fisher's fundamental theorem of natural selection, which is more general than the usual version, and phrased in terms of information geometry. I talked about gene regulatory networks with Sara Walker, who works there. A gene regulatory network is a very simple thing: a finite set of 'genes' with two kinds of arrows from some genes to some other genes: 'express' and 'repress'. It would be nice to study 'open' gene regulatory networks using the decorated cospan formalism. There seems to be a fairly simple semantics for these networks where time goes in integer steps and at each time each gene is either on (expressed) or off (repressed). I also had some conversations with Jim Crutchfield, who wants Blake Pollard to visit him up at U.C. Davis. 2) Tom Mifflin of Metron said that DARPA likes our work so much that they're starting two new programs related to these ideas. One will be at the Strategic Technology Office, and it will be big. Tom suggested I could get more money for grad students. The other is a small "seedling" program, just$500,000, for Metron to help develop autonomous vehicles for the Navy. 3) Blake noticed that Jason Erbele's paper [Categories in control](http://www.tac.mta.ca/tac/volumes/30/24/30-24abs.html) was cited by [Dominique Luzeaux](https://scholar.google.com/citations?user=8G2BKfEAAAAJ&hl=en) in a paper on the category-theoretic foundations of systems engineering. The main interesting thing about this paper is that Lzueaux is the Deputy Director of the Joint Directorate for Networks, Infrastructure and Information Systems, part of the French defense department. It may seem weird that the military is interested in category theory and operads, but it makes sense. They have a lot of money, they're willing to experiment to stay ahead of other countries, and they have huge organizational/strategic problems that involve complex networked systems. Thus, they've traditionally been at the forefront of "[systems of systems engineering](https://en.wikipedia.org/wiki/System_of_systems_engineering)".
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11.

27 February 2017:

Four pieces of news:

1) Brendan and Brandon made all the required changes to their paper "Corelations are the prop for extraspecial commutative Frobenius monoids", so this paper has now been published by Theory and Applications of Categories. You can see it here:

http://www.tac.mta.ca/tac/index.html#vol32

It's in the same issue as two papers by Voevodsky, inventor of homotopy type theory, along with papers by Clemens Berger, Michael Batanin and Jiri Adamek — all famous category theorists! So, congratulations!

2) Blake applied to give a talk on his work at the American Physical Society March Meeting in New Orleans on March 13-17. They asked him to present a poster. He applied to the Grad Student Association for funding to go on this trip, but it looks like he'll have to foot some of the bill himself. (Sadly, while the Metron project he and I are working on pays him enough money so he doesn't have to be a teaching assistant, it doesn't give us any travel money, and the Grad Student Association gives rather limited funds.)

3) Blake has also gotten an invitation from the famous information theorist Jim Crutchfield to speak about our work at U. C. Davis. I mentioned this earlier, but now it's really come true! I forget when Blake is going up there.

4) Not exactly academic, but: the Azimuth Climate Data Backup Project's Kickstarter campaign has succeeded! Our original goal was $5000. We got$20,427 of donations, and after Kickstarter took its cut we received $18,590.96. I wrote thank-you notes to all 627 contributors, and compiled a list thanking everyone who was okay with having their name made public. We are close to reaching our goal of backing up 40 terabytes of data. The next step will be to put copies in several secure locations. 3 places have volunteered to hold copies: the Princeton math department, a UNESCO center in France, and the U. C. Riverside department of computing and communications. It will take a while to accomplish these transfers. It looks like I'll be interviewed about this tomorrow by Amy Harmon of the New York Times. But reporters are often busy and distracted, so I'll believe it when it happens. Comment Source:27 February 2017: Four pieces of news: 1) Brendan and Brandon made all the required changes to their paper "Corelations are the prop for extraspecial commutative Frobenius monoids", so this paper has now been published by Theory and Applications of Categories. You can see it here: [http://www.tac.mta.ca/tac/index.html#vol32](http://www.tac.mta.ca/tac/index.html#vol32) It's in the same issue as two papers by Voevodsky, inventor of homotopy type theory, along with papers by Clemens Berger, Michael Batanin and Jiri Adamek — all famous category theorists! So, congratulations! 2) Blake applied to give a talk on his work at the American Physical Society March Meeting in New Orleans on March 13-17. They asked him to present a poster. He applied to the Grad Student Association for funding to go on this trip, but it looks like he'll have to foot some of the bill himself. (Sadly, while the Metron project he and I are working on pays him enough money so he doesn't have to be a teaching assistant, it doesn't give us any travel money, and the Grad Student Association gives rather limited funds.) 3) Blake has also gotten an invitation from the famous information theorist [Jim Crutchfield](http://csc.ucdavis.edu/~chaos/) to speak about our work at U. C. Davis. I mentioned this earlier, but now it's really come true! I forget when Blake is going up there. 4) Not exactly academic, but: the Azimuth Climate Data Backup Project's [Kickstarter campaign](https://www.kickstarter.com/projects/592742410/azimuth-climate-data-backup-project) has succeeded! Our original goal was$5000. We got $20,427 of donations, and after Kickstarter took its cut we received$18,590.96. I wrote thank-you notes to all 627 contributors, and compiled [a list thanking everyone who was okay with having their name made public](https://johncarlosbaez.wordpress.com/2017/02/18/azimuth-backup-project-part-4/). We are close to reaching our goal of backing up 40 terabytes of data. The next step will be to put copies in several secure locations. 3 places have volunteered to hold copies: the Princeton math department, a UNESCO center in France, and the U. C. Riverside department of computing and communications. It will take a while to accomplish these transfers. It looks like I'll be interviewed about this tomorrow by [Amy Harmon](https://www.nytimes.com/by/amy-harmon) of the _New York Times_. But reporters are often busy and distracted, so I'll believe it when it happens.
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12.

John, Congratulations on such a great response at Kickstarter and all the other good news!

Comment Source:John, Congratulations on such a great response at Kickstarter and all the other good news!


text:
# SE Comment Preview - An attempt to add real-time previewing for comments

## Screenshot

This script creates a new live preview directly beneath any comment you compose or edit.

When writing a comment with a lot of formatting, especially one containing some mathematical formulas, I often found myself pasting it into the "new answer" editor at the bottom of the page to see if everything was rendering properly. This script has made writing such comments a much nicer experience for me so I hope others enjoy it as well.

It works on all Stack Exchange sites (as far as I know). If the site happens to use MathJax for math formatting then the math is rendered in the new preview just like when you're writing an answer, if a little more slowly.

Check out the GitHub page for more info including a list of known issues.

## Installation

This script does not work in Firefox (yet?).

## Code

Markdown is processed using my fork of marked. Math removal/replacement is handled using Stack Exchange's methods. These are coodinated with marked using Megh Parikh's implementation. See the source code for more info.

If you're interested, there is also an experimental (buggy) version of this script which uses Pagedown instead of marked for Markdown processing: SE Comment Preview - Paged!

The Paged! version of the script only works on SE sites that use MathJax, but on those sites the new live comment previews behave very similarly to the built-in question/answer editors.

• This may conflict with other extensions. For example, the preview is removed when the return key is pressed (and the comment is >14 characters long), so it doesn't play well with Disable enter for submitting comments. Oct 19 '15 at 11:51
• When certain characters (<, >, &) are inside code blocks (text surrounded by backticks), they appear as the associated html entities (<, >, &) instead of the character that they are (and will be in the actual comment) in the preview. Jun 22 '18 at 3:52
• Also, it seems marked is throwing Uncaught Error: Infinite loop on byte: 10 when a new line (entered by pasting something with multiple lines, or pressing Shift+Enter) is in the comment field. Jun 22 '18 at 16:49
• The preview also doesn't show that urls will be automatically converted from http://domain.tld/path/to/page to a link that doesn't show the protocol like domain.tld/path/to/page. Jun 26 '18 at 19:00