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+1
+Modeling Effective Lifespan of Payment
+Channels
+Soheil Zibakhsh Shabgahi, Seyed Mahdi Hosseini, Seyed Pooya Shariatpanahi, Behnam Bahrak
+Abstract—While being decentralized, secure, and reliable, Bitcoin and many other blockchain-based cryptocurrencies suffer from
+scalability issues. One of the promising proposals to address this problem is off-chain payment channels. Since, not all nodes are
+connected directly to each other, they can use a payment network to route their payments. Each node allocates a balance that is frozen
+during the channel’s lifespan. Spending and receiving transactions will shift the balance to one side of the channel. A channel becomes
+unbalanced when there is not sufficient balance in one direction. In this case, we say the effective lifespan of the channel has ended.
+In this paper, we develop a mathematical model to predict the expected effective lifespan of a channel based on the network’s topology.
+We investigate the impact of channel unbalancing on the payment network and individual channels. We also discuss the effect of
+certain characteristics of payment channels on their lifespan. Our case study on a snapshot of the Lightning Network shows how the
+effective lifespan is distributed, and how it is correlated with other network characteristics. Our results show that central unbalanced
+channels have a drastic effect on the network performance.
+Index Terms—Bitcoin, Lightning Network, Payment Channel, Lifespan, Random Walk.
+!
+1
+INTRODUCTION
+B
+ITCOIN is the first decentralized cryptocurrency, in-
+troduced in 2008 which provides security, anonymity,
+transparency, and democracy without any trusted third
+party [1]. Most of these properties are achieved by using
+a blockchain as a distributed ledger. An inherent problem
+with using a blockchain over a network is that it sacrifices
+scalability [2], [3]. The reason is that all nodes, potentially
+tens of thousands, must exchange, store, and verify each
+and every transaction in the system [4]. Furthermore, each
+block has a limited size and blocks get generated at regular
+intervals (approximately every 10 minutes). This means that
+with the current blocksize of 1 MB the throughput of Bitcoin
+is about 4.6 transactions per second, which is much slower
+than centralized systems like Visa, WeChatPay, and PayPal
+[5]; making the use of Bitcoin in everyday transactions
+impractical.
+Another trade-off the Bitcoin consensus makes is that it
+ensures security by waiting for other miners to confirm a
+transaction by extending the block holding that transaction,
+which reduces the throughput. This way it makes sure that
+the double spending attack is highly improbable. Currently,
+the standard waiting time for a block to be confirmed is 6
+blocks, which is almost one hour [6].
+Bitcoin’s capacity limitations are being felt by users in
+the form of increased transaction fees and latency. With
+an increasing demand for making transactions, users need
+to pay more transaction fees in order to make sure that
+their transaction is more profitable for the miners; hence
+have a higher chance of making it into a block. Queuing of
+transactions and network bandwidth will lead to a longer
+delay time for a transaction to appear in the blockchain.
+There are many different proposals to solve the scala-
+bility problem. Most of the proposals fall into three cate-
+gories: Layer0, Layer1, and Layer2 solutions [7]. Layer0
+solutions try to enhance the infrastructure, like the network
+that connects the nodes. Layer1 solutions try to enhance
+the blockchain’s shortcomings by changing the consensus
+mechanism and protocols [8], [9]. Layer2 solutions propose
+ways to move away from the blockchain, and for this reason,
+they are also called off-chain solutions [10].
+In 2016 the idea of Lightning Network (LN) was pro-
+posed to move the transactions to the second layer (off-
+chain) [4]. The Lightning Network consists of payment
+channels in a P2P fashion. Payment channels allow two
+parties to exchange payments with negligible time and cost,
+but both parties must freeze an initial fund in the channel
+so no one can spend more money than they own and no
+double spending occurs. It is important to note that the sum
+of funds in each channel remains constant throughout the
+channel’s lifespan and only the channel’s balance changes.
+When two parties that do not have a direct channel want to
+exchange payments they can use other parties to route their
+payments. So a network of nodes is constructed and all the
+connected nodes can send each other payments.
+This system moves the cost of submitting a transaction
+off the blockchain. Only the final states between two nodes
+will eventually make it into the blockchain, which signifi-
+cantly increases throughput. Furthermore, no time is needed
+for the transaction to be confirmed and all transactions in a
+channel happen almost instantly.
+After several transactions through a channel, the channel
+starts to get unbalanced; meaning all of its funds have gone
+to one of the parties and the other node cannot route any
+more payments through the channel. In this case, it is best
+to close the unbalanced channel or open a new one.
+In this paper, we investigate the expected effective lifes-
+pan of a channel in a payment network. Our contributions
+can be summarized as follows:
+•
+We provide simulation evidence of how channel
+unbalancing impacts its throughput. Moreover, we
+show how the performance of the payment network
+arXiv:2301.01240v1  [cs.DC]  11 Sep 2022
+
+2
+can be affected if a number of channels become
+unbalanced.
+•
+We present a mathematical model of payment chan-
+nels to predict the expected time for a channel to get
+unbalanced considering the channel’s position in the
+network and its initial balance. We call this time the
+Effective Lifespan of the channel.
+•
+We evaluate our model through simulation, and
+observe how the Effective Lifespan of a channel is
+affected if we change any of its characteristics.
+•
+By analyzing a recent snapshot of the Lightning Net-
+work, we find the distribution of real-world chan-
+nel lifespans and its correlation with the network’s
+topological parameters. We also investigate the rela-
+tionship between the centrality of a channel in the
+network and its effective lifespan.
+2
+RELATED WORK
+While the LN white paper [4] does not discuss channel re-
+balancing, there exists some research on channel balances
+and their significance.
+The importance of channel balances is mainly discussed
+in four major areas; re-balancing, security, performance, and
+financial.
+Re-balancing: [11] proposes a method for re-balancing
+payment channels. This work allows arbitrary sets of users
+to securely re-balance their channels. However, this paper
+does not discuss the application of re-balancing, and how
+frequently it should be performed. [12] also proposes meth-
+ods for rebalancing LN channels, but does not discuss the
+frequency of rebalancing.
+Performance: In [13] the authors discuss why it is in
+the best interest of the network to have balanced channels.
+They propose a method to re-balance some channels to
+improve the network’s performance. [14] presents a method
+in which a node can make its channels balanced through
+circular subgraphs. It also develops a method for measuring
+imbalance in a payment network.
+Security: There has been some research on the security
+aspects of channel unbalancing. In [14], [15], and [16] the
+authors describe a method in which it is possible for an
+adversary to uncover channel balances. Having unbalanced
+channels poses the threat of griefing attacks. The incentive
+for honest behavior in the LN channels is the penalty for
+misbehavior. If a node cheats by publishing an old contract,
+it will be penalized and all of the channel funds can be
+claimed by the victim. When channels are unbalanced the
+penalty is less so there is less incentive for honest behav-
+ior. In [17] the authors discuss some countermeasures like
+watchtowers to keep the misbehaving nodes from closing
+the channel.
+Financial: Routing payments through a channel can
+make revenue for the owner. So payment channels can be
+looked at as investments. In [18] the authors do an in-depth
+financial analysis on how much should payment channels
+charge for routing payments. One of the key factors in this
+analysis is the lifespan of payment channels. In order to
+analyze investing in a payment channel, nodes should be
+able to have an estimate on how long the investment stays
+profitable and what is the impact of channel unbalancing on
+the profits of a channel. Branzei et al. [18] assume an equal
+probability of having a payment from each side in a channel
+and use the lifetime of channels for financial analyses. We
+will show how the lifespan of a channel could be affected
+by this probability.
+In this paper we focus on the details of estimating chan-
+nel lifespans; considering parameters such as the placement
+of the channel in the topology and payment rates between
+each pair and explain the importance of estimating channel
+lifespans. This gives us a better and more realistic estimation
+of the channel’s lifespan compared to existing work. More-
+over, we measure the impact of imbalanced channels on the
+network.
+Despite the importance of payment channel’s lifespan, to
+the best of our knowledge, the expected lifespan of channels
+in the payment network has not been discussed in detail.
+3
+TECHNICAL BACKGROUND
+In this section, we provide a technical background to under-
+stand the remainder of this paper thoroughly.
+Payment Channels
+Payment channel is a financial contract between two parties
+in a cryptocurrency like Bitcoin. The contract allocates a
+balance of funds from both parties. The contract is estab-
+lished by a 2-of-2 multisignature address which requires the
+cooperation of both parties to spend the funds.
+Payments are made off the blockchain by passing on
+a new version of the contract with a different balance of
+allocated funds on the spending transaction; which both
+parties have to sign. The channel is closed when one of
+the parties publishes the latest version of the contract to
+the blockchain. We define the payment direction to be the
+direction in which funds are moving during a transaction.
+In this paper, we call the sum of locked funds in a
+channel the channel’s capacity. When all of the funds of
+a channel are allocated to one of the parties, the channel
+becomes unbalanced. In this case payments can only be made
+from one side of the channel. A channel’s effective lifespan is
+the time from creation of a channel until the first imbalance
+occurs. A channel’s success probability is defined as the
+number of successful payments made through the channel
+divided by the total number of payment attempts.
+Several connected payment channels can form a pay-
+ment network, in the case of Bitcoin, this network is called
+the Lightning Network [4]. This network is used to route
+payments through intermediate channels between nodes
+who do not have a direct channel between them. We de-
+fine a network’s success probability as the total number of
+successful payments made on the network divided by the
+total attempts to route payments through the network [19].
+Random Walk
+The random walk model has been used in a wide variety
+of contexts to model the movement of objects in different
+spaces. This paper uses one-dimensional random walk to
+model the liquidity balance in a payment channel. Two
+endpoints on the left and right sides of the random walk
+are assumed to represent the channel imbalance condition.
+
+3
+Fig. 1. Relation of network success rate with percentage of unbalanced
+channels in the network.
+In our model, each payment corresponds to one step of
+the random walk model, and the direction of the payment
+determines the direction of that step. Suppose we take prob-
+abilities p and 1 − p as the probability of payment direction
+(i.e., step direction). We can find the expected number of
+payments (steps) needed for the channel (the random walk
+model)to get unbalanced (to reach one of the endpoints).
+Betweenness Centrality
+Betweenness centrality is a measure based on shortest paths
+for the importance of the location of a node or an edge
+in a graph. Betweenness centrality for an edge(a, b) in the
+network is defined as follows: �
+s,t∈V
+s̸=t
+σ(s,t|edge(a,b))
+σ(s,t)
+, where
+σ(s, t) is the total number of shortest paths between nodes
+s and t and σ(s, t|edge(a, b)) is the total number of shortest
+paths between s and t that pass through edge(a, b).
+4
+MOTIVATION
+One of the important characteristics of a payment network
+is reliability. Reliability can be defined as the probability of
+payment success [19].
+In this section, we analyze the payment routing failure
+probability of a singular channel after unbalancing, and the
+network’s success probability of routing a payment when
+some channels are unbalanced.
+4.1
+Singular Channel
+We ran a simulator of a single payment channel to see how
+much the failure rate increases after the first time that the
+channel becomes unbalanced. Fig. 2 shows the failure rate
+after the first time a channel becomes unbalanced. During
+the simulation, 5000 payments were being routed through
+an initially balanced channel. Then the simulator calculates
+the failure rate after the first time the channel becomes un-
+balanced. As Fig. 2 suggests, the probability of the direction
+of payments (p) is a key factor in determining how much
+the probability of payment success degrades after the first
+imbalance occurs. Channels capacity has little to no impact
+on how well it performs after unbalancing.
+These results show that the probability of payment di-
+rection (p), which depends on the network topology and
+Fig. 2. Failure rate after unbalancing.
+the network’s transaction flow, is one of the most impor-
+tant parameters in determining the channel’s lifespan; more
+importantly, shows the impact of unbalancing on channel
+success probability after the channel becomes unbalanced.
+4.2
+Network Performance
+Using the CLoTH simulator [19] we simulate and measure
+the performance of Lightning Network. In each iteration we
+take channels from the given LN snapshot and make them
+unbalanced, we then measure the success probability after
+attempting 5000 payments. Choosing more central channels
+as unbalanced channels is more reasonable, because they
+route more payments and thus have a higher probability
+of becoming unbalanced in the real world. We considered
+two scenarios for selecting channels to unbalance: choos-
+ing channels randomly and choosing channels that have
+a higher betweenness centrality. As illustrated in Fig. 1,
+as the percentage of unbalanced channels increases, the
+routing success rate decreases dramatically for both channel
+selection scenarios. In the random selection scenario, it is
+noticeable that the first 10 percent of unbalanced channels
+have less effect on the network performance than the last
+10 percent of unbalanced channels. We see that unbalancing
+channels with a higher betweenness centrality has a higher
+impact on the network performance in contrast with the
+random selection scenarios. Therefore, per any percentage of
+unbalanced channels, selection with betweenness centrality
+is more effective.
+Seres et al. [20] suggest that in the Lightning Network,
+the top 14% central channels will have the most significant
+impact on the network. In a different experiment we made
+15% of the network’s channels unbalanced, we first sort the
+channels by betweenness centrality and take a window of
+15% of the channels per experiment. We start with the 15%
+most central channels and move all the way up to 15%
+least central channels. It can be inferred from the results
+in Fig. 3 that more central channels have more impact on
+the network success rate when they become unbalanced. As
+we can see in Fig. 3, the top 15% central channels have the
+most significant effect on the success rate when they become
+unbalanced. This confirms the result from Fig. 1.
+
+100
+Random
+90
+Most central
+80
+rate
+70
+Success
+60
+50
+40
+30
+20
+20
+40
+60
+80
+100
+Percentage of initialy unbalanced channels0.40
+Capacity: 1.2 Msat
+Capacity: 1.8 Msat.
+0.35
+Capacity: 2.4 Msat.
+Capacity: 3.0 Msat.
+0.30
+Capacity: 3.6 Msat.
+0.20
+Fal
+0.15
+0.10
+0.05
+0.300
+0.325
+0.3500.3750.400
+0.425
+0.450
+0.475
+0.500
+probability of payment direction4
+Fig. 3. Per each data point the i-th to (i+4500)-th most central channels
+are unbalanced and the success rate of the network is measured. The
+total number of channels is 30457.
+5
+THE MODEL
+As we discussed in Section 4 channel balances have a sig-
+nificant effect on both channel, and network performance.
+In this section, we introduce a mathematical model to deter-
+mine the expected time for a channel to get unbalanced;
+we call this the channel’s expected lifespan. We model
+the dynamics of a payment channel with a random walk
+problem. Each payment passing through the channel will
+represent a step the random walker takes. We will first
+discuss our assumptions and describe the model in detail.
+We then discuss how to find the model parameters. We
+proceed by doing an analysis on how the expected lifespan
+is affected by changing channel’s characteristics.
+5.1
+Random Walk Model
+Take a payment channel between two nodes A and B, and
+take their initial balance allocated for the channel to be FA
+and FB, respectively. The goal is to determine the expected
+time it will take for this payment channel to become unbal-
+anced for the first time. We make the following assumptions:
+•
+All the payments have the same amount denoted
+with ω (PaymentSize).
+•
+The payments from each node come with a Poisson
+distribution.
+Since the number of nodes is large and the probability
+of sending a transaction for a given time is small, we can
+assume that transaction arrival for each channel is a Poisson
+process for moderate time windows [21]. Although the
+dynamics of the network will change over time, we make
+the assumption of having a fixed topology.
+We model the dynamics of a payment channel with a
+random walk problem. Each payment is simulated by a step
+the random walk takes. To simulate a payment channel, take
+the liquidity of node A as the distance of the random walk
+from the endpoint on the right hand side and the liquidity
+of node B as the distance from the endpoint on the left hand
+side.
+The payment direction determines the direction of that
+step. So the payment direction probability is the probability
+of going to the right or left for the random walk in each step.
+Fig. 4. Distribution of expected lifespan with 10000 random walk simula-
+tions with p = 1
+2 and a = b = 1.2 Msat.
+Let the random walk start at the origin of the number
+line. The two endpoints a and −b are ⌊ FA
+ω ⌋ and ⌊ FB
+ω ⌋,
+respectively.
+Since we assume that the payments from each side are
+made independently with a Poisson process, and the sum
+of two independent Poisson processes is itself a Poisson
+process, we can say that payments come to the channel with
+a Poisson distribution having:
+λpayment = λA,B + λB,A,
+(1)
+thus the relation between expected time and expected num-
+ber of random walk steps is:
+Etime =
+Esteps
+λpayment
+,
+(2)
+where Etime is the expected time until unbalancing and
+Esteps is the expected number of steps until unbalancing
+occurs.
+The expected number of payments until unbalancing
+occurs, can be a better metric depending on the application;
+when multiplied by average fee per payment, it gives the
+expected routing income, and when divided by λ it gives
+the expected lifespan.
+The objective is to determine the time it takes for a
+channel to become unbalanced. We first try to find the
+expected number of steps needed for the random walker
+to reach +a or −b for the first time.
+Lemma 1. The expected number of steps to reach +a or −b
+for the first time starting from zero considering the probability p
+for the positive direction and q = 1 − p for the negative direction
+is:
+Esteps =
+� apa(pb−qb)+bqb(qa−pa)
+(p−q)(pa+b−qa+b)
+p ̸= 1/2
+ab
+p = 1/2
+(3)
+.
+We provide the proof of Lemma 1 in Appendix A.
+We simulated a Random Walk which starts from point
+zero with the same probability of going to each side (p = 1
+2).
+The simulation ran 10000 times to find the distribution of the
+number of steps needed to reach +a or −b. Fig. 4 illustrates
+the result of the simulation. We can observe that most of
+the times the random walk reaches one of the bounds in
+less than 400 steps, but there are not many situations where
+
+98
+96
+Success rate
+94
+92
+90
+88
+86
+0
+5000
+10000
+15000
+20000
+25000350
+300
+250
+count
+200
+150
+100
+50
+0
+0
+500
+1000
+1500
+2000
+2500
+3000
+number of steps5
+it takes a huge number of steps to reach these bounds.
+However, the average number of steps needed to reach these
+bounds is 400.5 confirming 11.
+5.2
+Finding p
+In Section 5.1 we modeled the payment channel dynamics
+with a random walk and a parametric formula was con-
+structed according to Lemma 1.
+A payment network can be formally expressed by an
+unweighted directed graph. V
+represents the set of all
+nodes, and the set of edges is denoted by E. Each channel
+is represented using two edges from E each for one of the
+directions.
+We define MRates to be the matrix of payment rates
+between each two nodes. The rate of payments (i.e., number
+of payments per day) from node i to node j is denoted by
+MRatesij.
+λa,b represents the rate of payments transmitted over
+the edge(a, b). λa,b consists of the sum of portions of the
+payment rate between each pair of nodes that pass through
+edge(a, b). So we have:
+λa,b =
+�
+s,t∈V
+s̸=t
+σ(s, t|edge(a, b))
+σ(s, t)
+MRatesst,
+(4)
+where σ(s, t) is the number of shortest paths from node s
+to node t and σ(s, t|edge(a, b)) is the number of shortest
+paths from node s to node t passing through edge(a, b) in
+the directed graph G.
+Lemma 2. p
+q = λ(a,b)
+λ(b,a).
+According to Lemma 2:
+p =
+λ(a, b)
+λ(a, b) + λ(b, a)
+(5)
+Therefore we can find p based on the network topology.
+Lemma 3. If ∀s, t ∈ V
+: MRatesst = MRatests then
+p = 0.5.
+We provide the proofs of Lemmas 2 and 3 in Appendix A. If
+we assume that MRates is a symmetric matrix, according
+to Lemma 3, p is independent of MRates matrix and the
+network topology.
+5.3
+Model Analysis
+In this section we analyze the effect of channel parameters
+on the channel’s expected lifespan and perform a financial
+analysis for channel lifespan.
+For more realistic parameter values we used a recent
+snapshot of the Lightning Network taken on Feb2019 as
+a reference point. The average payment size is considered
+to be 60000 sat1 [22] and the average channel capacity is
+considered 2.4 Msat2 according to the snapshot.
+For simplicity we use ”lifespan” and ”expected number
+of payments until channel is unbalanced”, interchangeably.
+1. satoshi
+2. million satoshi
+Fig. 5. Effect of payment direction probability on balanced channels,
+according to different channel capacities.
+Fig. 6. The effect of payment direction probability on expected number
+of payments, according to different initial balance ratios. The channel
+capacity is considered 2.4 Msat.
+We first answer the question of how sensitive is a
+channel’s lifespan to the changes in p. As demonstrated in
+Fig.
+5; if the channel is initially balanced, the maximum
+lifespan happens on p = 1
+2. Also, lifespan is more sensitive
+to changes in p when the capacity is higher. From this
+result we can infer that it is an important consideration
+for a node to make sure the channel is placed in a way
+that p is close to
+1
+2, otherwise the channel’s lifespan is
+affected dramatically. A reasonable proposal for nodes who
+want to keep their channels active as long as possible is to
+charge routing fees in a way that encourages other nodes to
+route their payments through the node in order to achieve
+p = 0.5. Fig. 6 shows that if a channel is initially unbalanced,
+its maximum possible lifespan takes a hit. Although the
+maximum lifespan does not occur at exact p = 1
+2, it occurs at
+a point close to this value. So even if a channel is somewhat
+unbalanced, the nodes must try to keep p as close as possible
+to 50%.
+We now answer the question of how the lifespan is
+affected by the channel capacity. As Fig. 7 suggests, the
+channel lifespan increases with increasing its capacity. It is
+noteworthy that the slope of this graph is increasing. So if
+a node doubles its channel capacity, the channel’s lifespan
+will be more than doubled. Moreover, Fig. 7 shows the effect
+of a channel’s initial imbalance on its lifespan.
+Usually when a node wants to create a new channel
+
+Capacity: 1.80 Msat.
+600
+Capacity: 2.40 Msat.
+Capacity: 3.00 Msat.
+500
+400
+300
+200
+100
+0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Probability (p)400
+a/b: 1
+a/b: 5
+350
+a/b: 50
+300
+250
+200
+150
+100
+50
+0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Probability (p)6
+Fig. 7. Effect of channel capacity on the expected number of payments,
+according to initial balance ratios
+Fig. 8. For a fixed a = 1.2 Msat, the effect of channel b’s capacity on the
+maximum possible lifespan in any p.
+with another node in the network, the only parameter it has
+control over is the amount of funds it wants to put in the
+channel, not the funds its partner puts in the channel. This
+brings up the question: how will the channel’s lifespan be
+affected with the amount the other node wants to put in the
+channel if our fund stays at a fixed value. Figures (8) and (9)
+illustrate this effect. Fig. 8 shows the maximum achievable
+lifespan considering any p value and how it is affected by
+the fund that the other node commits to the channel. The
+maximum lifespan grows with the initial fund of the other
+edge in a linear fashion. Fig. 9 illustrates the effect of our
+edge capacity if the peer node’s capacity is fixed. Figure
+(9) shows that if p is in favor of payments in the direction
+of our edge (p ≥ 1
+2), the lifespan increases almost linearly;
+otherwise (p < 1
+2), the other edge becomes the bottleneck
+and the fund we put towards the channel will have little
+to no effect on the expected lifespan of the channel. If the
+funds we put towards the channel do not have an effect on
+the channel’s lifespan, we have wasted cost opportunities.
+6
+IMPLEMENTATION AND EVALUATION
+We provided a simulation proof of concept on a constructed
+Lightning Network to show the accuracy of the model
+discussed in Section 5. In this section we describe our
+methodology for creating data and calculating accuracy of
+Fig. 9. Having a fixed initial balance from peer node (b) analyzing the
+effect of our initial balance fund (a), according to different payment
+direction probabilities (p).
+our model. We later analyze the results to see under which
+conditions the model performs better.
+6.1
+Methodology
+The testing pipeline shown in Fig. 10 uses the following
+modules:
+6.1.1
+Network Generator
+For each test, a random network was generated using Net-
+workX’s [23] gnp random graph with the number of nodes
+being 50 and the channel existence probability being 20%
+(245 edges on average).
+6.1.2
+Mrates Generator
+As discussed in previous sections the Mrates matrix holds
+the rates in which each two nodes send payments to one
+another. The Mrates generator takes two main parameters:
+SC and SK. SC determines the sparseness of the Mrates
+matrix and SK determines the matrix skewness in relation to
+its main diagonal. Per each test, a new matrix is generated.
+In table (1) the sparse coefficient and the skew were changed
+to test how the model will perform in each scenario.
+6.1.3
+Lifespan Predictor
+The lifespan predictor takes the network and the Mrates
+matrix and using the model discussed in 5 gives the ex-
+pected time for each channel to become unbalanced.
+6.1.4
+Payment Generator
+Payment generator creates random payments in CLoTH
+simulator’s input format [19]. These payments follow the
+Mrates generator values on average.
+6.1.5
+Simulator
+We used a modified version of the CLoTH simulator [19].
+We modified CLoTH such that the simulator logs the un-
+balancing of channels and chooses paths randomly in cases
+where more than one shortest path exists.
+The payment generator and simulator run 100 iterations
+per test.
+
+10000
+a/b: 1
+f Payments
+a/b: 5
+a/b: 50
+8000
+6000
+4000
+2000
+0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+Capacity (sat.)
+1e7Maximum Expected Number of Payments
+5000
+4000
+3000
+2000
+1000
+0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+Other Edge Capacity (sat.)
+1e7800
+p: 0.40
+p: 0.45
+700
+p: 0.50
+600
+p: 0.55
+p: 0.60
+500
+400
+300
+200
+100
+0
+0.5
+1.0
+1.5
+2.0
+2.5
+Edge Capacity (sat.)
+1e67
+Fig. 10. Model evaluation pipeline.
+TABLE 1
+error of prediction and real lifetime
+SK
+1
+4
+6
+10
+0.9
+0.15
+0.10
+0.09
+0.10
+SC
+0.5
+0.11
+0.10
+0.08
+0.07
+0.3
+0.09
+0.07
+0.05
+0.06
+0
+0.11
+0.07
+0.07
+0.07
+6.1.6
+Lifespan Calculator
+This module aggregates the results of 100 iterations of the
+previous step and calculates the average lifespan and its
+error. This data will be used to determine the accuracy of
+the model.
+6.1.7
+Model Evaluation
+The error of each channel is calculated as |real−prediction|
+real
+.
+Because some channels are positioned in a way that almost
+no payments pass through them, only after a long while that
+most channels are unbalanced, some payments pass through
+them, we count these channels as abnormalities and do not
+consider them in the error calculations. These are usually
+the channels that are estimated to have a very long lifespan.
+The means of calculated errors are given in Table (1)
+considering different SC and SK values.
+As we see, better results are obtained with smaller SCs
+(meaning a busier Lightning Network). It is also notable that
+SK value has little to no effect on the model performance.
+This means that the model performs well in either case that
+p is close to 0.5 and p is far from 0.5.
+7
+LIGHTNING NETWORK ANALYSIS
+In this section we will provide an analysis on channel
+lifespans of a recent snapshot of the Lightning Network.
+The simulation is constituted by nodes and channels taken
+from a snapshot of the Lightning Network Mainnet [24] on
+Feb 2019.
+In Section 5, we proposed a model for a payment channel
+using a random walk and we derived a formula to predict
+expected channel lifespans. Moreover, the expected lifespan
+of a payment channel can be found if the rate of payments
+are known by using 2. Lemma 3 shows that if we have the
+same rate for every pair of nodes, the probability of going
+to each side is equal to 0.5.
+Because payment rates and channel balances usually are
+not public in the Lightning Network, we have to make
+assumptions on the distribution, the amount of payments,
+and channel balances. We assume that all payment rates
+have the same value r, which means that the rates matrix
+(MRates) is symmetric. Thus according to Lemma 3, p = 1
+2
+for every channel in the network. According to 11 the
+expected number of payments is equal to a × b, where
+a = ⌊ FA
+ω ⌋ and b = ⌊ FB
+ω ⌋ for a bidirectional channel between
+A and B. We assume that all channels are initially balanced,
+meaning a = b = C
+2ω , where C is the channel’s capacity.
+According to previous results in Section 5 (equations (1)
+and (4)) we have:
+λpayment = (
+�
+s,t∈V
+s̸=t
+σ(s, t|edge(a, b))
+σ(s, t)
++ σ(s, t|edge(b, a))
+σ(s, t)
+)r
+(6)
+We also know that �
+s,t∈V
+s̸=t
+σ(s,t|edge(a,b))
+σ(s,t)
+is equal to the
+edge betweenness centrality of edge(a, b) (EBC(a, b)) in
+directed graph G [25].
+Because all channels are bidirectional: ∀edge(j, i) −→
+∃edge(i, j), thus ∀s, t ∈ V :
+σ(s, t|edge(a, b))
+σ(s, t)
+= σ(t, s|edge(b, a))
+σ(t, s)
+.
+(7)
+Assuming G
+′ as an undirected graph that is derived
+from G we have:
+EBCG(a, b) = EBCG′ (a, b),
+(8)
+thus
+λpayment = 2 × EBCG′ (a, b) × r.
+(9)
+If we put all results in (2), we have:
+Etime =
+( C
+ω )2
+4 × 2 × EBCG′ (a, b) × r
+(10)
+In what follows, we first calculate all payment channels’
+lifespans in the LN snapshot using equation (10). Then we
+focus on the relation between edge betweenness centrality
+and lifespan of the channels.
+7.1
+Distribution of Channels Effective Lifespan
+Equation (10) shows that the lifespan of a channel can be
+calculated based on its edge betweenness centrality and
+initial fund. We assume that r = 0.0022 transactions per day
+[26] and ω = 60000 sat [22]. The distribution of channels
+lifespans in our snapshot is shown in Fig. 11. Much like the
+distribution of channel capacities that resemble the Power
+Law distribution, Fig. 11 shows that there are a lot of
+channels with a low lifespan and very few channels with
+a very high lifespan.
+According to Seres et.al. [20] the most effective channels
+are the channels with the highest betweenness centrality.
+This paper suggests that the top 14% of the channels have
+the most significant effect on the network’s performance.
+Table (2) gives average, standard deviation, and median,
+for all channels in the network and the top 14% central
+channels.
+
+network generation
+MRates generation
+payment generation
+loop for 100 run
+lifespan prediction
+simulation
+lifespan calculation
+model evaluation8
+Fig. 11. Histogram of expected lifespan for the LN snapshot in Feb 2019.
+TABLE 2
+Lifespan statistics of the LN snapshot (day).
+All Channels
+Central Channels
+average
+1833.2
+172.3
+STD
+7086.9
+587.2
+median
+27.0
+1.6
+7.2
+Betweenness-Lifespan Correlation
+As Seres et.al. [20] suggests, the most central channels have
+the most impact on the network. As Fig. 12 shows, more
+central channels have a shorter lifespan because they route
+more payments per unit of time. In Fig. 12 we took batches
+of the most central edges and calculated the average cen-
+trality and the average lifespan per batch. The result shows
+that in general the more central a channel is, the sooner it
+will get unbalanced. We see an exception to this statement
+in the middle of the plot, where betweenness has a positive
+correlation with the average lifespan. This is due to the fact
+that some very central edges have a large capacity so they
+can route more payment considering that lifespan increases
+with capacity quadratically.
+In Section 7 A we showed that channels with larger
+edge betweenness centrality values have a higher impact
+on the performance of the network. In this section, we have
+shown these central channels will have shorter lifespans.
+Therefore, the network success rate will decrease quickly
+Fig. 12. The relation between expected lifespan and betweenness cen-
+trality of channels in the LN snapshot.
+due to unbalancing.
+8
+CONCLUSION
+In this paper we modeled payment channel liquidity with
+a random walk to estimate how long it takes for a channel
+to become unbalanced and the effect of being unbalanced
+on a channel’s probability of successful routing. We also
+analyzed how unbalanced channels degrade the network’s
+performance, and the relation between a channel’s centrality
+and its lifespan. We showed that the network’s success
+probability is sensitive to the channels’ unbalancing.
+We also introduced a method to estimate the lifespan
+of a channel in a payment network which can be used for
+determining a good placement in the network. We provided
+a proof of concept for our model and showed the results are
+95% accurate.
+This work shows that just allocating more funds towards
+a channel does not lead to having a more successful channel.
+The results show the channel’s success in the network
+depends greatly on the network topology, transaction flow,
+and the amount of funds the peer node puts in the channel.
+We suggested the amount a node should invest in a
+channel to get the longest channel lifespan and maximize its
+return on investment. These results show that a misplaced
+channel can have a very short lifespan and lose up to 40%
+of its efficiency, so nodes could potentially create a market
+based on these criteria to sell each other good connections
+in the network.
+APPENDIX A
+PROOFS
+A.1
+Lemma 1. The expected number of steps to reach
++a or −b for the first time starting from zero is
+Esteps =
+� apa(pb−qb)+bqb(qa−pa)
+(p−q)(pa+b−qa+b)
+p ̸= 1/2
+ab
+p = 1/2
+(11)
+Consider Sx as the expected number of steps to reach +a or
+−b for the first time starting from x. Let p be the probability
+of going to the positive direction and q the probability of
+going in the negative direction (p + q = 1). Then we can say
+that if the Random Walk starts from x, he will go to x + 1
+with probability of p and x − 1 with probability of q. so we
+can infer this recurrence equation: sx = 1 + qsx−1 + psx+1
+where sx is the expected number of steps until reaching the
+end point starting from point x. For the boundary conditions
+we have: sa = s−b = 0 implying that the expected number
+of steps needed to reach +a or −b starting from +a or −b is
+zero.
+so:
+sx = 1
+psx−1 − q
+psx−2 − 1
+p
+(12)
+The characteristic equation of (12) is:
+(z2 − 1
+pz + q
+p)(z − 1) = 0
+(13)
+
+104
+103
+Count
+102
+101
+100
+0
+25000
+50000
+75000100000125000150000175000
+Lifespan (day)1750
+1500
+1250
+1000
+750
+500
+250
+0
+5000
+10000
+15000
+20000
+Average Edge Betweenness Centrality of Batch9
+if p = q = 1/2 we have ∆ = 0 therefore z1 = z2 = z3 =
+1 so the expected number of steps needed to reach +a or −b
+starting from x is:
+sx = (a − x)(b − x)
+(14)
+if p ̸= 1/2 we have
+√
+∆ = | 1−2p
+p
+| therefore z1 = z2 =
+1, z3 = q
+p and for the number of steps we have:
+sx =
+apa+b + bqa+b
+(2p − 1)(pa+b − qa+b) +
+1
+1 − 2p x+
+(a + b)paqb
+(2p − 1)(qa+b − pa+b) ( q
+p )x
+(15)
+we take x = 0 as this gives the expected number of steps to
+reach +a or −b starting from zero. so we have:
+S0 =
+� apa(pb−qb)+bqb(qa−pa)
+(p−q)(pa+b−qa+b)
+p ̸= 1/2
+ab
+p = 1/2
+(16)
+A.2
+Lemma 2. p
+q = λ(A,B)
+λ(B,A)
+In assumptions of Section 5 it is assumed that each node
+sends its payments to other nodes with a Poisson distri-
+bution. The parameter of the distribution for edge(a, b) is
+λ(a, b), which is the payment rate between nodes a and b.
+Assume the random variable of payments from a to b as X
+and the random variable of payments from b to a as Y .Thus
+we have:
+P(X = n) = e−λ(A,B)(λ(A, B))n
+n!
+(17)
+The total payment rate in each channel is the sum of rates of
+its two edges. It is known that the distribution of a random
+variable which is the sum of two random variables with a
+Poisson process is a Poisson process; the rate of this process
+equals the sum of rates.
+When we have a payment from two Poisson distribu-
+tions sending payments to the same channel; The probabil-
+ity for the payment to be a payment from node a to node b
+(p) is:
+p = P(X = 1|X + Y = 1) =
+e−λx(λx)1
+1!
+× e−λy (λy)0
+0!
+e−(λx+λy)(λx+λy)1
+1!
+(18)
+Thus:
+p =
+λx
+λx + λy
+=
+λ(a, b)
+λ(a, b) + λ(b, a)
+(19)
+A.3
+Lemma 3. If ∀s, t ∈ V : MRatesst = MRatests then
+p = 0.5.
+We know from lemma 2 that: p
+q = λ(a,b)
+λ(b,a) so we have:
+p
+q =
+� σ(s,t|edge(a,b))
+σ(s,t)
+MRatesst
+� σ(t,s|edge(b,a))
+σ(t,s)
+MRatests
+(20)
+Because
+all
+channels
+are
+bidirectional(∀edge(a, b)
+:
+∃edge(b, a)) we have ∀s, t ∈ V :
+σ(s, t|edge(a, b))
+σ(s, t)
+= σ(t, s|edge(b, a))
+σ(t, s)
+(21)
+In the other hand if we have ∀s, t ∈ V : MRatesst =
+MRatests, we can say:
+σ(s, t|edge(a, b))
+σ(s, t)
+×MRatesst = σ(T, S|edge(b, a))
+σ(t, s)
+×MRatests (22)
+then finally we have:
+λ(a, b) = λ(b, a)
+(23)
+so
+p = 1
+2
+(24)
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+Soheil Zibakhsh Shabgahi received his bach-
+elor’s degree in Computer Engineering from
+the Department of Electrical and Computer En-
+gineering, University of Tehran, Tehran, Iran.
+He is currently a research assistant at the
+Data lab under the supervision of professor
+Behnam Bahrak. His research interest are in
+blockchain systems, Theoretical Computer Sci-
+ence,distributed systems, and Machine Learn-
+ing.
+Seyed Mahdi Hosseini is currently an under-
+graduate student majoring in computer engi-
+neering at the School of Electrical and Computer
+Engineering at the College of Engineering of
+the University of Tehran. His research interest
+consists of blockchain, system networks, and
+distributed systems.
+Seyed
+Pooya
+Shariatpanahi
+received
+the
+B.Sc., M.Sc., and Ph.D. degrees from the De-
+partment of Electrical Engineering, Sharif Uni-
+versity of Technology, Tehran, Iran, in 2006,
+2008, and 2013, respectively. He is currently an
+Assistant Professor with the School of Electri-
+cal and Computer Engineering, College of En-
+gineering, University of Tehran. Before joining
+the University of Tehran, he was a Researcher
+with the Institute for Research in Fundamental
+Sciences (IPM), Tehran. His research interests
+include information theory, network science, wireless communications,
+and complex systems. He was a recipient of the Gold Medal at the
+National Physics Olympiad in 2001.
+Behnam Bahrak received his bachelor’s and
+master’s degrees, both in electrical engineering,
+from Sharif University of Technology, Tehran,
+Iran, in 2006 and 2008, respectively. He received
+the Ph.D. degree from the Bradley Department
+of Electrical and Computer Engineering at Vir-
+ginia Tech in 2013. He is currently an Assistant
+Professor of Electrical and Computer Engineer-
+ing at University of Tehran.
+

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+arXiv:2301.13776v1  [math.OC]  31 Jan 2023
+REAL FACTORIZATION OF POSITIVE SEMIDEFINITE MATRIX POLYNOMIALS
+SARAH GIFT AND HUGO J. WOERDEMAN
+Abstract. A real symmetric positive semidefinite matrix polynomial Q(x) with square determinant that
+is not identically zero can be factored as
+Q(x) = G(x)T G(x)
+where G(x) is itself a real square matrix polynomial with degree half that of Q(x). We provide a constructive
+proof of this fact, rooted in finding a skew-symmetric solution to a modified algebraic Riccati equation
+XSX − XR + RT X + P = 0,
+where P, R, S are real n × n matrices with P and S real symmetric.
+Keywords Positive semidefinite matrix polynomial, Algebraic Riccati equation, Matrix factorization
+MSCcodes 47A68 (Primary), 46C20, 15A48, 93B05
+1. Introduction
+The Fej´er-Riesz factorization was first shown for matrix polynomials by Rosenblatt [19] and Helson [14]. In
+particular, given a matrix polynomial Q(x) = �2m
+i=0 Qixi with Qi Hermitian and Q(x) positive semidefinite
+for all x ∈ R, we can factorize it as
+Q(x) = G(x)∗G(x)
+where G(x) = �m
+i=0 Gixi. In 1964, Gohberg generalized this factorization to certain operator-valued poly-
+nomials [8].
+Later it was further generalized to operator-valued polynomials in general form [20].
+The
+multivariable case has also been studied, e.g. in [17]. For an overview of the work done with the operator-
+valued Fej´er-Riesz theorem, see [5].
+In this paper, we provide a constructive proof of the real analog of the Fej´er-Riesz factorization of matrix-
+valued polynomials.
+In particular, we show that a matrix polynomial Q(x) = �2m
+i=0 Qixi with Qi real
+symmetric, Q(x) positive semidefinite for all x ∈ R, and det(Q(x)) equal to a nonzero square, admits the
+factorization
+Q(x) = G(x)T G(x)
+where G(x) is itself a real square matrix polynomial with degree half that of Q(x). This result was first
+shown by Hanselka and Sinn [13] using methods from projective algebraic geometry and number theory. We
+provide an alternative, linear algebraic proof, inspired by the proof of the Fej´er-Riesz factorization presented
+in Section 2.7 of [1]. That proof in turn, was taken from [6, 12]. For earlier work on factorizations of real
+symmetric matrix polynomials (not necessarily positive semidefinite) see e.g. [18].
+A key part of the Fej´er-Riesz factorization proof we follow requires finding a Hermitian solution to an
+algebraic Riccati equation
+(∗)
+XDX + XA + A∗X − C = 0,
+where D and C are Hermitian. Reducing a factorization problem to solving a Riccati equation is a technique
+that has been used in many other papers as well (see, e.g., [2], [7], [15, Chapter 19] and references therein).
+This technique is useful because Riccati equations have been studied extensively. Early work was done by
+Willems [21] and Coppel [3] in analyzing properties of solutions of continuous algebraic Riccati equations.
+Another key paper was [4] where Riccati equations were used to solve H∞-control problems. For an in depth
+analysis of algebraic Riccati equations, please see the book by Lancaster and Rodman [15].
+Both authors were supported by National Science Foundation grant DMS 2000037 .
+1
+
+2
+SARAH GIFT AND HUGO J. WOERDEMAN
+For the current real factorization problem, we end up needing to find a real skew-symmetric solution to
+an equation of the form
+(∗∗)
+XSX − XR + RT X + P = 0,
+where P and S are real symmetric. This is not quite an algebraic Riccati equation of the form eq. (∗) and
+thus we call it a modified algebraic Riccati equation. In general, to find a skew-Hermitian solution X, one
+often considers instead iX, which is Hermitian; however, we want real solutions and thus this method is
+not applicable here. Thus we instead follow the same steps presented in [15] for finding a real symmetric
+solution to the real algebraic Riccati equation and amend them to our current situation. This is the topic
+of Section 2, which culminates in giving sufficient conditions for a skew-symmetric solution of our modified
+Riccati equation eq. (∗∗). In Section 3 we provide additional background on matrix polynomials necessary
+for our main result, the real factorization of a symmetric positive semidefinite matrix polynomial, presented
+in Section 4. The major advantage of our proof compared to that by Hanselka and Sinn [13] is that ours is
+constructive. We end this paper with a couple examples illustrating the construction.
+2. A Modified Algebraic Riccati Equation
+The goal of this section is to provide necessary and sufficient conditions for which there is a real skew-
+symmetric solution X to the modified algebraic Riccati equation
+(1)
+XSX − XR + RT X + P = 0,
+where P, R, S are real n × n matrices with P and S real symmetric. In the book Algebraic Riccati Equations
+[15], Lancaster and Rodman show conditions for which there is a real symmetric solution X to the CARE
+equation
+XDX + XA + AT X − C = 0,
+where A, C, D are real n × n matrices with C and D real symmetric. We amend these results to the present
+situation. Define the 2n × 2n real matrices
+(2)
+Mr =
+�R
+−S
+P
+RT
+�
+,
+ˆHr =
+�0
+I
+I
+0
+�
+,
+Hr =
+�
+P
+RT
+R
+−S
+�
+.
+Then both ˆHr and Hr are real symmetric. Also
+ˆHrMr = M T
+r ˆHr
+and
+HrMr = M T
+r Hr.
+Using terminology from [15], we say Mr is both Hr-symmetric and ˆHr-symmetric.
+We can now give a
+condition for a real solution of eq. (1) to exist.
+Proposition 2.1. X is a real solution of eq. (1) if and only if
+G(X) := Im
+�
+I
+X
+�
+is Mr-invariant.
+Proof. If G(X) is Mr-invariant, then
+(3)
+�R
+−S
+P
+RT
+� � I
+X
+�
+=
+� I
+X
+�
+Z
+for some n × n matrix Z. The first block row gives Z = R − SX and the second gives P + RT X = XZ.
+Combining the two gives
+P + RT X = X(R − SX).
+Thus X solves eq. (1). Conversely, if X solves eq. (1), then eq. (3) holds for Z = R − SX and ths G(X) is
+Mr-invariant.
+□
+More than just a real solution, though, we want a skew-symmetric solution. Thus we next strive to give a
+condition for such a solution. For this we first need a definition (see [15]).
+Definition 2.1. Let H be a real invertible n × n matrix. A subspace M is called
+(1) H-nonnegative if ⟨Hx, x⟩ ≥ 0 for all x ∈ M
+(2) H-nonpositive if ⟨Hx, x⟩ ≤ 0 for all x ∈ M
+
+REAL FACTORIZATION OF POSITIVE SEMIDEFINITE MATRIX POLYNOMIALS
+3
+(3) H-neutral if ⟨Hx, x⟩ = 0 for all x ∈ M
+Proposition 2.2. Let X be a real solution of eq. (1). Then,
+(1) X is skew-symmetric if and only if G(X) is ˆHr-neutral.
+(2) G(X) is Hr-nonpositive if and only if (XT + X)(R − SX) ≤ 0.
+Proof.
+(1) Assume X is skew-symmetric. Let y ∈ G(X), so y =
+� I
+X
+�
+z for some z ∈ Rn. Then
+⟨Hry, y⟩ = zT �I
+XT � ˆHr
+� I
+X
+�
+z = zT (X + XT )z = 0.
+Thus G(X) is ˆHr-neutral. On the other hand, suppose G(X) is ˆHr-neutral. Then for all z ∈ Rn,
+zT(X + XT )z = 0, so X + XT = 0. Thus X is skew-symmetric.
+(2) G(X) is Hr-nonpositive if and only if for all z ∈ Rn,
+�
+Hr
+�
+I
+X
+�
+z,
+�
+I
+X
+�
+z
+�
+≤ 0
+zT �I
+XT � �P
+RT
+R
+−S
+� � I
+X
+�
+z ≤ 0
+zT[P + RT X + XT R − XT SX]z ≤ 0
+zT [XR − XSX + XT R − XT SX]z ≤ 0
+by eq. (1)
+zT (XT + X)(R − SX)z ≤ 0
+Thus G(X) is Hr-nonpositive if and only if (XT + X)(R − SX) ≤ 0.
+□
+proposition 2.2 shows that in order to get a real skew-symmetric solution X to the equation eq. (1), we need
+an ˆHr-neutral subspace G(X) of dimension n. We consider now conditions for such a subspace to exist. For
+this we first state a few known results (see [15]).
+Definition 2.2. Let A be a square matrix and λi be an eigenvalue of A. We call the sizes of the Jordan
+blocks of λi the partial multiplicities of λi.
+Theorem 2.1. [15, Part of Theorem 2.6.3]
+Let A be a real n × n H-symmetric matrix and the partial
+multiplicities of A corresponding to the real eigenvalues are all even. Then there exists an A-invariant H-
+neutral subspace of dimension k − p where k is the number of positive eigenvalues of H (counting algebraic
+multiplicities) and p is the number of distinct pairs of non-real complex conjugate eigenvalues of A with odd
+algebraic multiplicity.
+Lemma 2.1. [15, p. 56] Let H be a real symmetric matrix. A real subspace M is H-neutral if and only if
+⟨Hx, y⟩ = 0 for all x, y ∈ M.
+Proof. This follows from the relation
+⟨Hx, y⟩ = 1
+2 (⟨H(x + y), x + y⟩ − ⟨Hx, x⟩ − ⟨Hy, y⟩) .
+□
+Now we are ready for the new result.
+Lemma 2.2. Let eq. (2) hold with P and S real symmetric. Among the following statements, (iii) =⇒
+(ii) =⇒ (i)
+(i) There exists an n-dimensional Mr-invariant Hr-neutral subspace
+(ii) There exists an n-dimensional Mr-invariant ˆHr-neutral subspace
+(iii) All real eigenvalues of Mr have even partial multiplicities and all imaginary eigenvalues of Mr have
+even algebraic multiplicity.
+If in addition Mr is invertible, then we also have (i) =⇒ (ii).
+
+4
+SARAH GIFT AND HUGO J. WOERDEMAN
+Proof. By theorem 2.1, if every imaginary eigenvalue of Mr has even algebraic multiplicity (i.e. p = 0),
+then an n-dimensional Mr-invariant ˆHr-neutral subspace exists (since ˆHr has n positive and n negative
+eigenvalues so k = n here). Thus (iii) =⇒ (ii). Next, since Hr = ˆ
+HrMr, it follows by lemma 2.1 that (ii)
+implies (i). Finally, if Mr is invertible, ˆ
+Hr = HrM −1
+r
+and thus (i) implies (ii).
+□
+We need one more result before the main theorem of this section. For this result, we first recall a definition
+(see, e.g., [15]).
+Definition 2.3. Let A ∈ Rn×n and B ∈ Rn×m. The pair (A, B) is said to be controllable if
+rank
+�B
+AB
+A2B
+· · ·
+An−1B�
+= n.
+Lemma 2.3. Assume that S ≥ 0 (positive semidefinite) and the pair (R, S) is controllable. Let L be an
+n-dimensional Mr-invariant Hr-nonnegative subspace of R2n. Then L is a graph subspace, i.e.
+L = Im
+� I
+X
+�
+for some real n × n matrix X.
+Proof. For L as defined in the statement, write
+L = Im
+�
+X1
+X2
+�
+for some real n × n matrices X1 and X2. We shall show that X1 is invertible. First, since L is Mr-invariant,
+�
+R
+−S
+P
+RT
+� �
+X1
+X2
+�
+=
+�
+X1
+X2
+�
+T
+for some n × n matrix T . Thus,
+RX1 − SX2 = X1T
+(4)
+PX1 + RT X2 = X2T
+(5)
+Next, since L is Hr-nonnegative, we know
+(6)
+�XT
+1
+XT
+2
+� �
+P
+RT
+R
+−S
+� �
+X1
+X2
+�
+= XT
+1 PX1 + XT
+1 RT X2 + XT
+2 RX1 − XT
+2 SX2
+is positive semidefinite. Let K = ker X1. Since eq. (6) is positive semidefinite, for every x ∈ K,
+0 ≤ xT XT
+1 PX1x + xT XT
+1 RT X2x + xT XT
+2 RX1x − xT XT
+2 SX2x = −xT XT
+2 SX2x.
+Since S ≥ 0, X2x ∈ ker S, so
+X2K ⊂ ker S.
+Then, equation eq. (4) implies
+T K ⊂ K.
+Consequently, equation eq. (5) gives
+RT X2K ⊂ X2K.
+All together, we have
+RT X2K ⊂ ker S.
+By induction, we get
+(RT )rX2K ⊂ ker S,
+r = 0, 1, 2, . . .
+Now for every x ∈ K,
+
+
+S
+SRT
+...
+S(RT )n−1
+
+ (X2x) = 0
+
+REAL FACTORIZATION OF POSITIVE SEMIDEFINITE MATRIX POLYNOMIALS
+5
+Since (R, S) is controllable, we must have X2x = 0. The only n-dimensional vector x for which X1x =
+X2x = 0 is the zero vector (otherwise dim L < n). Thus K = {0} and X1 is invertible. Hence
+L = Im
+�
+I
+X
+�
+,
+where X = X2X−1
+1
+and so L is a graph subspace.
+□
+Now we put everything together to get sufficient conditions for a real skew-symmetric solution of eq. (1):
+Theorem 2.2. Let eq. (2) hold with P and S real symmetric. Assume that S ≥ 0 and the pair (R, S) is
+controllable. Then the conditions (i) - (ii) are equivalent and (iii) implies (ii)
+(i) The equation eq. (1) has a real skew-symmetric solution.
+(ii) There exists an n-dimensional Mr-invariant ˆ
+Hr-neutral subspace.
+(iii) All real eigenvalues of Mr have even partial multiplicities and all imaginary eigenvalues of Mr have
+even algebraic multiplicity.
+Proof. The (iii) =⇒ (ii) was part of lemma 2.2. By proposition 2.1 and proposition 2.2, we know that
+if X is a real skew-symmetric solution to eq. (1), then G(X) is an n-dimensional Mr-invariant ˆHr-neutral
+subspace.
+Thus (i)
+=⇒
+(ii).
+Finally, assume there exists an n-dimensional Mr-invariant ˆHr-neutral
+subspace, say L. Since Hr = ˆHrMr, L is also an n-dimensional Mr-invariant Hr-neutral subspace. Clearly
+L is an Hr-nonnegative subspace, so by lemma 2.3, L is a graph subspace. Since L = G(X) is Mr-invariant,
+X is a real solution of eq. (1) by proposition 2.1. Since L = G(X) is also ˆHr-neutral, by proposition 2.2 X
+is skew-symmetric. Thus (ii) =⇒ (i).
+□
+3. Matrix Polynomials
+Building toward our goal of factorizing a real symmetric positive semidefinite matrix polynomial, we next
+state a few relevant results on matrix polynomials (see [11]).
+Definition 3.1. For n × n matrices Pi, we define an n × n matrix polynomial P(x) of degree m by
+P(x) =
+m
+�
+i=0
+Pixi.
+• The matrix polynomial is called real if all Pi are real matrices.
+• The matrix polynomial is called monic if Pm = In.
+• The matrix polynomial is called self-adjoint if Pi = P ∗
+i , its conjugate transpose, for all i.
+• The matrix polynomial is called symmetric if Pi = P T
+i
+for all i.
+• The matrix polynomial is called positive semidefinite (also nonnegative) if for all x ∈ R, P(x) is
+positive semidefinite.
+• The matrix polynomial is called regular if det(P(x)) is not identically zero.
+As in [16], we define the spectrum and Jordan canonical form of a matrix polynomial in the following
+ways.1
+Definition 3.2. Let P(x) be a regular matrix polynomial. Then the set of eigenvalues of P, i.e. the spectrum,
+is
+σ(P) := {λ ∈ C : det(P(λ)) = 0}.
+Definition 3.3. Let P(x) be a monic matrix polynomial, i.e.
+P(x) = Inxm +
+m−1
+�
+i=0
+Pixi.
+1Note that the partial multiplicities of an eigenvalue of a matrix polynomial are often defined as powers of the elementary
+divisors; however, these partial multiplicities are the same as the sizes of the Jordan blocks of our companion matrix. See the
+Appendix of [10] for a more in depth understanding of matrix polynomial equivalences, linearizations, partial multiplicities and
+elementary divisors.
+
+6
+SARAH GIFT AND HUGO J. WOERDEMAN
+The Jordan canonical form for P(x) is defined to be that of the companion matrix
+Cp :=
+
+
+0
+In
+0
+· · ·
+0
+0
+0
+In
+· · ·
+0
+...
+...
+...
+...
+...
+0
+0
+· · ·
+0
+In
+−P0
+−P1
+· · ·
+−Pm−2
+−Pm−1
+
+
+Theorem 3.1. [11, part of Theorem 12.8] For a monic self-adjoint matrix polynomial P(x), the following
+statements are equivalent:
+(i) P(x) is nonnegative.
+(ii) The partial multiplicities of P(x) for real points of the spectrum are all even.
+In the previous section, we found that if all real eigenvalues of Mr have even partial multiplicities and all
+imaginary eigenvalues of Mr have even algebraic multiplicity, then our equation eq. (1) has the desired real
+skew-symmetric solution. Thus we now want to connect the eigenvalues of Mr and their partial multiplicities
+with the eigenvalues of a monic nonnegative matrix polynomial P(x). For this, we begin with a few definitions
+from [11].
+Definition 3.4. Two matrix polynomials M1(x) and M2(x) of size n × n are called equivalent if
+M1(x) = E(x)M2(x)F(x)
+for some n × n matrix polynomials E(x) and F(x) with constant nonzero determinants.
+Definition 3.5. Let P(x) be an n × n monic matrix polynomial of degree m. A linear matrix polynomial
+λI − A is called a linearization of P(x) if
+(7)
+λI − A ∼
+�
+P(x)
+0
+0
+In(m−1)
+�
+where ∼ means equivalence of matrix polynomials.
+Note that Cp, the companion matrix, is a linearization of P(x). For any linearization λI − A, the partial
+multiplicities in every eigenvalue of A and P(x) are the same. Thus to gain results about the multiplicities
+of the eigenvalues of Mr, it suffices to show λI − Mr is a linearization of a matrix polynomial P(x).
+Lemma 3.1. Let Q(x) = �2m
+j=0 Qjxj be an n × n real symmetric matrix polynomial of degree 2m with
+Q0 = In. Then for
+Mr =
+
+
+− 1
+2Q1
+−In
+In
+0
+0
+...
+...
+...
+In
+0
+0
+Q2 − 1
+4Q2
+1
+1
+2Q3
+− 1
+2Q1
+In
+1
+2Q3
+Q4
+...
+0
+...
+...
+...
+1
+2Q2m−1
+...
+In
+1
+2Q2m−1
+Q2m
+0
+
+
+,
+we have
+λI − Mr ∼
+�
+rev Q(λ)
+0
+0
+I
+�
+where
+rev Q(x) :=
+2m
+�
+i=0
+Q2m−ixi = x2mIn +
+2m−1
+�
+i=1
+Q2m−ixi.
+
+REAL FACTORIZATION OF POSITIVE SEMIDEFINITE MATRIX POLYNOMIALS
+7
+Proof. First permute the rows and columns as follows:
+M1:=
+
+
+0
+0
+· · ·
+0
+In
+0
+In
+...
+0
+0
+...
+...
+...
+...
+0
+In
+0
+· · ·
+0
+In
+0
+· · ·
+0
+0
+0
+0
+...
+In
+0
+...
+...
+0
+...
+...
+0
+· · ·
+0
+In
+0
+
+
+(λI − Mr)
+
+
+In
+...
+In
+0
+0
+In
+...
+In
+
+
+,
+=
+
+
+−Q2m
+− 1
+2Q2m−1
+0
+λIn
+λIn
+−In
+0
+...
+...
+...
+...
+...
+λIn
+−In
+0
+λIn + 1
+2Q1
+In
+− 1
+2Q3
+1
+4Q2
+1 − Q2
+λIn + 1
+2Q1
+−In
+...
+−Q4
+− 1
+2Q3
+λIn
+...
+...
+...
+...
+...
+...
+− 1
+2Q2m−1
+−Q2m−2
+− 1
+2Q2m−3
+λIn
+−In
+
+
+Define V1, . . . , Vm by
+V1 = −Q2m − 1
+2λQ2m−1
+V2 = −1
+2Q2m−1 − λQ2m−2 − 1
+2λ2Q2m−3
+V3 = −1
+2λQ2m−3 − λ2Q2m−4 − 1
+2λ3Q2m−5
+...
+Vm−1 = −1
+2λm−3Q5 − λm−2Q4 − 1
+2λm−1Q3
+Vm = −1
+2λm−2Q3 − λm−1Q2 − λmQ1 − λm+1In
+Then, multiply M1 on the left by
+
+
+In
+�m
+i=2 λi−2Vi
+�m
+i=3 λi−3Vi
+· · ·
+Vm−1 + λVm
+Vm
+−λm−1(λIn + 1
+2Q1)
+λm−1In
+· · ·
+λ2In
+λIn
+0
+In(m−1)
+0
+0
+0
+Inm
+
+.
+Noting that
+−Q2m − 1
+2λQ2m−1 + λ
+m
+�
+i=2
+λi−2Vi = − rev Q(λ),
+
+8
+SARAH GIFT AND HUGO J. WOERDEMAN
+we get
+λI − Mr ∼
+
+
+− rev Q(λ)
+−In
+0
+∗
+...
+−In
+0
+∗
+In
+−In
+0
+∗
+...
+−In
+
+
+Thus it is evident now that
+λI − Mr ∼
+�
+rev Q(λ)
+0
+0
+I
+�
+.
+□
+4. Real Factorization of Non-negative Matrix Polynomial
+We are now ready for the main result. While the following theorem was previously proven by Hanselka
+and Sinn [13], we provide a new constructive proof following that of the complex analogue presented in the
+monograph by Bakonyi and Woerdeman [1].
+Theorem 4.1. Let Q(x) = �2m
+j=0 Qjxj be an n × n real symmetric matrix polynomial of degree 2m with
+Q(x) ≥ 0 for all x ∈ R and Q0 > 0. Then det(Q(x)) is a square if and only if there exists an n × n real
+matrix polynomial G(x) = �m
+j=0 Gjxj of degree m such that
+Q(x) = G(x)T G(x).
+Proof. First, assume Q(x) = G(x)T G(x). Then det(Q(x)) = det(G(x))2.
+On the other hand, assume
+det(Q(x)) is a square. Without loss of generality, assume Q0 = In (otherwise, take ˜Q(x) := Q−1/2
+0
+Q(x)Q−1/2
+0
+).
+Consider the (m + 1)n × (m + 1)n real symmetric matrix
+F0 =
+
+
+In
+1
+2Q1
+1
+2Q1
+Q2
+...
+...
+...
+1
+2Q2m−1
+1
+2Q2m−1
+Q2m
+
+
+.
+Given an nm × nm real skew-symmetric matrix
+X =
+
+
+X1,1
+· · ·
+X1,m
+...
+...
+Xm,1
+· · ·
+Xm,m
+
+ ,
+let
+FX = F0 +
+� 0
+X
+0n
+0
+�
+−
+� 0
+0n
+X
+0
+�
+.
+It should be noted that in the above line, the matrix decompositions are different; e.g. the X block and the
+−X block overlap in general. We want to solve
+Xopt = arg min
+rank(FX)
+such that FX ≥ 0.
+Let
+A =
+
+
+0
+In
+0
+...
+...
+In
+0
+
+ ∈ Rnm×nm
+and
+B =
+
+
+In
+0
+...
+0
+
+ ∈ Rnm×n.
+
+REAL FACTORIZATION OF POSITIVE SEMIDEFINITE MATRIX POLYNOMIALS
+9
+Then (A, B) is controllable,
+�
+0
+0
+X
+0
+�
+=
+�
+0
+0
+XB
+XA
+�
+,
+and
+�
+0
+X
+0
+0
+�
+=
+��
+0
+BT X
+0
+AT X
+�
+.
+Split F0 into four blocks as follows,
+F0 =
+� In
+Γ12
+Γ21
+Γ22
+�
+.
+Noting ΓT
+12 = Γ21 and Γ22 is real symmetric, we can recast the condition FX ≥ 0 as
+�
+In
+Γ12 + BT X
+ΓT
+12 − XB
+Γ22 + AT X − XA
+�
+≥ 0
+Consider the Schur complement with respect to In,
+Γ22 + AT X − XA − (ΓT
+12 − XB)I−1
+n (Γ12 + BT X).
+Setting this equal to zero, we get the modified algebraic Riccati equation
+(8)
+P + RT X − XR + XSX = 0
+where
+P = Γ22 − ΓT
+12Γ12
+R = A − BΓ12
+S = BBT .
+Note P = P T and S = ST with S ≥ 0. In this case, the associated Mr matrix is as follows:
+Mr =
+� A − BΓ12
+−BBT
+Γ22 − ΓT
+12Γ12
+AT − ΓT
+12BT
+�
+=
+
+
+− 1
+2Q1
+−In
+In
+0
+0
+...
+...
+...
+In
+0
+0
+Q2 − 1
+4Q2
+1
+1
+2Q3
+− 1
+2Q1
+In
+1
+2Q3
+Q4
+...
+0
+...
+...
+...
+1
+2Q2m−1
+...
+In
+1
+2Q2m−1
+Q2m
+0
+
+
+By lemma 3.1, Mr is a linearization of rev Q(x). Since Q(x) ≥ 0 for all x ∈ R, rev Q(x) = x2mQ
+� 1
+x
+�
+≥ 0
+for all nonzero x ∈ R. By continuity, then rev Q(x) ≥ 0 for all x ∈ R. Then by theorem 3.1, the partial
+multiplicities of every real eigenvalue of rev Q(λ) are all even. Since Mr is a linearization of rev Q(x), all
+the partial multiplicities of every eigenvalue of Mr and rev Q(λ) are the same, so the partial multiplicities
+of every real eigenvalue of Mr are all even. Further, by the non-negativity of rev Q(x),
+det(λI − Mr) = det(rev Q(x))
+has all roots of even algebraic multiplicity. In particular, all non-real eigenvalues of Mr have even algebraic
+multiplicity. Hence by theorem 2.2, there is a skew-symmetric solution, ˜X of eq. (8). Then since the Schur
+complement with respect to In is zero, we know
+F ˜
+X =
+�
+In
+Γ12 + BT ˜X
+ΓT
+12 − ˜XB
+Γ22 + AT ˜X − ˜XA
+�
+≥ 0
+
+10
+SARAH GIFT AND HUGO J. WOERDEMAN
+and
+rank(F ˜
+X) = rank
+��
+In
+Γ12 + BT ˜X
+ΓT
+12 − ˜XB
+Γ22 + AT ˜X − ˜XA
+��
+= rank In = n.
+If we factorize
+F ˜
+X =
+�
+In
+Γ12 + BT ˜X
+ΓT
+12 − ˜XB
+Γ22 + AT ˜X − ˜XA
+�
+=
+
+
+GT
+0
+...
+GT
+m
+
+
+�
+G0
+· · ·
+Gm
+�
+with G0 = In and Gi, i = 1, 2, . . . , m, real n × n matrices, we have
+Q(x) =
+�In
+xIn
+· · ·
+xmIn
+�
+F ˜
+X
+
+
+In
+xIn
+...
+xmIn
+
+
+=
+�In
+xIn
+· · ·
+xmIn
+�
+
+
+GT
+0
+...
+GT
+m
+
+
+�G0
+· · ·
+Gm
+�
+
+
+In
+xIn
+...
+xmIn
+
+ .
+Thus for G(x) = �m
+j=0 Gjxj, Q(x) = G(x)T G(x).
+□
+theorem 4.1 required Q0 > 0. We can relax this condition as follows.
+Corollary 4.1. Let Q(x) = �2m
+j=0 Qjxj be an n × n real symmetric matrix polynomial of degree 2m with
+Q(x) ≥ 0 for all x ∈ R and Q(x0) > 0 for some x0 ∈ R. Then det(Q(x)) is a square if and only if there
+exists an n × n real matrix polynomial G(x) = �m
+j=0 Gjxj of degree m such that
+Q(x) = G(x)T G(x).
+Proof. First, assume Q(x) = G(x)T G(x). Then det(Q(x)) = det(G(x))2.
+On the other hand, assume
+det(Q(x)) = f(x)2 for some polynomial f of degree m. Consider
+P(x) := Q(x0 − x).
+Then P(x) is an n × n real symmetric matrix polynomial of degree 2m such that
+P(0) = Q(x0) > 0
+and
+det(P(x)) = det(Q(x0 − x)) = f(x0 − x)2.
+Thus by theorem 4.1, there is an n × n real matrix polynomial H(x) = �m
+j=0 Hjxj of degree m such that
+P(x) = H(x)T H(x).
+Define
+G(x) := H(x0 − x).
+Then G(x) = �m
+j=0 Gjxj is an n × n real matrix polynomial such that
+Q(x) = P(x0 − x) = H(x0 − x)T H(x0 − x) = G(x)T G(x).
+□
+The proof of theorem 4.1 is constructive. It hinges on finding the real skew-symmetric solution X to the
+modified algebraic Riccati equation. Back in Section 2, we found such a solution by constructing an mn-
+dimensional Mr-invariant Hr-neutral subspace. Following the proof of Theorem 2.6.2, found in [15], also to
+be found in Gohberg, Lancaster, and Rodman’s later book Indefinite Linear Algebra and Applications [9],
+we first convert Mr to its real Jordan form
+Mr = SJS−1,
+where
+J = Jr1(λ1) ⊕ · · · ⊕ Jrk(λk) ⊕ J2rk+1(λk+1 ± iµk+1) ⊕ · · · ⊕ J2rk+ℓ(λk+ℓ ± iµk+ℓ).
+
+REAL FACTORIZATION OF POSITIVE SEMIDEFINITE MATRIX POLYNOMIALS
+11
+For real eigenvalues λj, where 1 ≤ j ≤ α, we take the first rj/2 columns of S corresponding to the block
+Jrj (note we know each rj is even). For any imaginary eigenvalues λj ± iµj with rj even, we do the same.
+Finally, we must consider the imaginary eigenvalues λj ± iµj with rj odd. For such an eigenvalues, there
+must be an even number of Jordan blocks of odd size (since we know every imaginary eigenvalue has even
+algebraic multiplicity). Collect the Jordan blocks together in pairs
+Kj = J2rj(λj ± iµj) ⊕ J2rj+1(λj+1 ± iµj+1).
+Then take the first rj − 1 columns of S corresponding to J2rj(λj ± iµj), the first rj+1 − 1 columns of S
+corresponding to J2rj+1(λj+1 ± iµj+1), along with the following two vectors
+(1) The rjth column of S corresponding to J2rj(λj ± iµj) plus the rj+1 + 1st column of S corresponding
+to J2rj+1(λj+1 ± iµj+1).
+(2) The rj +1st column of S corresponding to J2rj(λj ±iµj) minus the rj+1st column of S corresponding
+to J2rj+1(λj+1 ± iµj+1).
+Let us illustrate the above ideas in a few examples. The first example is the real eigenvalue case. The second
+example is the imaginary eigenvalue case with odd rj.
+Example 4.1. Take
+Q(x) =
+�2x2 + 2x + 1
+−4x2 − 3x
+−4x2 − 3x
+8x2 + 4x + 1
+�
+=
+�1
+0
+0
+1
+�
++ x
+� 2
+−3
+−3
+4
+�
++ x2
+� 2
+−4
+−4
+8
+�
+.
+Then Mr(x) = SJS−1 for
+J =
+
+
+−3
+1
+0
+0
+0
+−3
+0
+0
+0
+0
+0
+1
+0
+0
+0
+0
+
+ , S ≈
+
+
+−0.2778
+0.3148
+0.2222
+0.6852
+0.2778
+−0.3704
+0.1111
+0.3704
+−0.1389
+0.3519
+−0.0556
+−0.3519
+−0.1389
+−0.1759
+0.1111
+0.1759
+
+ .
+We have J = J2 (−3) ⊕ J2 (0), so r1 = r2 = 2. Thus we take the 1st column corresponding to the first block
+as well as the 1st column corresponding to the second block.
+
+
+−0.2778
+0.2222
+0.2778
+0.1111
+−0.1389
+−0.0556
+0.1389
+0.1111
+
+ =:
+�
+X1
+X2
+�
+Our invariant subspace is thus
+Im
+�
+X1
+X2
+�
+= Im
+�
+I
+X2X−1
+1
+�
+.
+Here we have then
+X = X2X−1
+1
+=
+�
+0
+−0.5
+0.5
+0
+�
+.
+Thus
+Fopt = F0 +
+�
+0
+X
+0
+0
+�
+−
+�
+0
+0
+X
+0
+�
+=
+
+
+1
+0
+1
+−2
+0
+1
+−1
+2
+1
+−1
+2
+−4
+−2
+2
+−4
+8
+
+ .
+We factorize as
+F =
+
+
+1
+0
+0
+1
+1
+−1
+−2
+2
+
+
+�1
+0
+1
+−2
+0
+1
+−1
+2
+�
+.
+Thus
+G0 =
+�1
+0
+0
+1
+�
+,
+G1 =
+� 1
+−2
+−1
+2
+�
+,
+G(x) = G0 + xG1.
+We can verify Q(x) = G(x)T G(x).
+
+12
+SARAH GIFT AND HUGO J. WOERDEMAN
+Example 4.2. Take now
+Q(x) =
+�2x2 + 2x + 1
+x2 + 2x
+x2 + 2x
+13x2 + 4x + 1
+�
+=
+�1
+0
+0
+1
+�
++ x
+�2
+2
+2
+4
+�
++ x2
+�2
+1
+1
+13
+�
+.
+Then Mr(x) = SJS−1 for
+J = 1
+2
+
+
+−3
+√
+11
+0
+0
+−
+√
+11
+−3
+0
+0
+0
+0
+−3
+√
+11
+0
+0
+−
+√
+11
+−3
+
+ , S ≈
+
+
+0.5477
+0
+−0.5477
+0
+0.0913
+−0.3028
+−0.0913
+0.3028
+0.1826
+−0.6055
+−0.1826
+0.6055
+−1.0954
+0
+1.0954
+0
+
+ .
+We have J = J2
+�
+3±i
+√
+11
+2
+�
+⊕ J2
+�
+3±i
+√
+11
+2
+�
+, so r1 = r2 = 1. Thus we take the 1st column corresponding to the
+first block plus the 2nd column corresponding to the second block as well as the 2nd column corresponding
+to the first block minus the first column corresponding to the second block.
+
+
+0.5477 + 0
+0 − (−0.5477)
+0.0913 + 0.3028
+−0.3028 − (−0.0913)
+0.1826 + 0.6055
+−0.6055 − (−0.1826)
+−1.0954 + 0
+0 − 1.0954
+
+ =:
+�X1
+X2
+�
+Our invariant subspace is thus
+Im
+�
+X1
+X2
+�
+= Im
+�
+I
+X2X−1
+1
+�
+.
+Here we have then
+X = X2X−1
+1
+=
+� 0
+2
+−2
+0
+�
+.
+Thus
+Fopt = F0 +
+�0
+X
+0
+0
+�
+−
+� 0
+0
+X
+0
+�
+=
+
+
+1
+0
+1
+3
+0
+1
+−1
+2
+1
+−1
+2
+1
+3
+2
+1
+13
+
+ .
+We factorize as
+F =
+
+
+1
+0
+0
+1
+1
+−1
+3
+2
+
+
+�1
+0
+1
+3
+0
+1
+−1
+2
+�
+.
+Thus
+G0 =
+�1
+0
+0
+1
+�
+,
+G1 =
+� 1
+3
+−1
+2
+�
+,
+G(x) = G0 + xG1.
+We can verify Q(x) = G(x)T G(x).
+References
+[1] M. Bakonyi and H. J. Woerdeman, Matrix Completions, Moments, and Sums of Hermitian Squares, Princeton Series
+in Applied Mathematics, Princeton University Press, 2011, http://www.jstor.org/stable/j.ctt7rp1d.
+[2] T. Chen and B. A. Francis, Spectral and inner-outer factorizations of rational matrices, SIAM Journal on Matrix
+Analysis and Applications, 10 (1989), pp. 1–17, https://doi.org/10.1137/0610001.
+[3] W. A. Coppel, Matrix quadratic equations, Bulletin of the Australian Mathematical Society, 10 (1974), pp. 377 – 401,
+https://doi.org/10.1017/S0004972700041071.
+[4] J. Doyle, K. Glover, P. Khargonekar, and B. Francis, State space solution to standard H2 and H∞ control problem,
+IEEE Transactions on Automatic Control, 34 (1989), pp. 831 – 847, https://doi.org/10.1109/9.29425.
+[5] M. A. Dritschel and J. Rovnyak, The operator fej´er-riesz theorem, in A Glimpse at Hilbert Space Operators: Paul
+R. Halmos in Memoriam, S. Axler, P. Rosenthal, and D. Sarason, eds., Springer Basel, Basel, 2010, pp. 223–254, https:
+//doi.org/10.1007/978-3-0346-0347-8_14.
+[6] M. A. Dritschel and H. J. Woerdeman, Outer factorizations in one and several variables, Transactions of the American
+Mathematical Society, 357 (2005), pp. 4661–4679, http://www.jstor.org/stable/3845206.
+[7] B. A. Francis, A course in H∞ control theory, vol. 88 of Lecture Notes in Control and Information Sciences, Springer-
+Verlag, Berlin, 1987, https://doi.org/10.1007/BFb0007371.
+[8] I. Gohberg, The factorization problem for operator functions, Izvestiya Akademii Nauk SSSR Seriya Matematicheskaya,
+28 (1964), pp. 1055–1082.
+
+REAL FACTORIZATION OF POSITIVE SEMIDEFINITE MATRIX POLYNOMIALS
+13
+[9] I. Gohberg, P. Lancaster, and L. Rodman, Indefinite Linear Algebra and Applications, Birkh¨auser, 2005, https:
+//doi.org/10.1007/b137517.
+[10] I. Gohberg, P. Lancaster, and L. Rodman, Invariant Subspaces of Matrices with Applications, SIAM, 2006, https:
+//doi.org/10.1137/1.9780898719093.
+[11] I. Gohberg,
+P. Lancaster,
+and L. Rodman,
+Matrix Polynomials,
+SIAM, 2009,
+https://doi.org/10.1137/1.
+9780898719024.
+[12] Y. Hachez and H. J. Woerdeman, The fischer-frobenius transformation and outer factorization, in Operator theory,
+structured matrices, and dilations, vol. 10 of Theta Series in Advanced Mathematics, Theta, Bucharest, 2007, pp. 181–203.
+[13] C. Hanselka and R. Sinn, Positive semidefinite univariate matrix polynomials, Mathematische Zeitschrift, 292 (2019),
+pp. 83–101, https://doi.org/10.1007/s00209-018-2137-7.
+[14] H. Helson, Lectures on invariant subspaces, Academic Press, 1964, https://doi.org/10.1016/C2013-0-12454-3.
+[15] P. Lancaster and L. Rodman, Algebraic Riccati Equations, Oxford University Press, 1995.
+[16] P. Lancaster and I. Zaballa, Spectral theory for self-adjoint auadratic eigenvalue problems - a review, ILAS, 37 (2021),
+pp. 211–246, https://doi.org/10.13001/ela.2021.5361.
+[17] J. W. McLean and H. J. Woerdeman, Spectral factorizations and sums of squares representations via semidefinite
+programming, SIAM J. Matrix Anal. Appl., 23 (2001/02), pp. 646–655, https://doi.org/10.1137/S0895479800371177.
+[18] A. C. M. Ran and L. Rodman, Factorization of matrix polynomials with symmetries, SIAM Journal on Matrix Analysis
+and Applications, 15 (1994), pp. 845–864, https://doi.org/10.1137/S0895479892235502.
+[19] M. Rosenblatt, A multi-dimensional prediction problem, Arkiv f¨or Matematik, 3 (1958), pp. 407–424, https://doi.org/
+10.1007/BF02589495.
+[20] M. Rosenblum, Vectorial teoplitz operators and the fej´er-riesz theorem, Journal of Mathematical Analysis and Applica-
+tions, 23 (1968), pp. 139–147.
+[21] J. Willems, Least squares stationary optimal control and the algebraic riccati equation, IEEE Transactions on Automatic
+Control, 16 (1971), pp. 621–634, https://doi.org/10.1109/TAC.1971.1099831.
+

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+Electron spin secluded inside a bottom-up assembled standing metal-molecule
+nanostructure
+Taner Esat,1, 2, ∗ Markus Ternes,1, 2, 3 Ruslan Temirov,1, 2, 4 and F. Stefan Tautz1, 2, 5
+1Peter Gr¨unberg Institute (PGI-3), Forschungszentrum J¨ulich, 52425 J¨ulich, Germany
+2J¨ulich Aachen Research Alliance (JARA), Fundamentals of Future Information Technology, 52425 J¨ulich, Germany
+3Institute of Physics II B, RWTH Aachen University, 52074 Aachen, Germany
+4Institute of Physics II, University of Cologne, 50937 Cologne, Germany
+5Experimental Physics IV A, RWTH Aachen University, 52074 Aachen, Germany
+(Dated: January 30, 2023)
+Artificial nanostructures, fabricated by placing building blocks such as atoms or molecules in well-
+defined positions, are a powerful platform in which quantum effects can be studied and exploited on
+the atomic scale. Here, we report a strategy to significantly reduce the electron-electron coupling
+between a large planar aromatic molecule and the underlying metallic substrate. To this end, we
+use the manipulation capabilities of a scanning tunneling microscope (STM) and lift the molecule
+into a metastable upright geometry on a pedestal of two metal atoms. Measurements at millikelvin
+temperatures and magnetic fields reveal that the bottom-up assembled standing metal-molecule
+nanostructure has an S = 1/2 spin which is screened by the substrate electrons, resulting in a Kondo
+temperature of only 291 ± 13 mK. We extract the Land´e g-factor of the molecule and the exchange
+coupling Jρ to the substrate by modeling the differential conductance spectra using a third-order
+perturbation theory in the weak coupling and high-field regimes. Furthermore, we show that the
+interaction between the STM tip and the molecule can tune the exchange coupling to the substrate,
+which suggests that the bond between the standing metal-molecule nanostructure and the surface
+is mechanically soft.
+On the way to spin qubits based on single atoms or
+molecules, it is essential to minimize the interaction with
+the environment, since the latter leads to decoherence
+[1]. The scanning tunneling microscope (STM) is an ideal
+tool to study quantum properties of nanoscale structures,
+because it not only allows the magnetic states of individ-
+ual atoms and molecules to be read out [2] and coher-
+ently controlled [3–5], but also enables the environment
+to be changed directly.
+The ability to arrange atoms
+and molecules on surfaces with atomic precision allows
+for the fabrication and study of unprecedented artificial
+nanostructures [6, 7]. Moreover, the STM can be used
+to fabricate multiple absolutely identical qubits [8] from
+individual atoms and molecules, which can also be ar-
+ranged and coupled with each other as desired. Com-
+pared to mesoscopic qubits, the structural control down
+to the atomic level may offer advantages.
+Magnetic atoms and molecules with degenerate ground
+states on metallic surfaces typically show the Kondo ef-
+fect: the spin degree of freedom is quenched at temper-
+atures below a characteristic Kondo temperature TK by
+the formation of a many-electron singlet state with the
+electrons of the bath [9, 10]. Because TK depends di-
+rectly on the coupling with the metal, the Kondo effect
+itself can be used as a gauge of the interaction with the
+environment. The strong hybridization of the d-orbitals
+of magnetic atoms with states of the metal substrate
+leads to TK of typically 40 - 300 K [11]. For magnetic
+molecules, on the other hand, Kondo temperatures of
+only a few Kelvin have been observed on metal surfaces
+[12–14], which can be explained by the weaker hybridiza-
+tion of the molecular orbitals with the substrate, or the
+shielding of the magnetic atoms by the surrounding lig-
+ands of the molecule. However, long relaxation times T1
+of several hundred nanoseconds up to days [15, 16] and
+dephasing times T2 in the nanosecond range [3–5] were so
+far only achieved for atoms and molecules that were de-
+coupled from the metallic surface by an atomically thin
+insulating layer.
+The presence of the decoupling layer
+has also resulted in a significant reduction of TK to a few
+Kelvin for magnetic atoms [17, 18].
+In this work, we show that exploiting the third
+dimension for the bottom-up assembly of standing
+metal-molecule nanostructures offers an alternative ap-
+proach to tune the coupling with the metallic sub-
+strate. Specifically, we show that for a single 3,4,9,10-
+perylenetetracarboxylic dianhydride (PTCDA) in the
+standing configuration on a pedestal of two Ag adatoms
+(Fig. 1a), both the interaction with the Ag(111) sub-
+strate is drastically reduced compared to the flat-lying
+PTCDA and the coupling with the metal substrate can
+be tuned by stretching the bond between the molecule
+and surface, utilizing the attraction between the STM
+tip and the molecule. We report the fabrication of an
+S = 1/2 spin nanostructure based on this strategy, with a
+very weak coupling to the underlying substrate, resulting
+in a TK of only 291 ± 13 mK — to our knowledge, the
+smallest TK ever measured on a metallic substrate using
+STM. Comparably low TK have so far only been found
+in mesoscopic quantum dots [19, 20].
+The Ag(111) surface was prepared in ultra-high vac-
+uum (UHV) by repeated Ar+ sputtering and heating at
+800 K. A small coverage of PTCDA molecules was evap-
+orated onto the clean Ag(111) surface at room temper-
+arXiv:2301.11762v1  [cond-mat.mes-hall]  27 Jan 2023
+
+2
+(a)
+(b)
+(c)
+10 Å
+Ag(111)
+~17.5 Å 
+STM tip
+−6
+−4
+−2
+0
+Current (pA)
++
+−0.6
+−0.4
+−0.2
+0.0
+0.2
+0.4
+0.6
+Bias voltage (mV)
+5.5
+6.0
+6.5
+7.0
+7.5
+dI/dV (10-4 G0)
+FIG. 1.
+(a) Schematic view of a standing PTCDA + 2Ag
+nanostructure on the Ag(111) surface, including the STM
+tip above the molecule.
+The bar shows the tunneling cur-
+rent IT measured above the standing nanostructure at con-
+stant height. The white, grey and red spheres indicate hy-
+drogen, carbon and oxygen atoms, respectively, of PTCDA.
+(b) Constant-height STM image above a standing nanostruc-
+ture recorded at a tip height of z ≃ 17.5 ˚A above the surface.
+The bias voltage was V = −50 mV. The white cross marks
+the location where the dI/dV conductance spectra were mea-
+sured. The molecular plane is indicated by the dashed orange
+line. (c) dI/dV conductance spectrum (blue) on a standing
+metal-molecule nanostructure measured at T = 30 mK and
+B = 0 T (Vmod = 50 µV). The tip was stabilized at IT = 45 pA
+and V = −1 mV. The red curve shows the fit based on the
+Frota function (see text for details). The spectrum is shown
+in units of the conductance quantum G0 = 2e
+h .
+ature from a custom-built Knudsen cell. After evapora-
+tion, the sample was flashed at 480 K for 2 min and then
+cooled down to 100 K and transferred to the STM. All ex-
+periments were performed in the J¨ulich Quantum Micro-
+scope [21], a millikelvin scanning tunneling microscope
+which uses the adiabatic demagnetization of electronic
+magnetic moments in a magnetocaloric material to reach
+temperatures in the range between 30 mK and 1 K. In
+this instrument, B fields of up to 8 T perpendicular to the
+sample surface can also be applied. Differential conduc-
+tance (dI/dV ) spectra were measured using conventional
+lock-in techniques with the STM feedback loop switched
+off and an AC modulation amplitude Vmod = 20−100 µV
+and frequency fmod = 187 Hz. The PtIr tip was treated
+in-situ by applying controlled voltage pulses and indenta-
+tions into the clean silver surface until the spectroscopic
+signature of the Ag(111) surface state appeared.
+The standing PTCDA + 2Ag nanostructure was fabri-
+cated on the Ag(111) surface in three steps by controlled
+manipulation with the tip of the STM as described in
+Ref. [22]. First, two single Ag atoms were attached to
+the two carboxylic oxygens on one side of the flat-lying
+molecule by lateral manipulation with the tip.
+Then,
+one of the carboxylic oxygens on the opposite side was
+contacted and the PTCDA molecule was pulled up on a
+curved trajectory until it stood upright. The tip was then
+moved straight up until the bond between the molecule
+and the tip broke, leaving the molecule in the standing
+position on the two Ag adatoms [22]. The stability of
+the standing metal-molecule nanostructure arises from
+the balance between local covalent interactions and non-
+local long-range van der Waals forces [23, 24].
+In constant-height STM images, the standing metal-
+molecule nanostructure can be recognized by two fea-
+tures that are distributed symmetrically around the plane
+of the molecule (dashed orange line in Fig. 1b) and
+separated by a nodal plane perpendicular to the latter
+(Fig. 1b). It is interesting to note that the node of the π-
+orbital in the plane of the molecule could not be resolved.
+The two features coincide with the positions where the
+interaction with the tip is most pronounced [22]. In the
+standing configuration, there is only a weak overlap be-
+tween the wave functions of the metallic surface and the
+lowest unoccupied molecular orbital (LUMO) of PTCDA,
+because the lobes of the molecular π-orbital are oriented
+perpendicular to the plane of the molecule. This allows
+the standing nanostructure to function as a quantum dot
+and coherent field emitter [22].
+At mK temperatures, a peak at zero bias is evident
+in the dI/dV spectrum measured on a standing metal-
+molecule nanostructure (Fig. 1c). In fact, at these low
+temperatures we additionally observe a dip at zero bias
+due to the dynamical Coulomb blockade (DCB) [25]. We
+have thus corrected all dI/dV spectra on the standing
+nanostructure for the DCB dip (see Supplementary Ma-
+terial). Previous studies have hinted that the LUMO of
+the standing metal-molecule nanostructure must contain
+a single unpaired electron [22, 26]. Therefore, it is plausi-
+ble to assume that the zero-bias peak originates from the
+Kondo effect, in which the spin of this localized electron is
+screened by itinerant substrate electrons. To verify this,
+we measured dI/dV spectra at different B fields. Already
+at B ≈ 100 − 120 mT a Zeeman splitting of the zero-bias
+peak is discernable (Fig. 2a).
+At higher B fields, the
+Kondo effect is completely quenched and the spectrum is
+dominated by the symmetric steps arising from inelastic
+spin-flip excitations (Fig. 2b).
+To extract the precise energy of the Zeeman splitting
+∆, we calculated the numerical derivative of the dI/dV
+spectra and fitted the peak positions with a Gaussian (see
+Supplementary Material). As shown in Figs. 2c and d,
+the energies of the spin-flip excitations scale linearly with
+the external B field. Only close to the critical field BC,
+which is required to initially split the Kondo resonance,
+
+3
+−1
+0
+1
+Bias voltage (mV)
+1
+2
+3
+4
+dI/dV (10-4 G0)
+0
+0.1
+0.2
+B field (T)
+0.00
+0.01
+0.02
+0.03
+Δ (mV)
+(a)
+(b)
+(c)
+(d)
+1 T
+3 T
+5 T
+7 T
+2Δ
+−0.25
+0
+0.25
+Bias voltage (mV)
+15
+20
+25
+30
+35
+dI/dV (10-4 G0)
+0.08 T
+0.20 T
+0
+2.5
+5.0
+B field (T)
+0.0
+0.2
+0.4
+0.6
+0.8
+Δ (mV)
+g = 2.006 ± 0.007
+FIG. 2. (a)-(b) dI/dV spectra (blue) on a standing metal-
+molecule nanostructure, measured at different B fields at
+T
+≃ 50 mK (setpoints in panel (a) were IT
+= 100 pA,
+V = −10 mV, Vmod = 100 µV and IT = 100 pA, V = −1 mV,
+Vmod = 20 µV in panel(b)).
+In panel (a), the B field was
+changed in steps of 20 mT. The orange curves in panel (b)
+show the fits based on perturbation theory (see text for de-
+tails). The spectra are vertically displaced for clarity. (c)-(d)
+The Zeeman splitting ∆ extracted from the dI/dV spectra as
+a function of B. The gray dashed line in panel (c) serves as a
+guide for the eye. The red curve in panel (d) shows the linear
+fit for the Zeeman splitting.
+the Zeeman energy rises noticeably faster with increas-
+ing B field. To extract the Land´e factor g, we consider
+only the data points at B fields ≥ 1 T (Fig. 2b) since
+the Kondo effect in the strong coupling regime leads to
+renormalization of the g-factor. A linear fit of the form
+∆ = gµBB for the Zeeman effect, where µB is the Bohr
+magneton, yields a Land´e factor g = 2.006 ± 0.007. By
+interpolating the data points at low B fields, we obtain
+BC = 108±5 mT for the critical field. Using the relation
+[27]
+BC = 1
+2
+kBTK
+gµB
+,
+(1)
+valid for temperatures T < 0.25 TK, this gives an esti-
+mate of 291 ± 13 mK for the Kondo temperature TK.
+An independent estimate of the Kondo temperature
+TK can be obtained from the width of the Kondo reso-
+nance (Fig. 1c). However, it should be noted that this is
+only a rough estimate, since the width of the Kondo reso-
+nance is related to TK by a non-universal scaling constant
+[28]. To extract its width, we fitted the Kondo resonance
+with a Frota line shape [29]
+ρ(E)Frota = ℜ
+�
+iΓK
+E − E0 + iΓK
+.
+(2)
+Additional broadening effects due to the Fermi distri-
+bution and the modulation amplitude were taken into
+account.
+The best fit then yields a width of ΓK ≃
+43 µV and thus a TK = ΓK(2π × 0.103)/kB ≃ 320 mK
+[29], in good agreement with the above estimate from
+the B field dependence.
+We attribute the features in
+the dI/dV spectrum at approximately ±0.25 mV and
+±0.55 mV (Fig. 1c and Supplementary Material) to ei-
+ther molecular vibrations or frustrated translations of the
+standing metal-molecule nanostructure [23]. Note that a
+strong electron-vibrational coupling can also lead to a
+further decrease of TK [30].
+The low Kondo temperature of the standing nanostruc-
+ture in conjunction with the low base temperature and
+high energy resolution of our mK STM enable us to quan-
+titatively describe the interaction of the localized spin
+with its environment, also as a function of temperature.
+For this purpose, we employ the Anderson-Appelbaum
+model [31–33] and calculate the tunneling conductance
+from the Kondo Hamiltonian in a perturbative approach
+that includes processes up to third order in the exchange
+interaction J [34]. The model allows tunneling electrons
+to interact with the localized spin via spin-spin (tT S ˆσt·ˆS)
+or potential scattering (tT S U). Here tT S is the matrix
+element for a transition from the tip to the molecule or
+vice versa, and ˆσt and ˆS are the spin operators of the
+tunneling and localized electrons, respectively. In addi-
+tion, the model takes into account the spin-spin exchange
+scattering between the electrons of the substrate and the
+localized spin (Jρ ˆσs · ˆS), where ρ denotes the substrate’s
+electron density at the Fermi energy and ˆσs the spin oper-
+ator of itinerant electrons in the substrate. This approach
+provides the correct description under the following con-
+ditions: the magnetic impurity is predominantly coupled
+to one of the electrodes (here the substrate), the system
+is in equilibrium (limit of small bias voltages), and the
+system is in the weak coupling limit (T >∼ TK) or high-
+field regime (B >∼ kBTK).
+We performed least-square
+fits and extracted the dimensionless coupling strength
+Jρ between the substrate and the localized electron and
+its Land´e factor g.
+Before we focus on the temperature dependence, we
+first examine the influence of the B field on Jρ at 50 mK
+(Fig. 2b).
+As the B field increases, we see a decrease
+in the height of the peak structure on top of the steps
+originating from spin-flip excitations. Since those peak
+heights are proportional to Jρ [34], this indicates that
+|Jρ| decreases with increasing B field.
+This is clearly
+seen in Fig. 3a, where the fitted Jρ is plotted versus B
+field. It may at first sight seem surprising that we still
+observe a coupling between the localized and itinerant
+spins at high B fields.
+This behaviour is, however, in
+
+4
+−1
+0
+1
+Bias voltage (mV)
+1
+2
+3
+4
+5
+dI/dV (10-4 G0)
+(a)
+(b)
+(c)
+(d)
+1009 mK
+751 mK
+275 mK
+70 mK
+41 mK
+0
+2.5
+5.0
+B field (T)
+−0.10
+−0.09
+−0.08
+−0.07
+−0.06
+Jρ
+0
+500
+1000
+Temperature (mK)
+−0.06
+−0.05
+−0.04
+Jρ
+~TK
+0
+500
+1000
+Temperature (mK)
+1.90
+1.95
+2.00
+2.05
+g-factor
+FIG. 3. (a) Coupling strength Jρ as extracted from the fits in
+Fig. 2b as a function of B field. (b) dI/dV spectra (blue) on
+a standing metal-molecule nanostructure, measured at differ-
+ent temperatures in an external field B = 7 T (IT = 100 pA,
+V
+= −10 mV, Vmod = 100 µV). The orange curves show
+the fits based on perturbation theory (see text for details).
+The spectra are vertically displaced for clarity. (c) Coupling
+strength Jρ as extracted from the fits as a function of tem-
+perature. The dashed blue line indicates the Kondo energy
+scale TK as determined from the B-field data of Fig. 2. (d)
+Land´e g-factor estimated from the fits in panel (b) (black)
+and the effective g-factor geff after taking into account renor-
+malization effects due to the exchange interaction (red). The
+red line illustrates a linear fit and the red shaded area the
+corresponding confidence interval.
+good agreement with numerical renormalization group
+(NRG) calculations for an S = 1/2 Kondo impurity at
+finite temperatures and B fields [27], in which it was
+shown that the intensity of the split Kondo resonance
+varies even if µBB/kBTK ≫ 1, which corresponds to the
+present situation. In other words, we have even at high
+B fields access to bias driven Kondo correlations whose
+gradual emergence at decreasing temperatures drives |Jρ|
+up. This behavior can be readily observed by looking at
+the temperature-dependent data for constant B = 7 T
+(Fig. 3b). The fits reveal that |Jρ| increases with decreas-
+ing temperature (Fig. 3c). For B = 0, such an increase
+of |Jρ| would signal the progressive breakdown of the
+perturbation approach, yielding a divergence of Jρ and
+the crossover into the Kondo singlet as a new ground
+state [9, 10].
+However, here we are in the high-field
+regime and therefore will not reach the Kondo ground
+state even in the limit T → 0. We note that the tem-
+perature range in which the perturbation theory starts to
+0.5
+1.0
+1.5
+2.0
+Setpoint conductance (10-4 G0)
+−0.075
+−0.070
+−0.065
+−0.060
+Jρ
+zB,eq
+zB,st
+FIG. 4. Coupling strength Jρ as extracted from the fits of
+dI/dV spectra on a standing metal-molecule nanostructure
+that were measured for different setpoint conductances at T ≃
+45 mK and an external field of B = 7 T. The tip was initially
+stabilized at IT = 100 pA and V = −6 mV and then moved
+up by 1 ˚A in steps of 0.1 ˚A. The gray dashed line serves as a
+guide for the eye. Insets show schematically how the PTCDA
+molecule is pulled up by the tip.
+collapse (Fig. 3b) agrees very well with the Kondo energy
+scale of 291 ± 13 mK which was derived from the B-field
+behaviour at low temperatures T < TK.
+For the fitted Land´e factor g in Fig. 3d we also see a
+strong decrease with increasing temperature. This can
+be attributed to the energy renormalization [34]. Taking
+this into account, we obtain an effective gyromagnetic
+factor of geff = g(T)×(1+Jρ(T)) ≈ 1.91±0.01, which has
+no temperature dependence. Note that the deviation of
+the obtained g-factors from the B-field-dependent mea-
+surements and from the perturbative approach is <∼ 6%.
+Having demonstrated the sensitivity to changes of Jρ,
+we now explore the possibility to tune the exchange cou-
+pling mechanically. It was shown before that the standing
+metal-molecule nanostructure is susceptible to attractive
+forces from the tip, enabling controlled tilts, translations
+and rotations [22]. If it was possible to tune the verti-
+cal distance of the standing nanostructure from the sub-
+strate with attractive forces from the STM tip, it might
+also be possible to tune the exchange coupling Jρ. To
+explore this possibility, we measured dI/dV spectra on
+the standing metal-molecule nanostructure at B = 7 T
+and T ≃ 45 mK for different setpoint conductances, cor-
+responding to different distances between the tip and the
+molecule. In Fig. 4 the fitted Jρ are plotted as a function
+of setpoint conductances G. For G ≤ 0.6 · 10−4G0, the
+coupling is constant at (Jρ)eq ≃ −0.075. With increasing
+G, corresponding to decreasing tip-molecule distances,
+|Jρ| decreases and reaches (Jρ)st ≃ −0.060 for the high-
+est G. Measurements at even smaller distances (higher
+setpoints) are not feasible, because the resulting larger
+tunnel currents frequently induce sudden 30◦ rotations
+of the standing nanostructure around its vertical axis.
+Since we know that there are attractive forces acting be-
+
+5
+tween the molecule and the tip [22, 23], we interpret the
+decreasing |Jρ| as the result of an increased distance of
+the standing metal-molecule nanostructure from the sur-
+face. Under the assumption that the exchange interac-
+tion scales exponentially with the bond distance zB [35],
+J(zB) ∝ exp(−zB/dex), the vertical relaxation ∆zB of
+the bond between the standing molecule and substrate
+surface can be estimated. For a typical decay length of
+the exchange interaction, dex ≃ 0.4 ˚A [35], we obtain
+∆zB = dex ln(Jeq/Jst) ≃ 0.09 ˚A between the smallest
+and the largest G in Fig. 4.
+In conclusion, we have shown that in the standing
+configuration the exchange coupling between PTCDA
+within the assembled nanostructure and the Ag(111) sub-
+strate is strongly reduced, if compared to the flat-lying
+molecule.
+At B = 0 we observed a Kondo resonance
+with a width of only ΓK ≃ 43 µV at an experimental
+temperature of T = 30 mK. B-field-dependent measure-
+ments showed that the standing metal-molecule nanos-
+tructure is an S =
+1/2 system with a critical field of
+BC = 108 ± 5 mT. This corresponds to a Kondo tem-
+perature of only TK = 291 ± 13 mK. Furthermore, we
+demonstrated that, using attractive forces exerted by the
+STM tip, it is possible to tune the exchange coupling be-
+tween the localized spin in the nanostructure and the
+substrate. The combination of the small exchange cou-
+pling and the softness of the surface bond against verti-
+cal distortions makes the standing metal-molecule nanos-
+tructure an interesting candidate for STM-based electron
+spin resonance (STM-ESR) experiments. For STM-ESR
+experiments on individual atoms and molecules [1, 3–
+5, 7, 16, 18, 35–40], two important requirements have to
+be met [37, 41]: first, a sufficiently small coupling be-
+tween the object to be investigated and the substrate,
+in order to reach long relaxation and dephasing times,
+and second, the possibility to drive with a high-frequency
+electric field applied to the STM tip mechanical oscilla-
+tions of the object in the inhomogeneous B field of the
+tip, the latter being produced by a magnetic atom at
+the tip apex.
+Although a significant reduction of the
+interaction between the atomic or molecular object of
+interest with the metal substrate has been achieved on
+different atomically thin insulating layers [15, 42], ESR
+signals in the STM have been observed, somewhat sur-
+prisingly, mainly on a bilayer of magnesium oxide (MgO)
+film on Ag(001) surfaces [1, 3–5, 7, 16, 18, 35–40] and
+recently for the first time on two-monolayer NaCl films
+on Cu(100) [43]. In this situation, the standing metal-
+molecule nanostructure may be a promising specimen: its
+spin is more weakly coupled to the substrate than that
+of Cu atoms on MgO/Ag(001), which are ESR active
+[18], and the tip-induced displacement is about an order
+of magnitude larger than the displacements required for
+ESR-STM [37, 41]. It should be noted, however, that for
+ESR-STM a dynamic displacement driven by the high-
+frequency electric field is required, whereas in the present
+experiment we so far only tested the response to static
+forces between the molecule and the tip. But due to the
+strong molecular polarizability of PTCDA [26], we an-
+ticipate that the high-frequency electric field may have
+a similar effect. In upcoming experiments we will there-
+fore determine whether the dynamic displacement is suf-
+ficiently large, and whether relaxation times are suffi-
+ciently long for STM-ESR experiments. Finally, we note
+that standing metal-molecule nanostructure can also be
+prepared on the tip [26, 44, 45]. If it was indeed STM-
+ESR capable, the standing nanostructure could therefore
+be employed as a magnetic field sensor on the atomic
+scale [13, 46], in addition to being a sensor of electric sur-
+face potentials, as which it has already been used [44, 45].
+We thank Frithjof B. Anders (TU Dortmund) for
+fruitful discussions. The authors acknowledge financial
+support from the German Federal Ministry of Educa-
+tion and Research through the funding program ’quan-
+tum technologies - from basic research to market’, un-
+der Q-NL (project number 13N16032). M.T. acknowl-
+edges funding by the Heisenberg Program (TE 833/2-
+1) of the Deutsche Forschungsgemeinschaft (DFG).
+F.S.T. acknowledges funding by the DFG through SFB
+1083 ”Structure and Dynamics of Internal Interfaces”
+(223848855-SFB 1083).
+∗ Corresponding author: t.esat@fz-juelich.de
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+

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+arXiv:2301.04194v1  [math.OC]  10 Jan 2023
+ASYMPTOTICS OF IMPULSE CONTROL PROBLEM WITH
+MULTIPLICATIVE REWARD
+DAMIAN JELITO∗,† AND �LUKASZ STETTNER∗
+Abstract. We consider a long-run impulse control problem for a generic
+Markov process with a multiplicative reward functional. We construct a solu-
+tion to the associated Bellman equation and provide a verification result. The
+argument is based on the probabilistic properties of the underlying process
+combined with the Krein-Rutman theorem applied to the specific non-linear
+operator. Also, it utilises the approximation of the problem in the bounded
+domain and with the help of the dyadic time-grid.
+Keywords: impulse control, Bellman equation, risk-sensitive criterion, Markov
+process
+MSC2020 subject classifications: 93E20, 49J21, 49K21, 60J25
+1. Introduction
+Impulse control constitutes a versatile framework for controlling real-life stochas-
+tic systems. In this type of control, a decision-maker determines intervention times
+and instantaneous after-intervention states of the controlled process. By doing so,
+one can affect a continuous time phenomenon in a discrete time manner. Conse-
+quently, impulse control attracted considerable attention in the mathematical liter-
+ature; see e.g. [4, 9, 26] for classic contributions and [3, 10, 19, 21] for more recent
+results. In addition to generic mathematical properties, impulse control problems
+were studied with reference to specific applications including i.a. controlling ex-
+change rates, epidemics, and portfolios with transaction costs; see e.g. [18, 25, 27]
+and references therein.
+When looking for an optimal impulse control strategy, one must decide on the
+optimality criterion.
+Recently, considerable attention was paid to the so-called
+risk-sensitive functional given, for any γ ∈ R, by
+µγ(Z) :=
+�
+1
+γ ln E[exp(γZ)],
+γ ̸= 0,
+E[Z],
+γ = 0,
+(1.1)
+where Z is a (random) payoff corresponding to a chosen control strategy; see [14] for
+a seminal contribution. This functional with γ = 0 corresponds to the usual linear
+criterion and the case γ < 0 is associated with risk-averse preferences; see [5] for
+a comprehensive overview. Also, the functional with γ > 0 could be linked to the
+asymptotics of the power utility function; see [31] for details. Recent comprehensive
+discussion on the long-run version with µγ could be found in [6]. We refer also
+∗Institute of Mathematics, Polish Academy of Sciences, Warsaw, Poland
+E-mail addresses: djelito@impan.pl, l.stettner@impan.pl.
+†Corresponding author.
+1
+
+2
+ASYMPTOTICS OF IMPULSE CONTROL PROBLEM WITH MULTIPLICATIVE REWARD
+to [23] and references therein for a discussion on the connection between (1.1) and
+the duality of the large deviations-based criteria.
+In this paper we focus on the use of the functional µγ with γ > 0. More specif-
+ically, we consider the impulse control problem for some continuous time Markov
+process and construct a solution to the associated Bellman equation which char-
+acterises an optimal impulse control strategy. To do this, we study the family of
+impulse control problems in bounded domains and then extend the analysis to the
+generic locally compact state space. This idea was used in [1], where PDEs tech-
+niques were applied to obtain the characterisation of the controlled diffusions in the
+risks-sensitive setting. A similar approximation for the the average cost per unit
+time problem was considered in [32].
+The main contribution of this paper is a construction of a solution to the Bell-
+man equation associated with the problem, see Theorem 5.1 for details. It should
+be noted that we get a bounded solution even though the state space could be
+unbounded and we assume virtually no ergodicity conditions for the uncontrolled
+process. Also, note that present results for γ > 0 complement our recent findings
+on the impulse control with the risk-averse preferences; see [24] for the dyadic case
+and [15] for the continuous time framework. Nevertheless, it should be noted that
+the techniques for γ < 0 and γ > 0 are substantially different and it is not possible
+to directly transform the results in one framework to the other; see e.g. [16, 20] for
+further discussion.
+The structure of this paper is as follows. In Section 2 we formally introduce
+the problem, discuss the assumptions and, in Theorem 2.3, provide a verification
+argument. Next, in Section 3 we consider an auxiliary dyadic problem in a bounded
+domain and in Theorem 3.1 we construct a solution to the corresponding Bellman
+equation. This is used in Section 4 where we extend our analysis to the unbounded
+domain with the dyadic time-grid; see Theorem 4.2. Next, in Section 5 we finally
+construct a solution to the Bellman equation for the original problem; see Theo-
+rem 5.1. Finally, in Appendix A we discuss some properties of the optimal stopping
+problems that are used in this paper.
+2. Preliminaries
+Let X = (Xt)t≥0 be a continuous time standard Feller–Markov process on a
+filtered probability space (Ω, F, (Ft), P). The process X takes values in a locally
+compact separable metric space E endowed with a metric ρ and the Borel σ-field
+E. With any x ∈ E we associate a probability measure Px describing the evolution
+of the process X starting in x; see Section 1.4 in [28] for details. Also, we use Ex,
+x ∈ E, and Pt(x, A) := Px[Xt ∈ A], t ≥ 0, x ∈ E, A ∈ E, for the corresponding
+expectation operator and the transition probability, respectively.
+By Cb(E) we
+denote the family of continuous bounded real-valued functions on E.
+Also, to
+ease the notation, by T , Tx, and Tx,b we denote the families of stopping times,
+Px a.s. finite stopping times, and Px a.s. bounded stopping times, respectively.
+Also, for any δ > 0, by T δ ⊂ T , T δ
+x ⊂ Tx, and T δ
+x,b ⊂ Tx,b, we denote the
+respective subfamilies of dyadic stopping times, i.e.
+those taking values in the
+set {0, δ, 2δ, . . .} ∪ {∞}.
+Throughout this paper we fix some compact U ⊆ E and we assume that a
+decision-maker is allowed to shift the controlled process to U. This is done with
+the help of an impulse control strategy, i.e. a sequence V := (τi, ξi)∞
+i=1, where (τi)
+
+ASYMPTOTICS OF IMPULSE CONTROL PROBLEM WITH MULTIPLICATIVE REWARD
+3
+is an increasing sequence of stopping times and (ξi) is a sequence of Fτi-measurable
+after-impulse states with values in U. With any starting point x ∈ E and a strategy
+V we associate a probability measure P(x,V ) for the controlled process Y . Under this
+measure, the process starts at x and follows its usual (uncontrolled) dynamics up
+to the time τ1. Then, it is immediately shifted to ξ1 and starts its evolution again,
+etc. More formally, we consider a countable product of filtered spaces (Ω, F, (Ft))
+and a coordinate process (X1
+t , X2
+t , . . .). Then, we define the controlled process Y
+as Yt := Xi
+t, t ∈ [τi−1, τi) with the convention τ0 ≡ 0. Under the measure P(x,V )
+we get Yτi = ξi; we refer to Chapter V in [26] for the construction details; see
+also Appendix in [8] and Section 2 in [29]. A strategy V = (τi, ξi)∞
+i=1 is called
+admissible if for any x ∈ E we get P(x,V )[limn→∞ τn = ∞] = 1. The family of
+admissible impulse control strategies is denoted by V. Also, note that, to simplify
+the notation, by Yτ −
+i := Xi
+τi, i ∈ N∗, we denote the state of the process right before
+the ith impulse (yet, possibly, after the jump).
+In this paper we study the asymptotics of the impulse control problem given by
+sup
+V ∈V
+J(x, V ),
+x ∈ E,
+(2.1)
+where, for any x ∈ E and V ∈ V, we set
+J(x, V ) := lim inf
+T →∞
+1
+T ln E(x,V )
+�
+e
+� T
+0 f(Ys)ds+�∞
+i=1 1{τi≤T }c(Yτ−
+i
+,ξi)�
+,
+(2.2)
+with f denoting the running cost function and c being the shift-cost function,
+respectively. Note that this could be seen as a long-run standardised version of
+the functional (1.1) with γ > 0 applied to the impulse control framework. Here,
+the standardisation refers to the fact that we do not use directly the parameter γ
+(apart from its sign). Also, the problem is of the long-run type, i.e. the utility is
+averaged over time which improves the stability of the results.
+The analysis in this paper is based on the approximation of the problem in a
+bounded domain.
+Thus, we fix a sequence (Bm)m∈N of compact sets satisfying
+Bm ⊂ Bm+1 and E = �∞
+m=0 Bm. Also, we assume that U ⊂ B0. Next, we assume
+the following conditions.
+(A1) (Cost functions). The map f : E �→ R− is a continuous and bounded. Also,
+the map c : E × U �→ R− is continuous, bounded, and strictly non-positive,
+and satisfies the triangle inequality, i.e. for some c0 < 0, we have
+0 > c0 ≥ c(x, ξ) ≥ c(x, η) + c(η, ξ),
+x ∈ E, ξ, η ∈ U.
+Also, we assume that c satisfies the uniform limit at infinity condition
+lim
+∥x∥,∥y∥→∞ sup
+ξ∈U
+|c(x, ξ) − c(y, ξ)| = 0.
+(2.3)
+(A2) (Transition probability continuity). For any t, the transition probability Pt
+is continuous with respect to the total variation norm, i.e. for any sequence
+(xn) ⊂ E converging to x ∈ E, we get
+lim
+n→∞ sup
+A∈E
+|Pt(xn, A) − Pt(x, A)| = 0.
+(A3) (Distance control). For any compact set Γ ⊂ E, t0 > 0, and r0 > 0, we
+have
+lim
+r→∞ MΓ(t0, r) = 0,
+lim
+t→0 MΓ(t, r0) = 0,
+(2.4)
+
+4
+ASYMPTOTICS OF IMPULSE CONTROL PROBLEM WITH MULTIPLICATIVE REWARD
+where MΓ(t, r) := supx∈Γ Px[sups∈[0,t] ρ(Xs, x) ≥ r], t, r > 0.
+(A4) (Process irreducibility). For any m ∈ N, x ∈ Bm, δ > 0, and any open set
+O ⊂ Bm, we have
+Px [∪∞
+i=1{Xiδ ∈ O}] = 1.
+Also, we assume that for any x ∈ E, δ > 0, and m ∈ N, we have
+Px[τBm < ∞] = 1
+(2.5)
+where τBm := δ inf{k ∈ N: Xkδ /∈ Bm}.
+Before we proceed, let us comment on these assumptions. First, note that (A1)
+states typical cost-functions conditions. In particular, the non-positivity assump-
+tion for f is merely a technical normalisation. Indeed, for a generic ˜f ∈ Cb(E) we
+may set f(·) := ˜f(·) − ∥ ˜f∥ ≤ 0 to get
+Jf(x, V ) = J
+˜f(x, V ) − ∥ ˜f∥,
+x ∈ E, V ∈ V,
+where Jf denotes the version of the functional J from (2.2) corresponding to the
+running cost function f.
+Second, Assumption (A2) states that the transition probabilities Pt(x, ·) are
+continuous with respect to the total variation norm. Note that this directly implies
+that the transition semigroup associated to X is strong Feller, i.e. for any t > 0
+and a bounded measurable map h: E �→ R, the map x �→ Ex[h(Xt)] is continuous
+and bounded.
+Third, Assumption (A3) quantifies distance control properties of the underlying
+process. It states that, for a fixed time horizon, the process with a high probability
+stays close to its starting point and, with a fixed radius, with a high probability it
+does not leave the corresponding ball with a sufficiently short time horizon. Note
+that these properties are automatically satisfied if the transition semigroup is C0-
+Feller; see Proposition 2.1 in [21] and Proposition 6.4 in [2] for details.
+Finally, Assumption (A4) states a form of the irreducibility of the process X. It
+requires that the process visits a sufficiently rich family of sets with unit probability.
+To solve (2.1), we show the existence of a solution to the impulse control Bellman
+equation, i.e. a function w ∈ Cb(E) and a constant λ ∈ R satisfying
+w(x) = sup
+τ∈Tx,b
+ln Ex
+�
+exp
+�� τ
+0
+(f(Xs) − λ)ds + Mw(Xτ)
+��
+,
+x ∈ E,
+(2.6)
+where the operator M is given by
+Mh(x) := sup
+ξ∈U
+(c(x, ξ) + h(ξ)),
+h ∈ Cb(E), x ∈ E.
+We start with a simple observation giving a lower bound for the constant λ
+from (2.6). To do this, we define the semi-group type by
+r(f) := lim
+t→∞
+1
+t ln sup
+x∈E
+Ex
+�
+e
+� t
+0 f(Xs)ds�
+;
+(2.7)
+see e.g. Proposition 1 in [30] for a discussion on the properties of r(f).
+Lemma 2.1. Let (w, λ) be a solution to (2.6). Then, we get λ ≥ r(f).
+Proof. From (2.6), for any T ≥ 0, we get
+w(x) ≥ ln Ex
+�
+e
+� T
+0 (f(Xs)−λ)ds+Mw(XT )�
+.
+
+ASYMPTOTICS OF IMPULSE CONTROL PROBLEM WITH MULTIPLICATIVE REWARD
+5
+Thus, using the boundedness of w and Mw, we get
+∥w∥ ≥ sup
+x∈E
+ln Ex
+�
+e
+� T
+0 (f(Xs)−λ)ds�
+− ∥Mw∥.
+Consequently, dividing both hand-sides by T and letting T → ∞, we get 0 ≥
+r(f − λ), which concludes the proof.
+□
+Let us now link a solution to (2.6) with the optimal value and an optimal strategy
+for (2.1). To ease the notation, we recursively define the strategy ˆV := (ˆτi, ˆξi)∞
+i=1
+for i ∈ N \ {0} by
+�
+ˆτi
+:= inf{t ≥ ˆτi−1 : w(Xi
+t) = Mw(Xi
+t)},
+ˆξi
+:= arg maxξ∈U
+�
+c(Xi
+ˆτi, ξ) + w(ξ)
+�
+1{ˆτi<∞} + ξ01{ˆτi=∞},
+(2.8)
+where ˆτ0 := 0 and ξ0 ∈ U is some fixed point. First, we show that ˆV is a proper
+strategy.
+Proposition 2.2. The strategy ˆV given by (2.8) is admissible.
+Proof. To ease the notation, we define N(0, T ) := �∞
+i=1 1{ˆτi≤T }, T ≥ 0. We fix
+some T > 0 and x ∈ E, and show that we get
+P(x, ˆV )[N(0, T ) = ∞] = 0.
+(2.9)
+Recalling (2.8), on the event A := {limi→∞ ˆτi < +∞}, for any n ∈ N, n ≥ 1,
+we get w(Xn
+ˆτn) = Mw(Xn
+ˆτn) = c(Xn
+ˆτn, Xn+1
+ˆτn
+) + w(Xn+1
+ˆτn
+).
+Also, recalling that
+c(x, ξ) ≤ c0 < 0, x ∈ E, ξ ∈ U, for any n ∈ N, n ≥ 1, we have w(Xn+1
+ˆτn
+)−w(Xn
+ˆτn) =
+−c(Xn
+ˆτn, Xn+1
+ˆτn
+) ≥ −c0 > 0. Using this observation and Assumption (A3), we esti-
+mate the distance between consecutive impulses which will be used to prove (2.9).
+More specifically, for any k, m ∈ N, k, m ≥ 1, we get
+k+m−2
+�
+n=k
+(w(Xn+1
+ˆτn
+) − w(Xn+1
+ˆτn+1)) + (w(Xk+m
+ˆτk+m−1) − w(Xk+1
+ˆτk
+))
+= w(Xk+1
+ˆτk
+) +
+k+m−1
+�
+n=k+1
+(w(Xn+1
+ˆτn
+) − w(Xn
+ˆτn)) − w(Xk+1
+ˆτk
+)
+=
+k+m−1
+�
+n=k+1
+(w(Xn+1
+ˆτn
+) − w(Xn
+ˆτn)) ≥ −(m − 1)c0;
+(2.10)
+it should be noted that the specific values for k and m will be determined later.
+Using the continuity of w we may find K > 0 such that supx,y∈U(w(x) − w(y)) ≤
+K.
+Let m ∈ N be big enough to get −(m − 1) c0
+2
+> K.
+Thus, noting that
+Xk+m
+ˆτk+m−1, Xk+1
+ˆτk
+∈ U, we have (w(Xk+m
+ˆτk+m−1) − w(Xk+1
+ˆτk
+)) ≤ K < −(m − 1) c0
+2 . Con-
+sequently, recalling (2.10), on A, we get
+k+m−2
+�
+n=k
+(w(Xn+1
+ˆτn
+) − w(Xn+1
+ˆτn+1)) ≥ −(m − 1)c0
+2 .
+(2.11)
+Recalling the compactness of U and the continuity of w we may find r > 0 such
+that for any x ∈ U and y ∈ E satisfying ρ(x, y) < r we get |w(x) − w(y)| < − c0
+2 .
+
+6
+ASYMPTOTICS OF IMPULSE CONTROL PROBLEM WITH MULTIPLICATIVE REWARD
+Let us now consider the family of events
+Bk :=
+k+m−2
+�
+n=k
+{ρ(Xn+1
+ˆτn
+, Xn+1
+ˆτn+1) < r},
+k ∈ N, k ≥ 1,
+(2.12)
+and note that, for any k ∈ N, k ≥ 1, on Bk ∩ A we have �k+m−2
+n=k
+(w(Xn+1
+ˆτn
+) −
+w(Xn+1
+ˆτn+1)) < −(m − 1) c0
+2 . Thus, recalling (2.11), for any k ∈ N, k ≥ 1, we get
+P(x0, ˆV )[Bk ∩ A] = 0 and, in particular, we have
+P(x0, ˆV )[Bk ∩ {N(0, T ) = ∞}] = 0.
+(2.13)
+Let us now show that lim supk→∞ P(x0, ˆV )[Bc
+k ∩{N(0, T ) = ∞}] = 0. Noting that
+{N(0, T ) = ∞} = {limi→∞ ˆτi ≤ T }, for any t0 > 0 and k ∈ N, k ≥ 1, we get
+P(x0, ˆV ) [Bc
+k ∩ {N(0, T ) = ∞}]
+≤ P(x0, ˆV )
+��k+m−2
+�
+n=k
+{ρ(Xn+1
+ˆτn
+, Xn+1
+ˆτn+1) ≥ r} ∩ {ˆτn+1 − ˆτn ≤ t0}
+�
+∩ { lim
+i→∞ ˆτi ≤ T }
+�
++ P(x0, ˆV )
+��k+m−2
+�
+n=k
+{ρ(Xn+1
+ˆτn
+, Xn+1
+ˆτn+1) ≥ r} ∩ {ˆτn+1 − ˆτn > t0}
+�
+∩ { lim
+i→∞ ˆτi ≤ T }
+�
+≤ P(x0, ˆV )
+�k+m−2
+�
+n=k
+{ sup
+t∈[0,t0]
+ρ(Xn+1
+ˆτn
+, Xn+1
+ˆτn+t) ≥ r} ∩ { lim
+i→∞ ˆτi ≤ T }
+�
++ P(x0, ˆV )
+�k+m−2
+�
+n=k
+{ˆτn+1 − ˆτn > t0} ∩ { lim
+i→∞ ˆτi ≤ T }
+�
+.
+(2.14)
+Using Assumption (A3), for any ε > 0, we may find t0 > 0, such that
+sup
+x∈U
+Px
+�
+sup
+t∈[0,t0]
+ρ(X0, Xt) ≥ r
+�
+≤
+ε
+m − 1.
+(2.15)
+Thus, using the strong Markov property and noting that Xn+1
+ˆτn
+∈ U, for any k ∈ N,
+k ≥ 1, we get
+P(x0, ˆV )
+�k+m−2
+�
+n=k
+{ sup
+t∈[0,t0]
+ρ(Xn+1
+ˆτn
+, Xn+1
+ˆτn+t) ≥ r} ∩ { lim
+i→∞ ˆτi ≤ T }
+�
+≤
+k+m−2
+�
+n=k
+P(x0, ˆV )
+�
+{ sup
+t∈[0,t0]
+ρ(Xn+1
+ˆτn
+, Xn+1
+ˆτn+t) ≥ r} ∩ {ˆτn ≤ T }
+�
+=
+k+m−2
+�
+n=k
+P(x0, ˆV )
+�
+{ˆτn ≤ T }PXn+1
+ˆτn
+�
+sup
+t∈[0,t0]
+ρ(X0, Xt) ≥ r
+��
+≤ ε.
+(2.16)
+Recalling that ε > 0 was arbitrary, for any k ∈ N, k ≥ 1, we get
+P(x0, ˆV )
+�k+m−2
+�
+n=k
+{ sup
+t∈[0,t0]
+ρ(Xn+1
+ˆτn
+, Xn+1
+ˆτn+t) ≥ r} ∩ { lim
+i→∞ ˆτi ≤ T }
+�
+= 0.
+(2.17)
+
+ASYMPTOTICS OF IMPULSE CONTROL PROBLEM WITH MULTIPLICATIVE REWARD
+7
+Now, to ease the notation, let Ck := �∞
+n=k{ˆτn+1 − ˆτn > t0} ∩ {limi→∞ ˆτi ≤ T },
+k ∈ N, k ≥ 1, and note that Ck+1 ⊂ Ck, k ∈ N, k ≥ 1. We show that
+lim
+k→∞ P(x0, ˆV ) [Ck] = 0.
+For the contradiction, assume that limk→∞ P(x0, ˆV ) [Ck] > 0. Consequently, we get
+P(x0, ˆV ) [�∞
+k=1 Ck] > 0. Note that for any ω ∈ �∞
+k=1 Ck we have limi→∞ ˆτi(ω) ≤ T .
+In particular, we may find i0 ∈ N such that for any n ≥ i0 we get ˆτn+1(ω)− ˆτn(ω) ≤
+t0
+2 . This leads to the contradiction as from the fact that ω ∈ �∞
+k=1 Ck we also get
+ω ∈
+∞
+�
+k=1
+∞
+�
+n=k
+{ˆτn+1 − ˆτn > t0} ⊂
+∞
+�
+n=i0
+{ˆτn+1 − ˆτn > t0}.
+Consequently, we get limk→∞ P(x0, ˆV ) [Ck] = 0 and, in particular, we get
+lim sup
+k→∞
+P(x0, ˆV )
+�k+m−2
+�
+n=k
+{ˆτn+1 − ˆτn > t0} ∩ { lim
+i→∞ ˆτi ≤ T }
+�
+≤ lim
+k→∞ P(x0, ˆV ) [Ck] = 0.
+Hence, recalling (2.14) and (2.17), we get
+lim sup
+k→∞
+P(x0, ˆV ) [Bc
+k ∩ {N(0, T ) = ∞}] = 0.
+Thus, recalling (2.13), for any k ∈ N, k ≥ 1, we obtain
+P(x0, ˆV ) [N(0, T ) = ∞] = P(x0, ˆV ) [Bc
+k ∩ {N(0, T ) = ∞}] ,
+and letting k → ∞, we conclude the proof of (2.9).
+□
+Now, we show the verification result linking (2.6) with the optimal value and an
+optimal strategy for (2.1).
+Theorem 2.3. Let (w, λ) be a solution to (2.6) with λ > r(f). Then, we get
+λ = sup
+V ∈V
+J(x, V ) = J(x, ˆV ),
+x ∈ E,
+where the strategy ˆV is given by (2.8).
+Proof. The proof is based on the argument from Theorem 4.4 in [15] thus we show
+only an outline. First, we show that λ = J(x, ˆV ), x ∈ E, where the strategy ˆV is
+given by (2.8). Let us fix x ∈ E. Then, combining the argument used in Lemma
+7.1 in [2] and Proposition A.3, we get that the process
+e
+� ˆτ1∧T
+0
+(f(X1
+s )−λ)ds+w(X1
+ˆτ1∧T ),
+T ≥ 0,
+is a P(x, ˆV )-martingale. Noting that on the event {ˆτk+1 < T } we get w(Xk+1
+ˆτk+1) =
+Mw(Xk+1
+ˆτk+1) = c(Xk+1
+ˆτk+1, ˆξk+1) + w(ˆξk+1), k ∈ N, for any n ∈ N we recursively get
+ew(x) = E(x, ˆV )
+�
+e
+� ˆτ1∧T
+0
+(f(Ys)−λ)ds+w(X1
+ˆτ1∧T )�
+= E(x, ˆV )
+�
+e
+� ˆτ1∧T
+0
+(f(Ys)−λ)ds+1{ˆτ1<T }c(X1
+ˆτ1 ,X2
+ˆτ1 )+1{ˆτ1<T }w(X2
+ˆτ1 )+1{ˆτ1≥T }w(X1
+T )�
+= E(x, ˆV )
+�
+e
+� ˆτn∧T
+0
+(f(Ys)−λ)ds+�n
+i=1 1{ˆτi<T }c(Xi
+ˆτi ,Xi+1
+ˆτi
+)×
+×e
+�n
+i=1 1{ˆτi−1<T ≤ˆτi}w(Xi
+T )+1{ˆτn<T }w(Xn+1
+ˆτn
+)�
+.
+(2.18)
+
+8
+ASYMPTOTICS OF IMPULSE CONTROL PROBLEM WITH MULTIPLICATIVE REWARD
+Recalling Proposition 2.2 we get ˆτn → ∞ as n → ∞. Thus, letting n → ∞ in (2.18)
+and using Lebesgue’s dominated convergence theorem we get
+ew(x) = E(x, ˆV )
+�
+e
+� T
+0 (f(Ys)−λ)ds+�∞
+i=1 1{ˆτi<T }c(Xi
+ˆτi ,Xi+1
+ˆτi
+)+�∞
+i=1 1{ˆτi−1<T ≤ˆτi}w(Xi
+T )�
+.
+Thus, recalling the boundedness of w, taking the logarithm of both sides, dividing
+by T , and letting T → ∞ we obtain
+λ = lim inf
+T →∞ E(x, ˆV )
+�
+e
+� T
+0 f(Ys)ds+�∞
+i=1 1{ˆτi<T }c(Xi
+ˆτi ,Xi+1
+ˆτi
+)�
+.
+Second, let us fix some x ∈ E and an admissible strategy V = (ξi, τi)∞
+i=1 ∈
+V. We show that λ ≥ J(x, V ). Using the argument from Lemma 7.1 in [2] and
+Proposition A.3, we get that the process
+e
+� τ1∧T
+0
+(f(X1
+s )−λ)ds+w(X1
+τ1∧T ),
+T ≥ 0,
+is a P(x,V )-supermartingale. Noting that on the event {τk+1 < T } we have
+w(Xk+1
+τk+1) ≥ Mw(Xk+1
+τk+1) ≥ c(Xk+1
+τk+1, ξk+1) + w(ξk+1),
+k ∈ N,
+for any n ∈ N we recursively get
+ew(x) ≥ E(x,V )
+�
+e
+� τ1∧T
+0
+(f(Ys)−λ)ds+w(X1
+τ1∧T )�
+≥ E(x,V )
+�
+e
+� τ1∧T
+0
+(f(Ys)−λ)ds+1{τ1<T }c(X1
+τ1 ,X2
+τ1 )+1{τ1<T }w(X2
+τ1 )+1{τ1≥T }w(X1
+T )�
+≥ E(x,V )
+�
+e
+� τn∧T
+0
+(f(Ys)−λ)ds+�n
+i=1 1{τi<T }c(Xi
+τi ,Xi+1
+τi
+)×
+×e
+�n
+i=1 1{τi−1<T ≤τi}w(Xi
+T )+1{τn<T }w(Xn+1
+τn
+)�
+.
+(2.19)
+Recalling the admissibility of V , we get τn → ∞ as n → ∞. Thus, letting n → ∞
+in (2.19) and using Fatou’s lemma, we get
+ew(x) ≥ E(x,V )
+�
+e
+� T
+0 (f(Ys)−λ)ds+�∞
+i=1 1{τi<T }c(Xi
+τi ,Xi+1
+τi
+)+�n
+i=1 1{τi−1<T ≤τi}w(Xi
+T )�
+.
+Thus, taking the logarithm of both sides, dividing by T , and letting T → ∞, we
+get
+λ ≥ lim inf
+T →∞ E(x,V )
+�
+e
+� T
+0 f(Ys)ds+�∞
+i=1 1{τi<T }c(Xi
+τi ,Xi+1
+τi
+)�
+,
+which concludes the proof.
+□
+In the following sections we construct a solution to (2.6). In the construction we
+approximate the underlying problem using the dyadic time-grid. Also, we consider
+a version of the problem in the bounded domain.
+3. Dyadic impulse control in a bounded set
+In this section we consider a version of (2.1) with a dyadic-time-grid and obliga-
+tory impulses when the process leaves some compact set. In this way, we construct
+a solution to the bounded-domain dyadic counterpart of (2.6). More specifically,
+let us fix some δ > 0 and m ∈ N. We show the existence of a map wm
+δ ∈ Cb(Bm)
+and a constant λm
+δ ∈ R satisfying
+wm
+δ (x) = sup
+τ∈T δ
+x,b
+ln Ex
+�
+e
+� τ∧τBm
+0
+(f(Xs)−λm
+δ )ds+Mwm
+δ (Xτ∧τBm )
+�
+,
+x ∈ Bm.
+(3.1)
+
+ASYMPTOTICS OF IMPULSE CONTROL PROBLEM WITH MULTIPLICATIVE REWARD
+9
+In fact, we start with the analysis of an associated one-step equation. More specif-
+ically, we show the existence of a constant λm
+δ
+∈ R and a map wm
+δ
+∈ Cb(Bm)
+satisfying
+wm
+δ (x) = max
+�
+ln Ex
+�
+e
+� δ
+0 (f(Xs)−λm
+δ )ds+1{Xδ∈Bm}wm
+δ (Xδ)+1{Xδ /
+∈Bm}Mwm
+δ (Xδ)�
+,
+Mwm
+δ (x)
+�
+,
+x ∈ Bm,
+wm
+δ (x) = Mwm
+δ (x),
+x /∈ Bm;
+(3.2)
+see Theorem 3.1 for details. Also, note that we link (3.2) with (3.1) in Theorem 3.4.
+Theorem 3.1. There exists a constant λm
+δ
+> 0 and a map wm
+δ
+∈ Cb(Bm) such
+that (3.2) is satisfied and we get supξ∈U wm
+δ (ξ) = 0.
+Proof. The idea of the proof is to use the Krein-Rutman theorem to get a pos-
+itive eigenvalue with a non-negative eigenvector of the suitable operator.
+More
+specifically, we consider a cone of non-negative continuous and bounded functions
+C+
+b (Bm) ⊂ Cb(Bm) and, for any h ∈ C+
+b (Bm), we define the operators
+˜
+Mh(x) := sup
+ξ∈U
+ec(x,ξ)h(ξ),
+x ∈ E,
+˜P m
+δ h(x) := Ex
+�
+e
+� δ
+0 f(Xs)ds �
+1{Xδ∈Bm}h(Xδ) + 1{Xδ /∈Bm} ˜
+Mh(Xδ)
+��
+,
+x ∈ Bm,
+˜T m
+δ h(x) := max
+�
+˜P m
+δ h(x), ˜
+M ˜P m
+δ h(x)
+�
+,
+x ∈ Bm.
+Now, we use the Krein-Rutman theorem to show that ˜T m
+δ
+admits a positive eigen-
+value and a non-negative eigenfunction; see Theorem 4.3 in [7] for details. We start
+with verifying the assumptions. First, note that ˜T m
+δ
+is positively homogeneous,
+monotonic increasing, and we have
+˜T m
+δ
+1(x) ≥ e−δ∥f∥−∥c∥
+1(x),
+x ∈ Bm,
+where
+1 denotes the function identically equal to 1 on Bm. Also, using Assump-
+tion (A2), we get that ˜T m
+δ
+transforms C+
+b (Bm) into itself and it is continuous with
+respect to the supremum norm. Let us now show that ˜T m
+δ
+is in fact completely
+continuous. To see this, let (hn)n∈N ⊂ C+
+b (Bm) be a bounded (by some constant
+K > 0) sequence; using Arzel`a-Ascoli Theorem we show that it is possible to find
+a convergent subsequence of ( ˜T m
+δ hn)n∈N. Note that, for any n ∈ N, we get
+∥ ˜T m
+δ hn∥ ≤ eδ∥f∥K,
+hence ( ˜T m
+δ hn) is uniformly bounded. Next, let us fix some ε > 0, x ∈ Bm, and
+(xk) ⊂ Bm such that xk → x as k → ∞. Also, to ease the notation, for any n ∈ N,
+we set Hn(x) := 1{x∈Bm}hn(x) + 1{x/∈Bm} ˜
+Mhn(x), x ∈ E and note that Hn are
+measurable functions bounded by 2K uniformly in n ∈ N. Then, for any n, k ∈ N,
+we get
+| ˜T m
+δ hn(x) − ˜T m
+δ hn(xk)| ≤
+���Ex
+�
+e
+� δ
+0 f(Xs)dsHn(Xδ)
+�
+− Exk
+�
+e
+� δ
+0 f(Xs)dsHn(Xδ)
+����
++ | ˜
+M ˜P m
+δ hn(x) − ˜
+M ˜P m
+δ hn(xk)|.
+(3.3)
+Also, using Assumption (A1), we may find k ∈ N big enough such that, for any
+n ∈ N, we obtain
+| ˜
+M ˜P m
+δ hn(x) − ˜
+M ˜P m
+δ hn(xk)| ≤ eδ∥f∥K sup
+ξ∈U
+|ec(x,ξ) − ec(xk,ξ)| ≤ ε
+2.
+(3.4)
+
+10 ASYMPTOTICS OF IMPULSE CONTROL PROBLEM WITH MULTIPLICATIVE REWARD
+Next, note that for any u ∈ (0, δ) and n, k ∈ N, we get
+����Ex
+�
+e
+� δ
+0 f(Xs)dsHn(Xδ)
+�
+− Exk
+�
+e
+� δ
+0 f(Xs)dsHn(Xδ)
+����
+≤
+���Ex
+��
+e
+� δ
+0 f(Xs)ds − e
+� δ
+u f(Xs)ds�
+Hn(Xδ)
+����
++
+���Exk
+��
+e
+� δ
+0 f(Xs)ds − e
+� δ
+u f(Xs)ds�
+Hn(Xδ)
+����
++
+���Exk
+�
+e
+� δ
+u f(Xs)dsHn(Xδ)
+�
+− Ex
+�
+e
+� δ
+u f(Xs)dsHn(Xδ)
+���� .
+(3.5)
+Also, using the inequality |ey − ez| ≤ emax(y,z)|y − z|, y, z ∈ R, we may find u > 0
+small enough such that, for any n, k ∈ N, we get
+���Exk
+��
+e
+� δ
+0 f(Xs)ds − e
+� δ
+u f(Xs)ds�
+Hn(Xδ)
+���� ≤ 2Keδ∥f∥u∥f∥ ≤ ε
+6.
+(3.6)
+Next, setting F u
+n (x) := Ex
+�
+e
+� δ−u
+0
+f(Xs)dsHn(Xδ−u)
+�
+, n ∈ N, x ∈ E, and using the
+Markov property combined with Assumption (A2), we may find k ∈ N big enough
+such that for any n ∈ N, we get
+���Exk
+�
+e
+� δ
+u f(Xs)dsHn(Xδ)
+�
+− Ex
+�
+e
+� δ
+u f(Xs)dsHn(Xδ)
+����
+= |Exk[F u
+n (Xu)] − Ex[F u
+n (Xu)]|
+≤ 2Keδ∥f∥ sup
+A∈E
+|Pu(xk, A) − Pu(x, A)| ≤ ε
+6.
+Thus, recalling (3.5)–(3.6), we get that for k ∈ N big enough and any n ∈ N,
+we get
+���Ex
+�
+e
+� δ
+0 f(Xs)dsHn(Xδ)
+�
+− Exk
+�
+e
+� δ
+0 f(Xs)dsHn(Xδ)
+���� ≤
+ε
+2. This combined
+with (3.3)–(3.4) shows | ˜T m
+δ hn(x) − ˜T m
+δ hn(xk)| ≤ ε for k ∈ N big enough and any
+n ∈ N, which proves the equicontinuity of the family ( ˜T m
+δ hn)n∈N. Consequently,
+using Arzel`a-Ascoli, we may find a uniformly (in x ∈ Bm) convergent subsequence
+of ( ˜T m
+δ hn)n∈N and the operator ˜T m
+δ
+is completely continuous.
+Thus, using the
+Krein-Rutman theorem we conclude that there exists a constant ˜λm
+δ
+> 0 and a
+non-zero map hm
+δ ∈ C+
+b (Bm) such that
+˜T m
+δ hm
+δ (x) = ˜λm
+δ hm
+δ (x),
+x ∈ Bm.
+(3.7)
+After a possible normalisation, we assume that supξ∈U hm
+δ (ξ) = 1.
+Let us now show that hm
+δ (x) > 0, x ∈ Bm. To see this, let us define D :=
+e−δ∥f∥ 1
+˜λm
+δ
+and let Oh ⊂ Bm be an open set such that
+inf
+x∈Oh hm
+δ (x) > 0;
+(3.8)
+note that this set exists thanks to the continuity of hm
+δ
+and the fact that hm
+δ
+is
+non-zero. Next, using (3.7), we have
+hm
+δ (x) ≥ DEx
+�
+1{Xδ∈Oh}hm
+δ (Xδ) + 1{Xδ∈Bm\Oh}hm
+δ (Xδ)
+�
+,
+x ∈ Bm.
+
+ASYMPTOTICS OF IMPULSE CONTROL PROBLEM WITH MULTIPLICATIVE REWARD 11
+Then, for any n ∈ N, we inductively get
+hm
+δ (x) ≥ DEx[1{Xδ∈Oh}hm
+δ (Xδ)]
++
+n
+�
+i=2
+DiEx
+�
+1{Xδ∈Bm\Oh,X2δ∈Bm\Oh,...,X(i−1)δ∈Bm\Oh,Xiδ∈Oh}hm
+δ (Xiδ)
+�
++ DnEx
+�
+1{Xδ∈Bm\Oh,X2δ∈Bm\Oh,...,Xiδ∈Bm\Oh}hm
+δ (Xnδ)
+�
+, x ∈ Bm.
+Thus, letting n → ∞ and using Assumption (A4) combined with (3.8), we show
+hm
+δ (x) > 0 for any x ∈ Bm.
+Next, we define wm
+δ (x) := ln hm
+δ (x), x ∈ Bm, and λm
+δ
+:=
+1
+δ ln ˜λm
+δ .
+Thus,
+from (3.7), we get that the pair (wm
+δ , λm
+δ ) satisfies
+˜T m
+δ ewm
+δ (x) = eδλm
+δ ewm
+δ (x),
+x ∈ Bm,
+and
+sup
+ξ∈U
+wm
+δ (ξ) = 0.
+In fact, using Assumption (A1) and the argument from Theorem 3.1 in [15], we
+have
+wm
+δ (x) = max
+�
+ln Ex
+�
+e
+� δ
+0 (f(Xs)−λm
+δ )ds+1{Xδ∈Bm}wm
+δ (Xδ)+1{Xδ /
+∈Bm}Mwm
+δ (Xδ)�
+,
+Mwm
+δ (x)
+�
+,
+x ∈ Bm.
+Finally, we extend the definition of wm
+δ to the full space E by setting
+wm
+δ (x) := Mwm
+δ (x),
+x /∈ Bm;
+note that the definition is correct since, at the right-hand side, we need to evaluate
+wm
+δ only at the points from U ⊂ B0 ⊂ Bm and this map is already defined there.
+□
+As we show now, Equation (3.2) may be linked to a specific martingale charac-
+terisation.
+Proposition 3.2. Let (wm
+δ , λm
+δ ) be a solution to (3.2). Then, for any x ∈ Bm, we
+get that the process
+zm
+δ (n) := e
+� (nδ)∧τBm
+0
+(f(Xs)−λm
+δ )ds+wm
+δ (X(nδ)∧τBm ),
+n ≥ 0,
+is a Px-supermartingale. Also, the process
+zm
+δ (n ∧ (ˆτ m
+δ /δ)),
+n ∈ N,
+is a Px-martingale, where ˆτ m
+δ := δ inf{k ∈ N: wm
+δ (Xkδ) = Mwm
+δ (Xkδ)}.
+Proof. To ease the notation, we show the proof only for δ = 1; the general case
+follows the same logic. Let us fix m, n ∈ N and x ∈ Bm. Then, using the fact
+wm
+1 (y) = Mwm
+1 (y), x /∈ Bm, and the inequality
+ewm
+1 (y) ≥ Ey
+�
+e
+� 1
+0 (f(Xs)−λm
+1 )ds+1{X1∈Bm}wm
+1 (X1)+1{X1 /
+∈Bm}Mwm
+1 (X1)�
+,
+y ∈ Bm,
+
+12 ASYMPTOTICS OF IMPULSE CONTROL PROBLEM WITH MULTIPLICATIVE REWARD
+we have
+Ex[zm
+1 (n + 1)|Fn]
+= 1{τBm≤n}e
+� τBm
+0
+(f(Xs)−λm
+1 )ds+wm
+1 (XτBm )
++ 1{τBm>n}e
+� n
+0 (f(Xs)−λm
+1 )ds×
+× EXn[e
+� n+1
+n
+(f(Xs)−λm
+1 )ds+1{X1∈Bm}wm
+1 (X1)+1{X1 /
+∈Bm}wm
+1 (X1)|Fn]
+= 1{τBm≤n}e
+� n∧τBm
+0
+(f(Xs)−λm
+1 )ds+wm
+1 (Xn∧τBm )
++ 1{τBm>n}e
+� n∧τBm
+0
+(f(Xs)−λm
+1 )ds×
+× EXn[e
+� n+1
+n
+(f(Xs)−λm
+1 )ds+1{X1∈Bm}wm
+1 (X1)+1{X1 /
+∈Bm}Mwm
+1 (X1)|Fn]
+≤ e
+� n∧τBm
+0
+(f(Xs)−λm
+1 )ds+wm
+1 (Xn∧τBm ) = zm
+1 (n),
+which shows the supermartingale property of (zm
+1 (n)). Next, note that on the set
+{τBm ∧ ˆτ m
+1 > n} we get
+ewm
+1 (Xn) = EXn
+�
+e
+� 1
+0 (f(Xs)−λm
+1 )ds+1{X1∈Bm}wm
+1 (X1)+1{X1 /
+∈Bm}Mwm
+1 (X1)�
+.
+Thus, we have
+Ex[zm
+1 ((n + 1) ∧ ˆτ m
+1 )|Fn]
+= 1{τBm∧ˆτ m
+1 ≤n}e
+� τBm ∧ˆτm
+1
+0
+(f(Xs)−λm
+1 )ds+wm
+1 (XτBm ∧ˆτm
+1 )
++ 1{τBm∧ˆτ m
+1 >n}e
+� n
+0 (f(Xs)−λm
+1 )ds×
+× EXn[e
+� n+1
+n
+(f(Xs)−λm
+1 )ds+1{X1∈Bm}wm
+1 (X1)+1{X1 /
+∈Bm}Mwm
+1 (X1)|Fn]
+= e
+� n∧τBm ∧ˆτm
+1
+0
+(f(Xs)−λm
+1 )ds+wm
+1 (Xn∧τBm ∧ˆτm
+1 ) = zm
+1 (n ∧ ˆτ m
+1 ),
+which concludes the proof.
+□
+Let us denote by Vδ,m the family of impulse control strategies with impulse times
+in the time-grid {0, δ, 2δ, . . .} and obligatory impulses when the controlled process
+exits the set Bm at some multiple of δ. Using a martingale characterisation of (3.2),
+we get that λm
+δ is the optimal value of the impulse control problem with impulse
+strategies from Vδ,m. To show this result, we introduce a strategy ˆV := (ˆτi, ˆξi)∞
+i=1 ∈
+Vδ,m defined recursively, for i = 1, 2, . . ., by
+ˆτi := ˆσi ∧ τ i
+Bm,
+ˆσi := δ inf{n ≥ ˆτi−1/δ: n ∈ N, wm
+δ (Xi
+nδ) = Mwm
+δ (Xi
+nδ)},
+τ i
+Bm := δ inf{n ≥ ˆτi−1/δ: n ∈ N, Xi
+nδ /∈ Bm},
+ˆξi := arg max
+ξ∈U
+(c(Xi
+ˆτi, ξ) + wm
+δ (ξ))1{ˆτi<∞} + ξ01{ˆτi=∞},
+(3.9)
+where ˆτ0 := 0 and ξ0 ∈ U is some fixed point.
+Theorem 3.3. Let (wm
+δ , λm
+δ ) be a solution to (3.2). Then, for any x ∈ Bm, we get
+λm
+δ =
+sup
+V ∈Vδ,m
+lim inf
+n→∞
+1
+nδ ln E(x,V )
+�
+e
+� nδ
+0
+f(Ys)ds+�∞
+i=1 1{τi≤nδ}c(Yτ−
+i
+,ξi)�
+.
+Also, the strategy ˆV defined in (3.9) is optimal.
+
+ASYMPTOTICS OF IMPULSE CONTROL PROBLEM WITH MULTIPLICATIVE REWARD 13
+Proof. The proof follows the lines of the proof of Theorem 2.3 and is omitted for
+brevity.
+□
+Next, we link (3.2) with an infinite horizon optimal stopping problem under the
+non-degeneracy assumption.
+Theorem 3.4. Let (wm
+δ , λm
+δ ) be a solution to (3.2) with λm
+δ > r(f). Then, we get
+that (wm
+δ , λm
+δ ) satisfies (3.1).
+Proof. As in the proof of Proposition 3.2, we consider only δ = 1; the general case
+follows the same logic.
+First, note that for any x ∈ Bm, n ∈ N, and τ ∈ T δ
+x , using Proposition 3.2 and
+Doob’s optional stopping theorem, we have
+ewm
+1 (x) ≥ Ex
+�
+e
+� n∧τ∧τBm
+0
+(f(Xs)−λm
+1 )ds+wm
+1 (Xn∧τ∧τBm )
+�
+.
+Also, recalling the boundedness of wm
+1 and Proposition A.2, and letting n → ∞,
+we get
+ewm
+1 (x) ≥ Ex
+�
+e
+� τ∧τBm
+0
+(f(Xs)−λm
+1 )ds+wm
+1 (Xτ∧τBm )
+�
+.
+Next, noting that wm
+1 (Xτ∧τBm) ≥ Mwm
+1 (Xτ∧τBm), and taking the supremum over
+τ ∈ T δ
+x , we get
+ewm
+1 (x) ≥ sup
+τ∈T δ
+x
+Ex
+�
+e
+� τ∧τBm
+0
+(f(Xs)−λm
+1 )ds+Mwm
+1 (Xτ∧τBm )
+�
+.
+Second, using again Proposition 3.2, for any x ∈ Bm and n ∈ N, we get
+wm
+1 (x) = ln Ex
+�
+e
+� n∧ˆτm
+δ ∧τBm
+0
+(f(Xs)−λm
+1 )ds+wm
+1 (Xn∧ˆτm
+δ
+∧τBm )
+�
+.
+Using again the boundedness of wm
+1 and Proposition A.2, and letting n → ∞, we
+get
+wm
+1 (x) = ln Ex
+�
+e
+� ˆτm
+δ ∧τBm
+0
+(f(Xs)−λm
+1 )ds+wm
+1 (Xˆτm
+δ
+∧τBm )
+�
+.
+In fact, noting that wm
+1 (Xˆτ m
+δ ∧τBm) = Mwm
+1 (Xˆτ m
+δ ∧τBm), we obtain
+wm
+1 (x) = ln Ex
+�
+e
+� ˆτm
+δ ∧τBm
+0
+(f(Xs)−λm
+1 )ds+Mwm
+1 (Xˆτm
+δ
+∧τBm )
+�
+,
+thus we get
+ewm
+1 (x) = sup
+τ∈T δ
+x
+Ex
+�
+e
+� τ∧τBm
+0
+(f(Xs)−λm
+1 )ds+Mwm
+1 (Xτ∧τBm )
+�
+.
+Finally, using Proposition A.4, we have
+ewm
+1 (x) = sup
+τ∈T δ
+x,b
+Ex
+�
+e
+� τ∧τBm
+0
+(f(Xs)−λm
+1 )ds+Mwm
+1 (Xτ∧τBm )
+�
+,
+which concludes the proof.
+□
+Remark 3.5. In Theorem 3.4 we showed that, if λm
+δ
+> r(f), a solution to the
+one-step equation (3.2) is uniquely characterised by the optimal stopping value
+function (3.1).
+If λm
+δ
+≤ r(f), the problem is degenerate and, in particular, we
+cannot use the uniform integrability result from Proposition A.2. In fact, in this
+
+14 ASYMPTOTICS OF IMPULSE CONTROL PROBLEM WITH MULTIPLICATIVE REWARD
+case it is even possible that the one-step Bellman equation admits multiple solutions
+and the optimal stopping characterisation does not hold; see e.g. Theorem 1.13
+in [22] for details.
+4. Dyadic impulse control
+In this section we consider a dyadic full-domain version of (2.1). We construct
+a solution to the associated Bellman equation which will be later used to find
+a solution to (2.6).
+The argument uses a bounded domain approximation from
+Section 3. More specifically, throughout this section we fix some δ > 0 and show
+the existence of a function wδ ∈ Cb(E) and a constant λδ ∈ R, which are a solution
+to the dyadic Bellman equation of the form
+wδ(x) = sup
+τ∈T δ
+x,b
+ln Ex
+�
+e
+� τ
+0 (f(Xs)−λδ)ds+Mwδ(Xτ )�
+,
+x ��� E.
+(4.1)
+In fact, we set
+λδ := lim
+m→∞ λm
+δ ;
+(4.2)
+note that this constant is well-defined as, from Theorem 3.3, recalling that Bm ⊂
+Bm+1, we get λm
+δ ≤ λm+1
+δ
+, m ∈ N.
+First, we state the lower bound for λδ.
+Lemma 4.1. Let (wδ, λδ) be a solution to (4.1). Then, we get λδ ≥ r(f).
+Proof. The proof follows the lines of the proof of Lemma 2.1 and is omitted for
+brevity.
+□
+Next, we show the existence of a solution to (4.1) under the non-degeneracy
+assumption λδ > r(f).
+Theorem 4.2. Let λδ be given by (4.2) and assume that λδ > r(f). Then, there
+exists wδ ∈ Cb(E) such that (4.1) is satisfied and we get supξ∈U wδ(ξ) = 0.
+Proof. We start with some general comments and an outline of the argument. First,
+note that from Theorem 3.1, for any m ∈ N, we get a solution (wm
+δ , λm
+δ ) to (3.2)
+satisfying supξ∈U wm
+δ (ξ) = 0. Also, from the assumption λδ > r(f) we get λm
+δ >
+r(f) for m ∈ N sufficiently big (for simplicity, we assume that λ0
+δ > r(f)). Thus,
+using Theorem 3.4, we get that, for any m ∈ N, the pair (wm
+δ , λm
+δ ) satisfies (3.1).
+Second, to construct a function wδ, we use Arzel`a-Ascoli Theorem. More specif-
+ically, recalling that supξ∈U wm
+δ (ξ) = 0 and using the fact that −∥c∥ ≤ c(x, ξ) ≤ 0,
+x ∈ E, ξ ∈ U, for any m ∈ N and x ∈ E, we get
+−∥c∥ ≤ Mwm
+δ (x) ≤ 0.
+Also, note that, for any m ∈ N and x, y ∈ E, we have
+|Mwm
+δ (x) − Mwm
+δ (y)| ≤ sup
+ξ∈U
+|c(x, ξ) − c(y, ξ)|.
+Consequently, the sequence (Mwm
+δ )m∈N is uniformly bounded and equicontinuous.
+Thus, using Arzel`a-Ascoli Theorem combined with a diagonal argument, we may
+find a subsequence (for brevity still denoted by (Mwm
+δ )m∈N) and a map φδ ∈ Cb(E)
+such that Mwm
+δ (x) converges to φδ(x) as m → ∞ uniformly in x from any compact
+set. In fact, using Assumption (A1) and the argument from the first step of the
+
+ASYMPTOTICS OF IMPULSE CONTROL PROBLEM WITH MULTIPLICATIVE REWARD 15
+proof of Theorem 4.1 in [15], we get that the convergence is uniform in x ∈ E.
+Then, we define
+wδ(x) := sup
+τ∈T δ
+x,b
+ln Ex
+�
+e
+� τ
+0 (f(Xs)−λδ)ds+φδ(Xτ )�
+,
+x ∈ E.
+(4.3)
+To complete the construction, we show that wm
+δ
+converges to wδ uniformly on
+compact sets. Indeed, in this case we have
+|Mwm
+δ (x) − Mwδ(x)| ≤ sup
+ξ∈U
+|wm
+δ (ξ) − wδ(ξ)| → 0,
+m → ∞,
+thus φδ ≡ Mwδ and from (4.3) we get that (4.1) is satisfied. Also, recalling that
+from Theorem 3.1 we get supξ∈U wm
+δ (ξ) = 0, m ∈ N, we also get supξ∈U wδ(ξ) = 0.
+Finally, to show the convergence, we define the auxiliary functions
+wm,1
+δ
+(x) := sup
+τ∈T δ
+x,b
+ln Ex
+�
+e
+� τ∧τBm
+0
+(f(Xs)−λm
+δ )ds+φδ(Xτ∧τBm )
+�
+,
+x ∈ E,
+(4.4)
+wm,2
+δ
+(x) := sup
+τ∈T δ
+x,b
+ln Ex
+�
+e
+� τ∧τBm
+0
+(f(Xs)−λδ)ds+φδ(Xτ∧τBm )
+�
+,
+x ∈ E.
+(4.5)
+We split the rest of the proof into three steps: (1) proof that |wm
+δ (x)−wm,1
+δ
+(x)| → 0
+as m → ∞ uniformly in x ∈ E; (2) proof that |wm,1
+δ
+(x) − wm,2
+δ
+(x)| → 0 as m → ∞
+uniformly in x ∈ E; (3) proof that |wm,2
+δ
+(x) − wδ(x)| → 0 as m → ∞ uniformly in
+x from compact sets.
+Step 1. We show |wm
+δ (x) − wm,1
+δ
+(x)| → 0 as m → ∞ uniformly in x ∈ E. Note
+that, for any x ∈ E and m ∈ N, we have
+wm,1
+δ
+(x) ≤ sup
+τ∈T δ
+x,b
+ln
+�
+Ex
+�
+e
+� τ∧τBm
+0
+(f(Xs)−λm
+δ )ds+Mwm
+δ (Xτ∧τBm )
+�
+e∥φδ−Mwm
+δ ∥
+�
+= wm
+δ (x) + ∥φδ − Mwm
+δ ∥.
+Similarly, we get wm
+δ (x) ≤ wm,1
+δ
+(x) + ∥φδ − Mwm
+δ ∥, thus
+sup
+x∈E
+|wm
+δ (x) − wm,1
+δ
+(x)| ≤ ∥φδ − Mwm
+δ ∥.
+Recalling the fact that φδ is a uniform limit of Mwm
+δ as m → ∞, we conclude the
+proof of this step.
+Step 2. We show that |wm,1
+δ
+(x) − wm,2
+δ
+(x)| → 0 as m → ∞ uniformly in x ∈ E.
+Recalling that λm
+δ ↑ λδ, we get wm,1
+δ
+(x) ≥ wm,2
+δ
+(x) ≥ −∥φδ∥, x ∈ E. Thus, using
+the inequality | ln y − ln z| ≤
+1
+min(y,z)|y − z|, y, z > 0, we get
+0 ≤ wm,1
+δ
+(x) − wm,2
+δ
+(x) ≤ e∥φδ∥(ewm,1
+δ
+(x) − ewm,2
+δ
+(x)),
+x ∈ E.
+(4.6)
+Then, noting that φδ(·) ≤ 0, for any m ∈ N and x ∈ E, we obtain
+0 ≤ ewm,1
+δ
+(x) − ewm,2
+δ
+(x) ≤ sup
+τ∈T δ
+x,b
+�
+Ex
+�
+e
+� τ∧τBm
+0
+(f(Xs)−λm
+δ )ds+φδ(Xτ∧τBm )
+�
+− Ex
+�
+e
+� τ∧τBm
+0
+(f(Xs)−λδ)ds+φδ(Xτ∧τBm )
+��
+≤ sup
+τ∈T δ
+x,b
+Ex
+�
+e
+� τ
+0 f(Xs)ds �
+e−λm
+δ τ − e−λδτ��
+.
+(4.7)
+
+16 ASYMPTOTICS OF IMPULSE CONTROL PROBLEM WITH MULTIPLICATIVE REWARD
+Also, recalling that λ0
+δ ≤ λm
+δ ≤ λδ, m ∈ N, for any x ∈ E and T ≥ 0, we get
+0 ≤ sup
+τ∈Tx,b
+Ex
+�
+e
+� τ
+0 f(Xs)ds �
+e−λm
+δ τ − eλδτ��
+≤ sup
+τ∈Tx,b
+Ex
+��
+1{τ≤T } + 1{τ>T }
+�
+e
+� τ
+0 f(Xs)ds �
+e−λm
+δ τ − e−λδτ��
+≤ sup
+τ<T
+τ∈Tx,b
+eT ∥f∥Ex
+��
+e−λm
+δ τ − e−λδτ��
++ sup
+τ≥T
+τ∈Tx,b
+Ex
+�
+e
+� τ
+0 (f(Xs)−λ0
+δ)ds�
+.
+(4.8)
+Recalling λ0
+δ > r(f) and using Lemma A.1, for any ε > 0, we may find T ≥ 0, such
+that
+0 ≤ sup
+x∈E
+sup
+τ≥T
+τ∈Tx,b
+Ex
+�
+e
+� τ
+0 (f(Xs)−λ0
+δ)ds�
+≤ ε.
+Also, using the inequality |ex − ey| ≤ emax(x,y)|x − y|, x, y ≥ 0, we obtain
+sup
+τ<T
+Ex
+��
+e−λm
+δ τ − e−λδτ��
+≤ sup
+τ<T
+Ex
+�
+emax(−λm
+δ τ,−λδτ)τ(λδ − λm
+δ )
+�
+≤ e|λm
+δ |T T (λδ − λm
+δ ).
+(4.9)
+Thus, for fixed T ≥ 0, we find m ≥ 0, such that e|λm
+δ |T T (λδ − λm
+δ ) ≤ ε. Hence,
+recalling (4.6)–(4.8), for any x ∈ E and T, m big enough, we get
+0 ≤ wm,1
+δ
+(x) − wm,2
+δ
+(x) ≤ e∥φδ∥2ε.
+Recalling that ε > 0 was arbitrary, we conclude the proof of this step.
+Step 3. We show that |wm,2
+δ
+(x) − wδ(x)| → 0 as m → ∞ uniformly in x from
+compact sets. First, we show that wm,2
+δ
+(x) ≤ wδ(x) for any m ∈ N and x ∈ E. Let
+ε > 0 and τ ε
+m ∈ T δ
+x,b be an ε-optimal stopping time for wm,2
+δ
+(x). Then, we get
+wδ(x) ≥ ln Ex
+�
+e
+� τεm∧τBm
+0
+(f(Xs)−λδ)ds+φδ(Xτεm∧τBm )
+�
+≥ wm,2
+δ
+(x) − ε.
+As ε > 0 was arbitrary, we get wm,2
+δ
+(x) ≤ wδ(x), m ∈ N, x ∈ E. In fact, using
+a similar argument, for any x ∈ E, we may show that the map m �→ wm,2
+δ
+(x) is
+non-decreasing.
+Second, let ε > 0 and τε ∈ T δ
+x,b be an ε-optimal stopping time for wδ(x). Then,
+we obtain
+0 ≤ wδ(x) − wm,2
+δ
+(x) ≤ ln Ex
+�
+e
+� τε
+0
+(f(Xs)−λδ)ds+φδ(Xτε )�
++ ε
+− ln Ex
+�
+e
+� τε∧τBm
+0
+(f(Xs)−λδ)ds+φδ(Xτε∧τBm )
+�
+.
+(4.10)
+Noting that τBm ↑ +∞ as m → ∞ and using the quasi left-continuity of X combined
+with Lemma A.2 and the boundedness of φδ, we get
+lim
+m→∞ Ex
+�
+e
+� τε∧τBm
+0
+(f(Xs)−λδ)ds+φδ(Xτε∧τBm )
+�
+= Ex
+�
+e
+� τε
+0
+(f(Xs)−λδ)ds+φδ(Xτε )�
+.
+Thus, using (4.10) and recalling that ε > 0 was arbitrary, we get limm→∞ wm,2
+δ
+(x) =
+wδ(x). Also, noting that by Proposition A.3 and Proposition A.4, the maps x �→
+
+ASYMPTOTICS OF IMPULSE CONTROL PROBLEM WITH MULTIPLICATIVE REWARD 17
+wδ(x) and x �→ wm,2
+δ
+(x) are continuous, and using the monotonicity of m �→
+wm,2
+δ
+(x), from Dini’s Theorem we get that wm,2
+δ
+(x) converges to wδ(x) uniformly
+in x from compact sets, which concludes the proof.
+□
+We conclude this section with a verification result related to (4.1).
+Theorem 4.3. Let (wδ, λδ) be a solution to (4.1) with λδ > r(f). Then, we get
+λδ := sup
+V ∈Vδ lim inf
+n→∞
+1
+n ln E(x,V )
+�
+e
+� nδ
+0
+f(Ys)ds+�∞
+i=1 1{τi≤nδ}c(Yτ−
+i
+,ξi)�
+,
+where Vδ is a family of impulse control strategies with impulse times on the dyadic
+time-grid {0, δ, 2δ, . . .}.
+Proof. The proof follows the lines of the proof of Theorem 2.3 and is omitted for
+brevity.
+□
+5. Existence of a solution to the Bellman equation
+In this section we construct a solution (w, λ) to (2.6), which together with The-
+orem 2.3 provides a solution to (2.1). The argument uses a dyadic approximation
+and the results from Section 4. More specifically, we consider a family of dyadic
+time steps δk :=
+1
+2k , k ∈ N. First, we specify the value of λ. In fact, we define
+λ := lim inf
+k→∞ λδk,
+(5.1)
+where λδk is a constant given by (4.2), corresponding to δk. Note that, if for some
+k0 ∈ N we get λδk0 > r(f), then using Theorem 4.3, we get that λδk ≤ λδk+1,
+k ≥ k0, and the limit inferior could be replaced by the usual limit.
+Theorem 5.1. Let λ be given by (5.1) and assume that λ > r(f). Then, there
+exists w ∈ Cb(E) such that (2.6) is satisfied.
+Proof. The argument is partially based on the one used in Theorem 4.2 thus we
+discuss only the main points. From the fact that λ > r(f) we get λδk > r(f) for
+sufficiently big k ∈ N; to simplify the notation, we assume λδ0 > r(f). Thus, using
+Theorem 4.2, for any k ∈ N, we get the existence of a map wδk ∈ Cb(E) satisfying
+wδk(x) = sup
+τ∈T
+δk
+x,b
+ln Ex
+�
+e
+� τ
+0 (f(Xs)−λδk )ds+Mwδk (Xτ )�
+,
+x ∈ E
+and such that supξ∈U wδk(ξ) = 0. Thus, we get
+−∥c∥ ≤ Mwδk(x) ≤ 0,
+k ∈ N, x ∈ E,
+and the family (Mwδk)k∈N is uniformly bounded. Also, it is equicontinuous as we
+have
+|Mwδk(x) − Mwδk(y)| ≤ sup
+x∈U
+|c(x, ξ) − c(y, ξ)|,
+x, y ∈ E.
+Thus, using Arzel`a-Ascoli theorem, we may choose a subsequence (for brevity still
+denoted by (Mwδk)), such that (Mwδk) converges uniformly on compact sets to
+some map φ. In fact, using Assumption (A1) and the argument from the first step
+of the proof of Theorem 4.1 from [15], we get that Mwδk(x) converges to φ(x) as
+k → ∞ uniformly in x ∈ E. Next, let us define
+w(x) := sup
+τ∈Tx,b
+ln Ex
+�
+e
+� τ
+0 (f(Xs)−λ)ds+φ(Xτ )�
+,
+x ∈ E.
+(5.2)
+
+18 ASYMPTOTICS OF IMPULSE CONTROL PROBLEM WITH MULTIPLICATIVE REWARD
+In the following, we show that wδk converges to w uniformly in compact sets as
+k → ∞. Then, we get that Mwδk converges to Mw, hence Mw ≡ φ and (2.6) is
+satisfied.
+To show the convergence, we define
+w1
+δk(x) := sup
+τ∈T
+δk
+x,b
+ln Ex
+�
+e
+� τ
+0 (f(Xs)−λδk)ds+φ(Xτ )�
+,
+k ∈ N, x ∈ E.
+In the following, we show that |w(x) − w1
+δk(x)| → 0 and |w1
+δk(x) − wδk(x)| → 0 as
+k → ∞ uniformly in x from compact sets. In fact, to show the first convergence,
+we note that
+w0
+δk(x) ≤ w1
+δk(x) ≤ w2
+δk(x),
+k ∈ N, x ∈ E,
+where
+w0
+δk(x) := sup
+τ∈T
+δk
+x,b
+ln Ex
+�
+e
+� τ
+0 (f(Xs)−λ)ds+φ(Xτ )�
+,
+k ∈ N, x ∈ E,
+w2
+δk(x) := sup
+τ∈Tx,b
+ln Ex
+�
+e
+� τ
+0 (f(Xs)−λδk )ds+φ(Xτ )�
+,
+k ∈ N, x ∈ E.
+Thus, to prove |w(x) − w1
+δk(x)| → 0 it is enough to show |w(x) − w0
+δk(x)| → 0 and
+|w(x) − w2
+δk(x)| → 0 as k → ∞.
+For transparency, we split the rest of the proof into three parts: (1) proof that
+|w(x) − w0
+δk(x)| → 0 as k → ∞ uniformly in x from compact sets; (2) proof that
+|w(x) − w2
+δk(x)| → 0 as k → ∞ uniformly in x ∈ E; (3) proof that |w1
+δk(x) −
+wδk(x)| → 0 as k → ∞ uniformly in x ∈ E.
+Step 1. We show that |w(x) − w0
+δk(x)| → 0 as k → ∞ as k → ∞ uniformly in x
+from compact sets. First, note that we have w0
+δk(x) ≤ w(x), k ∈ N, x ∈ E. Next,
+for any x ∈ E and ε > 0, let τε ∈ Tx,b be an ε-optimal stopping time for w(x) and
+let τ k
+ε be its T δk
+x,b approximation given by
+τ k
+ε := inf
+�
+τ ∈ T δk
+x,b : τ ≥ τε
+�
+=
+∞
+�
+j=1
+1{ j−1
+2k <τε≤
+j
+2m }
+j
+2k .
+Then, we get
+0 ≤ w(x) − w0
+δk(x)
+≤ Ex
+�
+e
+� τε
+0
+(f(Xs)−λ)ds+φ(Xτε )�
+− Ex
+�
+e
+� τk
+ε
+0
+(f(Xs)−λ)ds+φ(Xτk
+ε )
+�
++ ε.
+Also, using Proposition A.2 and letting k → ∞, we have
+lim
+k→∞ Ex
+�
+e
+� τk
+ε
+0
+(f(Xs)−λ)ds+φ(Xτk
+ε )
+�
+= Ex
+�
+e
+� τε
+0
+(f(Xs)−λ)ds+φ(Xτε )�
+.
+Consequently, recalling that ε > 0 was arbitrary, we obtain limk→∞ w0
+δk(x) =
+w(x) for any x ∈ E.
+In fact, using the monotonicity of the sequence (w0
+δk)k∈N
+combined with Proposition A.3, Proposition A.4, and Dini’s theorem, we get that
+the convergence is uniform on compact sets, which concludes the proof of this step.
+
+ASYMPTOTICS OF IMPULSE CONTROL PROBLEM WITH MULTIPLICATIVE REWARD 19
+Step 2. We show that |w(x) − w2
+δk(x)| → 0 as k → ∞ uniformly in x ∈ E.
+First, note that −∥φ∥ ≤ w(x) ≤ w2
+δk(x), k ∈ N, x ∈ E. Thus, using the inequality
+| ln y − ln z| ≤
+1
+min(y,z)|y − z|, y, z > 0, we get
+0 ≤ w2
+δk(x) − w(x) ≤ e∥φ∥(ew2
+δk (x) − ew(x)),
+k ∈ N, x ∈ E.
+Also, recalling that φ(·) ≤ 0, for any k ∈ N and x ∈ E, we obtain
+0 ≤ ew2
+δk (x) − ew(x) ≤ sup
+τ∈Tx,b
+Ex
+�
+e
+� τ
+0 f(Xs)ds �
+e−λδkτ − e−λτ��
+.
+Thus, repeating the argument from the second step of the proof of Theorem 4.2,
+we get w2
+δk(x) → w(x) as k → ∞ uniformly in x ∈ E, which concludes the proof of
+this step.
+Step 3. We show that |w1
+δk(x) − wδk(x)| → 0 as k → ∞ uniformly in x ∈ E.
+In fact, recalling that ∥Mwδk − φ∥ → 0 as k → ∞, the argument follows the lines
+of the one used in the first step of the proof of Theorem 4.2. This concludes the
+proof.
+□
+Appendix A. Properties of optimal stopping problems
+In this section we discuss some properties of the optimal stopping problems that
+are used in this paper.
+Throughout this section we consider g, G ∈ Cb(E) and
+assume G(·) ≤ 0 and r(g) < 0, where r(g) is the type of the semigroup given
+by (2.7) corresponding to the map g. We start with a useful result related to the
+asymptotic behaviour of the running cost function g.
+Lemma A.1. Let a be such that r(g) < a < 0. Then,
+(1) The map x �→ U g−a
+0
+1(x) := Ex
+�� ∞
+0
+e
+� t
+0 (g(Xs)−a)dsdt
+�
+is continuous and
+bounded.
+(2) We get
+lim
+T →∞ sup
+x∈E
+sup
+τ≥T
+τ∈Tx
+Ex
+�
+e
+� τ
+0 g(Xs)ds�
+= 0.
+Proof. For transparency, we prove the claims point by point.
+Proof of (1). First, we show the boundedness of x �→ U g−a
+0
+1(x). Let ε <
+a − r(g).
+Using the definition of r(g − a) we may find t0 ≥ 0, such that for
+any t ≥ t0 we get supx∈E Ex
+�
+e
+� t
+0 (g(Xs)−a)ds�
+≤ et(r(g)−a+ε). Then, using Fubini’s
+theorem and noting that r(g) − a + ε < 0, for any x0 ∈ E, we get
+0 ≤ U g−a
+0
+1(x0) ≤
+� ∞
+0
+sup
+x∈E
+Ex
+�
+e
+� t
+0 (g(Xs)−a)ds�
+dt
+=
+� t0
+0
+sup
+x∈E
+Ex
+�
+e
+� t
+0 (g(Xs)−a)ds�
+dt +
+� ∞
+t0
+sup
+x∈E
+Ex
+�
+e
+� t
+0 (g(Xs)−a)ds�
+dt
+≤
+� t0
+0
+et(∥g∥−a)dt +
+� ∞
+t0
+et(r(g)−a+ε)dt < ∞,
+which concludes the proof of the boundedness of x �→ U g−a
+0
+1(x).
+
+20 ASYMPTOTICS OF IMPULSE CONTROL PROBLEM WITH MULTIPLICATIVE REWARD
+For the continuity, note that using Assumption (A2) and repeating the argument
+used in Lemma 4 in Section II.5 of [13], we get that x �→ Ex
+�
+e
+� t
+0 (g(Xs)−a)dsdt
+�
+is
+continuous for any t ≥ 0. Also, as in the proof of the boundedness, we may show
+0 ≤ sup
+x∈E
+Ex
+�
+e
+� t
+0 (g(Xs)−a)ds�
+≤ et(∥g∥−a)1{t∈[0,t0]} + et(r(g)−a+ε)1{t>t0}
+and the upper bound is integrable (with respect to t).
+Thus, using Lebesgue’s
+dominated convergence theorem, we get the continuity of the map x �→ U g−a
+0
+1(x) =
+� ∞
+0
+Ex
+�
+e
+� t
+0 (g(Xs)−a)ds�
+dt, which concludes the proof of this step.
+Proof of (2). Noting that U g−a
+0
+1(x) ≥
+� 1
+0 e−t(∥g∥−a)dt, x ∈ E, we may find
+d > 0, such that U g−a
+0
+1(x) ≥ d > 0, x ∈ E. Thus, recalling that a < 0, we get
+0 ≤ sup
+τ≥T
+τ∈Tx
+Ex
+�
+e
+� τ
+0 g(Xs)ds�
+≤ sup
+τ≥T
+τ∈Tx
+Ex
+�
+e
+� τ
+0 (g(Xs)−a)dseaτU g−a
+0
+1(Xτ)1
+d
+�
+≤ eaT
+d
+sup
+τ≥T
+τ∈Tx
+Ex
+�
+e
+� τ
+0 (g(Xs)−a)dsU g−a
+0
+1(Xτ)
+�
+= eaT
+d
+sup
+τ≥T
+τ∈Tx
+Ex
+�� ∞
+0
+e
+� t+τ
+0
+(g(Xs)−a)dsdt
+�
+= eaT
+d
+sup
+τ≥T
+τ∈Tx
+Ex
+�� ∞
+τ
+e
+� t
+0 (g(Xs)−a)dsdt
+�
+≤ eaT
+d Ex
+�� ∞
+0
+e
+� t
+0 (g(Xs)−a)dsdt
+�
+≤ eaT
+d ∥U g−a
+0
+1∥ → 0,
+T → ∞,
+which concludes the proof.
+□
+Using Lemma A.1 we get the uniform integrability of a suitable family of random
+variables.
+This result is extensively used throughout the paper as it simplifies
+numerous limiting arguments.
+Proposition A.2. For any x ∈ E, the family {e
+� τ
+0 g(Xs)ds}τ∈Tx is Px-uniformly
+integrable.
+Proof. Let us fix some x ∈ E and, for any τ ∈ Tx and n ∈ N, define the event
+Aτ
+n := {� τ
+0 g(Xs)ds ≥ n}. Note that for any T ≥ 0, we get
+sup
+τ∈Tx
+Ex[1Aτne
+� τ
+0 g(Xs)ds] ≤ sup
+τ≤T
+τ∈Tx
+Ex[1Aτne
+� τ
+0 g(Xs)ds] + sup
+τ>T
+τ∈Tx
+Ex[1Aτne
+� τ
+0 g(Xs)ds]
+≤ sup
+τ≤T
+τ∈Tx
+eT ∥g∥Px[Aτ
+n] + sup
+τ>T
+τ∈Tx
+Ex[e
+� τ
+0 g(Xs)ds].
+Next, for any ε > 0, using Lemma A.1, we may find T > 0 big enough to get
+sup
+τ>T
+τ∈Tx
+Ex[e
+� τ
+0 g(Xs)ds] < ε.
+
+ASYMPTOTICS OF IMPULSE CONTROL PROBLEM WITH MULTIPLICATIVE REWARD 21
+Also, noting that for τ ≤ T , we get Aτ
+n ⊂ {T ∥g∥ ≥ n}, for any n > T ∥g∥, we also
+get
+sup
+τ≤T
+τ∈Tx
+Px[Aτ
+n] = 0.
+Consequently, recalling that ε > 0 was arbitrary, we obtain
+lim
+n→∞ sup
+τ∈Tx
+Ex[Aτ
+ne
+� τ
+0 g(Xs)ds] = 0,
+which concludes the proof.
+□
+Next, we consider an optimal stopping problem of the form
+u(x) := sup
+τ∈Tx,b
+ln Ex
+�
+exp
+�� τ
+0
+g(Xs)ds + G(Xτ)
+��
+,
+x ∈ E;
+(A.1)
+note that here the non-positivity assumption for G is only a normalisation as for a
+generic ˜G we may set G(·) = ˜G(·) − ∥ ˜G∥ to get G(·) ≤ 0.
+The properties of the map (A.1) are summarised in the following proposition.
+Proposition A.3. Let the map u be given by (A.1). Then, x �→ u(x) is continuous
+and bounded. Also, we get
+u(x) = sup
+τ∈Tx
+ln Ex
+�
+exp
+�� τ
+0
+g(Xs)ds + G(Xτ)
+��
+,
+x ∈ E.
+(A.2)
+Moreover, the process
+z(t) := e
+� t
+0 g(Xs)+u(Xt),
+t ≥ 0,
+is a supermartingale and the process z(t ∧ ˆτ), t ≥ 0, is a martingale, where
+ˆτ := inf{t ≥ 0 : u(Xt) ≤ G(Xt)}.
+(A.3)
+Proof. For transparency, we split the proof into two steps: (1) proof of the conti-
+nuity of x �→ u(x) and identity (A.2); (2) proof of the martingale properties of the
+process z.
+Step 1. We show that the map x �→ u(x) is continuous and the identity (A.2)
+holds. For any T ≥ 0 and x ∈ E, let us define
+ˆu(x) := sup
+τ∈Tx
+ln Ex
+�
+exp
+�� τ
+0
+g(Xs)ds + G(Xτ)
+��
+;
+(A.4)
+uT (x) := sup
+τ≤T
+ln Ex
+�
+exp
+�� τ
+0
+g(Xs)ds + G(Xτ)
+��
+.
+(A.5)
+Using Assumption (A3) and following the proof of Proposition 10 and Proposition
+11 in [16], we get that the map (T, x) �→ uT (x) is jointly continuous and bounded;
+see also Remark 12 therein. We show that uT (x) → ˆu(x) as T → ∞ uniformly in
+x ∈ E. Noting that
+−∥G∥ ≤ uT (x) ≤ u(x),
+T ≥ 0, x ∈ E,
+and using the inequality | ln y−lnz| ≤
+1
+min(y,z)|y−z|, y, z > 0, to show uT (x) → ˆu(x)
+as T → ∞ uniformly in x ∈ E it is enough to show euT (x) → eˆu(x) as T → ∞
+
+22 ASYMPTOTICS OF IMPULSE CONTROL PROBLEM WITH MULTIPLICATIVE REWARD
+uniformly in x ∈ E. Then, using Lemma A.1, for any ε > 0, we may find T ≥ 0
+such that for any x ∈ E, we obtain
+0 ≤ eˆu(x) − euT (x) ≤ sup
+τ∈Tx
+Ex
+�
+e
+� τ
+0 g(Xs)ds+G(Xτ ) − e
+� τ∧T
+0
+g(Xs)ds+G(Xτ∧T )�
+≤ sup
+τ∈Tx
+Ex
+�
+1{τ≥T }
+�
+e
+� τ
+0 g(Xs)ds+G(Xτ ) − e
+� T
+0 g(Xs)ds+G(XT )��
+≤ sup
+τ∈Tx
+Ex
+�
+1{τ≥T }e
+� τ
+0 g(Xs)ds+G(Xτ )�
+≤ sup
+τ≥T
+τ∈Tx
+Ex
+�
+e
+� τ
+0 g(Xs)ds�
+≤ ε.
+Thus, letting ε → 0, we get euT (x) → eˆu(x) as T → ∞ uniformly in x ∈ E and
+consequently uT (x) → ˆu(x) as T → ∞ uniformly in x ∈ E.
+Thus, from the
+continuity of x �→ uT (x), T ≥ 0, we get that the map x �→ ˆu(x) is continuous.
+Now, we show that u ≡ ˆu. First, we show that limT →∞ uT (x) = ˜u(x), where
+˜u(x) := supτ∈Tx lim infT →∞ ln Ex
+�
+e
+� τ∧T
+0
+g(Xs)ds+G(Xτ∧T )�
+, x ∈ E. For any T ≥ 0
+and x ∈ E, we get
+uT (x) = sup
+τ≤T
+lim inf
+S→∞ ln Ex
+�
+e
+� τ∧S
+0
+g(Xs)ds+G(Xτ∧S)�
+≤ ˜u(x),
+thus we get limT →∞ uT (x) ≤ ˜u(x). Also, for any x ∈ E, ˜τ ∈ Tx, and T ≥ 0, we get
+ln Ex
+�
+e
+� ˜τ∧T
+0
+g(Xs)ds+G(X˜τ∧T )�
+≤ uT(x).
+Thus, letting T → ∞ and taking supremum over ˜τ ∈ Tx we get limT →∞ uT (x) =
+˜u(x), x ∈ E. Also, using the argument from Lemma 2.2 from [17] we get ˜u ≡ u.
+Thus, we get u(x) = limT →∞ uT (x) = ˆu(x), x ∈ E, hence the map x �→ u(x) is
+continuous. Also, we get (A.2).
+Step 2. We show the martingale properties of z. First, we focus on the stopping
+time ˆτ. Let us define
+τT := inf{t ≥ 0 : uT −t(Xt) ≤ G(Xt)}.
+Using the argument from Proposition 11 in [16] we get that τT is an optimal stopping
+time for uT . Also, noting that the map T �→ uT (x), x ∈ E, is increasing, we get
+that T �→ τT is also increasing, thus we may define ˜τ := limT →∞ τT . We show that
+˜τ ≡ ˆτ.
+Let A := {˜τ < ∞}. First, we show that ˜τ ≡ ˆτ on A. On the event A, we get
+uT −τT (XτT ) = G(XτT ). Thus, letting T → ∞, we get u(X˜τ) = G(X˜τ), hence we
+get ˆτ ≤ ˜τ. Also, noting that uS(x) ≤ u(x), x ∈ E, S ≥ 0, on the set {ˆτ ≤ T } we
+get uT −ˆτ(Xˆτ) ≤ u(Xˆτ) ≤ G(Xˆτ), hence
+τT ≤ ˆτ.
+(A.6)
+Thus, recalling that ˆτ ≤ ˜τ < ∞ and letting T → ∞ in (A.6), we get ˜τ ≤ ˆτ, which
+shows ˜τ ≡ ˆτ on A.
+Now, we show that ˜τ ≡ ˆτ on Ac. Let ω ∈ Ac and suppose that ˆτ(ω) < ∞. Then,
+we may find T ≥ 0 such that ˆτ(ω) < T . Also, for any S ≥ T we get
+uS−ˆτ(ω)(Xˆτ(ω)(ω)) ≤ u(Xˆτ(ω)(ω)) ≤ G(Xˆτ(ω)(ω)).
+
+ASYMPTOTICS OF IMPULSE CONTROL PROBLEM WITH MULTIPLICATIVE REWARD 23
+Thus, we get τS(ω) ≤ ˆτ(ω) for any S ≥ T . Consequently, letting S → ∞ we get
+˜τ(ω) < ∞, which contradicts the choice of ω ∈ Ac. Consequently, on Ac we have
+˜τ = ∞ = ˆτ.
+Finally, we show the martingale properties. Let us define the processes
+zT (t) := e
+� t∧T
+0
+g(Xs)ds+uT −t∧T (Xt∧T ),
+T, t ≥ 0,
+z(t) := e
+� t
+0 g(Xs)ds+u(Xt),
+t ≥ 0.
+Using standard argument we get that for any T ≥ 0, the process zT (t), t ≥ 0, is a
+supermartingale and zT (t ∧ τT ), t ≥ 0, is a martingale; see e.g. [11, 12] for details.
+Also, recalling that from the first step we get uT (x) → u(x) as T → ∞ uniformly
+in x ∈ E, for any t ≥ 0, we get that zT (t) → z(t) and zT (t ∧ τT ) → z(t ∧ ˆτ) as
+T → ∞. Consequently, using Lebesgue’s dominated convergence theorem, we get
+that the process z(t) is a supermartingale and z(t∧ ˆτ), t ≥ 0, is a martingale, which
+concludes the proof.
+□
+Next, we consider an optimal stopping problem in a compact set and dyadic
+time-grid. More specifically, let δ > 0, let B ⊂ E be compact and assume that
+Px[τB < ∞] = 1, x ∈ B, where τB := δ inf{n ∈ N: Xnδ /∈ B}.
+Within this
+framework, we consider an optimal stopping problem of the form
+uB(x) := sup
+τ∈T δ ln Ex
+�
+exp
+�� τ∧τB
+0
+g(Xs)ds + G(Xτ∧τB)
+��
+,
+x ∈ E.
+(A.7)
+The properties of (A.7) are summarised in the following proposition.
+Proposition A.4. Let uB be given by (A.7). Then, we get
+uB(x) = sup
+τ∈T δ
+x,b
+ln Ex
+�
+exp
+�� τ∧τB
+0
+g(Xs)ds + G(Xτ∧τB)
+��
+,
+x ∈ E.
+(A.8)
+Also, the map x �→ uB(x) is continuous and bounded. Moreover, the process
+zδ(n) := e
+� nδ
+0
+g(Xs)+u(Xnδ),
+n ∈ N,
+is a supermartingale and the process z(n ∧ ˆτ/δ), n ∈ N, is a martingale, where
+ˆτ := δ inf{n ∈ N: uB(Xnδ) ≤ G(Xnδ)}.
+(A.9)
+Proof. To ease the notation, let us define
+ˆuB(x) := sup
+τ∈T δ
+x,b
+ln Ex
+�
+exp
+�� τ∧τB
+0
+g(Xs)ds + G(Xτ∧τB)
+��
+,
+x ∈ E,
+un
+B(x) := sup
+τ∈T δ
+τ≤nδ
+ln Ex
+�
+exp
+�� τ∧τB
+0
+g(Xs)ds + G(Xτ∧τB)
+��
+,
+n ∈ N, x ∈ E,
+and note that we get un
+B(x) ≤ ˆuB(x) ≤ uB(x),
+x ∈ E. Next, note that using
+the boundedness of G and Proposition A.2, by Lebesgue’s dominated convergence
+theorem, we obtain
+uB(x) = sup
+τ∈T
+lim
+n→∞ ln Ex
+�
+exp
+�� τ∧(nδ)∧τB
+0
+g(Xs)ds + G(Xτ∧(nδ)∧τB)
+��
+,
+x ∈ E.
+
+24 ASYMPTOTICS OF IMPULSE CONTROL PROBLEM WITH MULTIPLICATIVE REWARD
+Also, for any n ∈ N, x ∈ E, and τ ∈ T δ, we get
+un
+B(x) ≥ ln Ex
+�
+exp
+�� τ∧(nδ)∧τB
+0
+g(Xs)ds + G(Xτ∧(nδ)∧τB)
+��
+,
+x ∈ E.
+Thus, letting n → ∞ and taking the supremum with respect to τ ∈ T δ, we get
+limn→∞ un
+B(x) �� uB(x), x ∈ E. Consequently, we have
+lim
+n→∞ un
+B(x) = ˆuB(x) = uB(x),
+x ∈ E,
+which concludes the proof of (A.8).
+Let us now show the continuity of the map x �→ uB(x) and the martingale
+characterisation. To see this, note that using a standard argument one may show
+that, for any n ∈ N and x ∈ B, we get
+u0
+B(x) = G(x), x ∈ B,
+eun+1
+B
+(x) = max(eG(x), Ex
+�
+1{Xδ∈B}e
+� δ
+0 g(Xs)ds+un
+B(Xδ) + 1{Xδ /∈B}e
+� δ
+0 g(Xs)ds+G(Xδ)�
+,
+and, for any n ∈ N and x /∈ B, we get un
+B(x) = G(x); see e.g. Section 2.2 in [28]
+for details. Thus, letting n → ∞, for x ∈ B, we have
+euB(x) = max(eG(x), Ex
+�
+1{Xδ∈B}e
+� δ
+0 g(Xs)ds+uB(Xδ) + 1{Xδ /∈B}e
+� δ
+0 g(Xs)ds+G(Xδ)�
+,
+while for x /∈ B, we get uB(x) = G(x). Also, using Assumption (A2), we get that
+the process X is strong Feller. Thus, repeating the argument used in Lemma 4
+from Chapter II.5 in [13], we get that, for any bounded and measurable function
+h: E �→ R, the map
+E ∋ x �→ Ex
+�
+e
+� δ
+0 g(Xs)dsh(Xt)
+�
+is continuous and bounded. Applying this observation to h(x) := 1{x∈B}euB(x) and
+h(x) := 1{x/∈B}eG(x), x ∈ E, we get the continuity of x �→ uB(x). Also, using the
+argument from Proposition 3.2 we get that zδ(n), n ∈ N is a supermartingale and
+z(n ∧ ˆτ/δ), n ∈ N, is a martingale, which concludes the proof.
+□
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+Statements and Declarations
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+The authors have no relevant financial or non-financial interests to disclose.
+The authors contributed equally to this work.
+

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+arXiv:2301.02858v1  [eess.SP]  7 Jan 2023
+1
+Three Efficient Beamforming Methods for Hybrid
+IRS-aided AF Relay Wireless Networks
+Xuehui Wang, Feng Shu, Mengxing Huang, Fuhui Zhou, Riqing Chen, Cunhua Pan, Yongpeng Wu,
+and Jiangzhou Wang, Fellow, IEEE
+Abstract—Due to the “double fading” effect caused by con-
+ventional passive intelligent reflecting surface (IRS), the signal
+via the reflection link is weak. To enhance the received signal,
+active elements with the ability to amplify the reflected signal
+are introduced to the passive IRS forming hybrid IRS. In
+this paper, a hybrid IRS-aided amplify-and-forward (AF) relay
+wireless network is considered, where an optimization problem
+is formulated to maximize signal-to-noise ratio (SNR) by jointly
+optimizing the beamforming matrix at AF relay and the reflecting
+coefficient matrices at IRS subject to the constraints of transmit
+power budgets at the source/AF relay/hybrid IRS and that of
+unit-modulus for passive IRS phase shifts. To achieve high rate
+performance and extend the coverage range, a high-performance
+method based on semidefinite relaxation and fractional program-
+ming (HP-SDR-FP) algorithm is presented. Due to its extremely
+high complexity, a low-complexity method based on successive
+convex approximation and FP (LC-SCA-FP) algorithm is put
+forward. To further reduce the complexity, a lower-complexity
+method based on whitening filter, general power iterative and
+generalized Rayleigh-Ritz (WF-GPI-GRR) is proposed, where
+different from the above two methods, it is assumed that the
+amplifying coefficient of each active IRS element is equal, and
+the corresponding analytical solution of the amplifying coefficient
+can be obtained according to the transmit powers at AF relay
+and hybrid IRS. Simulation results show that the proposed three
+methods can greatly improve the rate performance compared to
+the existing networks, such as the passive IRS-aided AF relay and
+only AF relay network. In particular, a 50.0% rate gain over the
+existing networks is approximately achieved in the high power
+budget region of hybrid IRS. Moreover, it is verified that the
+proposed three efficient beamforming methods have an increasing
+This work was supported in part by the National Natural Science Foundation
+of China (Nos.U22A2002, 62071234 and 61972093), the Major Science
+and Technology plan of Hainan Province under Grant ZDKJ2021022, and
+the Scientific Research Fund Project of Hainan University under Grant
+KYQD(ZR)-21008.(Corresponding authors: Feng Shu).
+Xuehui Wang and Mengxing Huang are with the School of Information
+and Communication Engineering, Hainan University, Haikou, 570228, China.
+Feng Shu is with the School of Information and Communication Engi-
+neering, Hainan University, Haikou 570228, China, and also with the School
+of Electronic and Optical Engineering, Nanjing University of Science and
+Technology, Nanjing 210094, China (e-mail: shufeng0101@163.com).
+Fuhui Zhou is with the College of Electronic and Information Engineering,
+Nanjing University of Aeronautics and Astronautics, Nanjing 210000, China,
+also with the Key Laboratory of Dynamic Cognitive System of Electromag-
+netic Spectrum Space, Nanjing University of Aeronautics and Astronautics,
+Nanjing 210000, China, and also with the Ministry of Industry and Informa-
+tion Technology, Nanjing 211106, China (e-mail: zhoufuhui@ieee.org).
+Riqing Chen is with the Digital Fujian Institute of Big Data for Agriculture,
+Fujian Agriculture and Forestry University, Fuzhou 350002, China (e-mail:
+riqing.chen@fafu.edu.cn).
+Cunhua Pan is with National Mobile Communications Research Laboratory,
+Southeast University, Nanjing 211111, China (e-mail: cpan@seu.edu.cn).
+Yongpeng Wu is with the Shanghai Key Laboratory of Navigation and
+Location Based Services, Shanghai Jiao Tong University, Minhang 200240,
+China. (e-mail: yongpeng.wu2016@gmail.com).
+Jiangzhou Wang is with the School of Engineering, University of Kent,
+Canterbury CT2 7NT, U.K. Email: (e-mail: j.z.wang@kent.ac.uk).
+order in rate performance: WF-GPI-GRR, LC-SCA-FP and HP-
+SDR-FP.
+Index Terms—Double fading, intelligent reflecting surface,
+active elements, hybrid IRS, AF relay.
+I. INTRODUCTION
+With the rapid expansion of Internet-of-Things (IoT), the
+smart devices and data traffic explosively grows [1]–[3]. There
+are more stringent requirements for IoT in terms of massive
+connectivity, extended coverage, low-latency, low-power, and
+low-cost [4]–[6]. Because of high hardware cost and energy
+consumption, some existing technologies [7], such as mil-
+limeter wave (mmWave), massive multiple-input multiple-out
+(MIMO), coordinated multi-point, wireless network coding,
+are far away from meeting the demands, e.g., autonomous,
+ultra-large-scale, highly dynamic and fully intelligent services
+[8]. In the existing wireless networks, adding relay nodes can
+not only save the number of base stations, but also realize
+the cooperation of multiple communication nodes, so as to
+improve the throughput and reliability [9], [10]. However, the
+relay is an active device, which needs much energy to process
+signals. Therefore, it is imperative to develop a future wireless
+network, which is innovative, efficient and resource saving.
+Owing to the advantages of low circuit cost, low energy con-
+sumption, programmability and easy deployment, intelligent
+reflecting surface (IRS) is attractive, which has gained much
+research attention from both academia and industry [11]–[13].
+IRS is composed of a large number of passive electromagnetic
+units, which are dynamically controlled to reflect incident
+signal forming an intelligent wireless propagation environment
+in a software-defined manner [14]–[17]. From the perspective
+of electromagnetic theory, radiation pattern and physics nature
+of IRS unit, the free-space path loss models for IRS-assisted
+wireless communications were well introduced in [18]. A
+IRS-aided dual-hop visible light communication (VLC)/radio
+frequency (RF) system was proposed in [19] , where the
+performance analysis related to the outage probability and bit
+error rate (BER) were presented. Because of reconfigurability,
+IRS has been viewed as an enabling and potential technology
+to achieve performance enhancement, spectral and energy
+efficiency improvement. With more and more research on IRS,
+IRS has been widely applied to the following scenarios, physi-
+cal layer security [20]–[22], simultaneous wireless information
+and power transfer (SWIPT) [23], [24], multicell MIMO
+communications [25], [26], covert communications [27], [28],
+and wireless powered communication network (WPCN) [29]–
+[31]. To maximize secrecy rate for IRS-assisted multi-antenna
+
+2
+systems in [22], where an efficient alternating algorithm was
+developed to jointly optimize the transmit covariance of the
+source and the phase shift matrix of the IRS. For multicell
+communication systems [25], IRS was deployed at the cell
+boundary. While a method of jointly optimizing the active
+precoding matrices at the base stations (BSs) and the phase
+shifts at the IRS was proposed to maximize the weighted sum
+rate of all users. A IRS-aided secure MIMO WPCN is consid-
+ered in [31], by jointly optimizing the downlink (DL)/uplink
+(UL) time allocation, the energy transmit covariance matrix
+of hybrid access point (AP), the transmit beamforming matrix
+of users and the phase shifts of IRS, the maximum secrecy
+throughput of all users was achieved.
+Combining the advantages of IRS and relay is interesting,
+which can strike a good balance among cost, energy and per-
+formance. Recently, there were some related research works on
+the combination of IRS and relay appeared, which proved the
+combination could well serve for the wireless communication
+network in terms of coverage extension [32], [33], energy
+efficiency [34], spectral efficiency [35] and rate performance
+[36], [37]. The authors proposed an IRS-assisted dual-hop free
+space optical and radio frequency (FSO-RF) communication
+system with a decode-and-forward (DF) relaying protocol,
+and derived the exact closed-form expressions for the outage
+probability and bit error rate (BER) [32]. The simulation
+results verified that the combination can improve the coverage.
+An IRS-aided multi-antenna DF relay network was proposed
+in [36], where three methods, an alternately iterative structure,
+null-space projection plus maximum ratio combining (MRC)
+and IRS element selection plus MRC, were put forward to
+improve the rate performance. Obviously, the rate performance
+was improved by optimizing beamforming at relay and phase
+shifts at IRS. Moreover, compared with only IRS network, the
+hybrid network consisting of an IRS and a single-antenna DF
+relay can achieve the same rate performance with less IRS
+elements [37].
+However, the above existing research work focused on
+conventional passive IRS. Since the received signal via the
+reflecting channel link experiences large-scale fading twice
+(i.e., “double fading” effect), the received signal is weak in
+fact. Aiming at eliminating the “double fading” effect, the
+active IRS with extra power supply emerges, which can reflect
+and amplify the incident signals for obvious performance ad-
+vancement. In [38], the authors proposed the concept of active
+IRS and came up with a joint transmit and reflect precoding al-
+gorithm to solve the problem of capacity maximization, which
+existed in a signal model for active IRS. It was verified that the
+proposed active IRS could achieve a noticeable capacity gain
+compared to the existing passive IRS, which showed that the
+“double fading” effect could be broken by active IRS. With
+the same overall power budget, a fair performance comparison
+between active IRS and passive IRS was made theoretically
+in [39], where it proved that the active IRS surpassed passive
+IRS in the case of a small or medium number of IRS elements
+or sufficient power budget. Accordingly, a novel active IRS-
+assisted secure wireless transmission was proposed in [40],
+where the non-convex secrecy rate optimization problem was
+solved by jointly optimizing the beamformer at transmitter
+and reflecting coefficient matrix at IRS. It was demonstrated
+that with the aid of active IRS, a significantly higher secrecy
+performance gain could be obtained compared with existing
+solutions with passive IRS and without IRS design.
+Considering that active IRS has the ability to amplify signal,
+and in order to achieve higher rate performance or save more
+passive IRS elements of the combination network of IRS and
+relay, we propose that adding active IRS elements to passive
+IRS, thereby a combination network of hybrid IRS and relay is
+generated, which makes full use of the advantages of passive
+IRS, active IRS and relay to strike a good balance among
+circuit cost, energy efficiency and rate performance. To our
+best knowledge, it is lack of little research work on the hybrid
+IRS-aided amplify-and-forward (AF) relay network.
+In this case, using the criterion of Max SNR, three efficient
+beamforming methods are proposed to improve the rate perfor-
+mance of the proposed hybrid IRS-aided AF relay network or
+dramatically extend its coverage range. The main contributions
+of the paper are summarized as follows:
+1) To achieve a high rate, a high-performance method based
+on semidefinite relaxation and fractional programming
+(HP-SDR-FP) algorithm is presented to jointly optimize
+the beamforming matrix at AF relay and the reflecting
+coefficient matrices at IRS by optimizing one and fixing
+the other two. However, it is difficult to directly solve
+the non-convex optimization problem with fractional and
+non-concave objective function and non-convex con-
+straints. To address this issue, some operations such as
+vectorization, Kronecker product and Hadamard product
+are applied to simplify the non-convex optimization
+problem, then SDR algorithm, Charnes-Cooper trans-
+formation of FP algorithm and Gaussian randomization
+method are adopted to obtain the optimization variable.
+The proposed HP-SDR-FP method can harvest up to
+80% rate gain over the passive IRS-aided AF relay
+network as the number of active IRS elements tends to
+large. Additionally, its convergence rate is fast, and its
+highest order of computational complexity is M 13 and
+N 6.5 FLOPs.
+2) To reduce the extremely high computational complexity
+of the proposed HP-SDR-FP method, a low-complexity
+method based on successive convex approximation and
+FP (LC-SCA-FP) algorithm is presented. For the non-
+convex optimization problem, Dinkelbach’s transforma-
+tion of FP algorithm is firstly performed to simplify the
+objective function. Then by utilizing the first-order Tay-
+lor approximation of the simplified objective function
+and relaxing the unit-modulus constraint for passive IRS
+phase shifts, the non-convex optimization problem is
+transformed to convex, and can be solved. The proposed
+LC-SCA-FP method performs much better than passive
+IRS-aided AF relay network, passive IRS-aided AF
+relay network with random phase and only AF relay
+network in terms of rate. Its rate is 60% higher than
+that of passive IRS-aided AF relay system. Furthermore,
+it is convergent, and its highest order of computational
+complexity is M 6 and N 3 FLOPs, which is much lower
+than that of HP-SDR-FP method.
+
+3
+3) To further reduce the computational complexity of the
+above two methods, a lower-complexity method based
+on whitening filter, general power iterative algorithm
+and generalized Rayleigh-Ritz theorem (WF-GPI-GRR)
+is put forward, where it is assumed that the amplifying
+coefficient of each active IRS element is equal in the first
+time slot or the second time slot. To exploit the colored
+property of noise, whitening filter operation is performed
+to the received signal. In line with the transmit power at
+AF relay and hybrid IRS, the analytical solution of the
+amplifying coefficient can be obtained. Moreover, the
+closed-form expression of beamforming matrix at AF
+relay is derived by utilizing maximum-ratio combining
+and maximum-ratio transmission (MRC-MRT) scheme,
+GPI and GRR are respectively applied to obtain the
+phase shift matrices at IRS for the first time slot and
+the second time slot. Compared with passive IRS-aided
+AF relay network, its rate can be improved by 49%. Its
+highest order of computational complexity is M 3 and
+N 3 FLOPs, which is lower than the above two methods.
+The remainder of this paper is organized as follows. In
+Section II, a hybrid IRS-aided AF relay network is described.
+In Section III, we propose a high-performance method. Section
+IV describes a a low-complexity method. A lower-complexity
+method is presented in Section V. We present our simulation
+results in Section VI, and draw conclusions in Section VII.
+Notation: Scalars, vectors and matrices are respectively
+represented by letters of lower case, bold lower case, and bold
+upper case. (·)∗, (·)T , (·)H, and (·)−1 stand for matrix conju-
+gate, transpose, conjugate transpose, and inverse, respectively.
+E{·}, | · |, ∥ · ∥, tr(·), and arg(·) denote expectation operation,
+the modulus of a scalar, 2-norm, the trace of a matrix, and the
+phase of a complex number, respectively. ⊗ and ⊙ respectively
+denote Kronecker product and Hadamard product. The sign IN
+is the N × N identity matrix.
+II. SYSTEM MODEL
+A. Signal Model
+Fig. 1. System model for a hybrid IRS-aided AF relay wireless network.
+Fig. 1 sketches a hybrid IRS-aided AF relay network
+operated in a time division half-duplex scenario, where source
+(S) and destination (D) are respectively equipped with a single
+antenna, a AF relay is with M antennas, and an IRS includes
+N elements consisting of K active elements and L passive
+elements, i.e., N = K + L. The active elements reflect the
+incident signal by adjusting the amplitude and phase, while the
+passive elements reflect the incident signal only by shifting the
+phase. Let us define EN , EK and EL as the sets of N elements,
+K active elements and L passive elements, respectively. Fur-
+thermore, EN = EK ∪ EL and EK ∩ EL = ∅. The reflecting
+coefficient matrices of EN , EK and EL are respectively denoted
+by Θ, Φ and Ψ, where Θ = diag(α1, · · · , αN), and the
+reflecting coefficients of ith element in Θ is expressed by
+αi =
+� |βi|ejθi
+i ∈ EK,
+(1a)
+ejθi
+i ∈ EL,
+(1b)
+where |βi| and θi ∈ (0, 2π] are amplifying coefficient and
+phase shift of the ith element. For convenience of derivation
+below, we have the following definitions
+Θ = Φ + Ψ,
+Φ = EKΘ,
+Ψ = EKΘ,
+(2)
+where EK + EK = IN and EKEK = 0N. Φ, Ψ, EK and EK
+are sparse diagonal matrices. EK ∈ RN×N and EK ∈ RN×N
+are respectively depended on the location distribution of K
+active and L passive elements in the IRS. In other words, the
+kth non-zero value of the diagonal corresponding to the kth
+active element is 1, thus there are K values being 1 and the
+rest L values being 0 on the diagonal of EK. Additionally, EK
+is similar to EK. It is assumed that the direct channel between
+S and D is blocked, and the power of signals reflected by the
+IRS twice or more are such weak that they can be ignored. In
+the first time slot, the received signal at IRS can be denoted
+as
+yr
+1i =
+�
+Pshsix + n1i,
+(3)
+where x and Ps are the transmit signal and power from
+S, E{xHx} = 1. We assume all channels follow Rayleigh
+fading, hsi ∈ CN×1 is the channel from S to IRS. n1i
+represents the additive white Gaussian noise (AWGN) at IRS
+with distribution n1i ∼ CN(0, σ2
+1iEKIN), which is caused by
+K active elements. The received signal at AF relay is given
+by
+yr =
+�
+Ps(hsr + HirΘ1hsi)x + HirEKΘ1n1i + nr,
+(4)
+where hsr ∈ CM×1 and Hir ∈ CM×N are the channels from
+S to AF relay and IRS to AF relay. Θ1 = diag(α11, · · · , α1N)
+and Φ1
+=
+diag(φ11, · · · , φ1N) are the reflecting coeffi-
+cient matrices of EN and EK in the first time slot. nr ∼
+CN(0, σ2
+rIM) is the AWGN at AF relay. After performing
+receive and transmit beamforming, the transmit signal at AF
+relay can be expressed as
+yt = Ayr,
+(5)
+where A ∈ CM×M is the beamforming matrix. In the second
+time slot, the received signal at IRS is written by
+yr
+2i = HH
+iryt + n2i,
+(6)
+
+Hybrid IRS
+First time slot
+Active element
+-> Second time slot
+Passive element
+H
+nid
+H
+:Hir
+Hir
+Destination
+(S)
+(D)
+H
+nrd
+AF relay4
+SNR =
+γs|(hH
+rd + hH
+idΘ2HH
+ir)A(hsr + HirΘ1hsi)|2
+∥(hH
+rd + hH
+idΘ2HH
+ir)AHirEKΘ1∥2 + ∥(hH
+rd + hH
+idΘ2HH
+ir)A∥2 + ∥hH
+idEKΘ2∥2 + 1.
+(10)
+where HH
+ir ∈ CN×M is the channel from AF relay to IRS.
+n2i ∼ CN(0, σ2
+2iEKIN) is the noise. The received signal at
+D is as follows
+yd = (hH
+rd + hH
+idΘ2HH
+ir)yt + hH
+idEKΘ2n2i + nd,
+(7)
+where hH
+rd ∈ C1×M and hH
+id ∈ C1×N are the channels from
+AF relay to D and IRS to D. Θ2 = diag(α21, · · · , α2N) and
+Φ2 = diag(φ21, · · · , φ2N) are the reflecting coefficient matri-
+ces of EN and EK in the second time slot. nd ∼ CN(0, σ2
+d) is
+the AWGN at D. Substituting (4) and (5) into (7) yields
+yd =
+�
+Ps(hH
+rd + hH
+idΘ2HH
+ir)A(hsr + HirΘ1hsi)x
++ (hH
+rd + hH
+idΘ2HH
+ir)A(HirEKΘ1n1i + nr)
++ hH
+idEKΘ2n2i + nd.
+(8)
+It is assumed that σ2
+1i = σ2
+2i = σ2
+r = σ2
+d = σ2 and γs = Ps
+σ2 ,
+the achievable system rate can be defined as
+R = 1
+2 log2(1 + SNR),
+(9)
+where SNR can be formulated as (10), as shown at the top of
+next page.
+B. Problem Formulation
+To enhance the system rate performance, it is necessary
+to maximize system rate. Maximizing rate is equivalent to
+maximize SNR due to the fact that the log function is
+a monotone increasing function of SNR. The optimization
+problem is casted as
+max
+Θ1,Θ2,A SNR
+(11a)
+s.t. |Θ1(i, i)| = 1, |Θ2(i, i)| = 1, for i ∈ EL,
+(11b)
+γs∥EKΘ1hsi∥2 + ∥EKΘ1∥2
+F ≤ γi,
+(11c)
+γs∥A(hsr + HirΘ1hsi)∥2
++ ∥AHirEKΘ1∥2
+F + ∥A∥2
+F ≤ γr,
+(11d)
+γs∥EKΘ2HH
+irA(hsr + HirΘ1hsi)∥2
++ ∥EKΘ2HH
+irAHirEKΘ1∥2
+F
++ ∥EKΘ2HH
+irA∥2
+F + ∥EKΘ2∥2
+F ≤ γi, (11e)
+where γi = Pi
+σ2 and γr = Pr
+σ2 , Pi and Pr respectively denote
+the transmit power budgets at IRS and AF relay. Since the
+IRS is hybrid consisting of active and passive elements, it
+is difficult to solve the optimization problem. To enhance the
+rate performance, three efficient beamforming methods: 1) HP-
+SDR-FP; 2) LC-SCA-FP; and 3) WF-GPI-GRR, are proposed
+to optimize AF relay beamforming matrix A, IRS reflecting
+coefficient matrices Θ1 and Θ2.
+III. PROPOSED A HIGH-PERFORMANCE SDR-FP-BASED
+MAX-SNR METHOD
+In this section, a HP-SDR-FP method is proposed to solve
+problem (11). To facilitate processing, problem (11) is de-
+coupled into three subproblems by optimizing one and fixing
+the other two. For each subproblem, we relax it as an SDR
+problem, and combine Charnes-Cooper transformation of FP
+algorithm to transform the SDR problem into an semidefinite
+programming (SDP) problem. Furthermore, Gaussian random-
+ization method is applied to recover the rank-1 solution.
+A. Optimization of A Given Θ1 and Θ2
+Given Θ1 and Θ2, the optimization problem is reduced to
+max
+A
+SNR
+s.t.
+(11d),
+(11e).
+(12)
+Let us define a = vec(A) ∈ CM2×1, SNR can be translated
+to
+SNR =
+γsaHB1a
+aH(B2 + B3)a + ∥hH
+idEKΘ2∥2 + 1,
+(13)
+where
+B1 = [(hsr + HirΘ1hsi)∗(hsr + HirΘ1hsi)T ]
+⊗ [(hH
+rd + hH
+idΘ2HH
+ir)H(hH
+rd + hH
+idΘ2HH
+ir)], (14a)
+B2 = [(HirEKΘ1)∗(HirEKΘ1)T ]
+⊗ [(hH
+rd + hH
+idΘ2HH
+ir)H(hH
+rd + hH
+idΘ2HH
+ir)], (14b)
+B3 = IM ⊗ [(hH
+rd + hH
+idΘ2HH
+ir)H(hH
+rd + hH
+idΘ2HH
+ir)]. (14c)
+In the same manner, the constraints (11d) and (11e) can be
+respectively converted to
+aH(γsC1 + C2 + IM2)a ≤ γr,
+(15a)
+aH(γsD1 + D2 + D3)a + ∥EKΘ2∥2
+F ≤ γi,
+(15b)
+where
+C1 = [(hsr + HirΘ1hsi)∗(hsr + HirΘ1hsi)T ] ⊗ IM, (16a)
+C2 = [(HirEKΘ1)∗(HirEKΘ1)T ] ⊗ IM,
+(16b)
+D1 = [(hsr + HirΘ1hsi)∗(hsr + HirΘ1hsi)T ]
+⊗ [(EKΘ2HH
+ir)H(EKΘ2HH
+ir)],
+(16c)
+D2 = [(HirEKΘ1)∗(HirEKΘ1)T ]
+⊗ [(EKΘ2HH
+ir)H(EKΘ2HH
+ir)],
+(16d)
+D3 = IM ⊗ [(EKΘ2HH
+ir)H(EKΘ2HH
+ir)].
+(16e)
+Let us define �A = aaH ∈ CM2×M2, in accordance with the
+rank inequality: rank P ≤ min{m, n}, where P ∈ Cm×n, we
+
+5
+can get rank(�A) ≤ rank(a) = 1. The optimization problem
+can be recast as
+max
+�A
+γstr(B1�A)
+tr{(B2 + B3)�A} + ∥hH
+idEKΘ2∥2 + 1
+(17a)
+s.t.
+tr{(γsC1 + C2 + IM2)�A} ≤ γr,
+(17b)
+tr{(γsD1 + D2 + D3)�A} + ∥EKΘ2∥2
+F ≤ γi,
+(17c)
+�A ⪰ 0,
+rank(�A) = 1,
+(17d)
+which is a non-convex problem because of rank-one constraint.
+After removing rank(�A) = 1 constraint, we have the SDR
+problem of (17) as follows
+max
+�A
+γstr(B1�A)
+tr{(B2 + B3)�A} + ∥hH
+idEKΘ2∥2 + 1
+(18a)
+s.t.
+(17b),
+(17c),
+�A ⪰ 0.
+(18b)
+The objective function (18a) is a linear fractional function with
+respect to �A, which is a quasi-convex function with the denom-
+inator > 0, so problem (18) is a quasi-convex problem with
+convex constraints. It is necessary to apply Charnes-Cooper
+transformation, which helps convert the optimization problem
+from quasi-convex to convex. Introducing a slack variable m
+and defining m = (tr{(B2 + B3)�A} + ∥hH
+idEKΘ2∥2 + 1)−1,
+the above problem (18) is further rewritten as follows
+max
+�A,m
+γstr{B1�A}
+(19a)
+s.t. tr{(γsC1 + C2 + IM2)�A} ≤ mγr,
+(19b)
+tr{(γsD1 + D2 + D3)�A} + m∥EKΘ2∥2
+F ≤ mγi, (19c)
+tr{(B2 + B3)�A} + m∥hH
+idEKΘ2∥2 + m = 1,
+(19d)
+�A ⪰ 0, m > 0,
+(19e)
+where �A = m�A. Clearly, the above optimization problem has
+become a SDP problem, which is directly solved by CVX.
+The solution to problem (18) is �A = �A/m. However, the
+rank-one constraint rank(�A) = 1 is not considered in the SDR
+problem. Since the obtained solution �A is not generally rank-
+one matrix, the Gaussian randomization method is applied to
+achieve a rank-one solution �A, thereby, AF relay beamforming
+matrix A is achieved.
+B. Optimization of Θ1 Given A and Θ2
+Given that A and Θ2 are fixed, the optimization problem
+can be represented by as follows
+max
+Θ1
+γs|hH
+rid(hsr + HirΘ1hsi)|2
+∥hH
+ridHirEKΘ1∥2 + ∥hH
+rid∥2 + ∥hH
+idEKΘ2∥2 + 1
+(20a)
+s.t.
+|Θ1(i, i)| = 1,
+for i ∈ EL,
+(20b)
+(11c),
+(11d),
+(11e),
+(20c)
+where
+hrid
+=
+[(hH
+rd + hH
+idΘ2HH
+ir)A]H.
+In
+order
+to
+further
+simplify
+the
+objective
+function
+and
+constraints
+of
+the
+optimization
+problem,
+let
+us
+define
+u1
+=
+[α11, · · · , α1N]T , we have hsr + HirΘ1hsi = Hsirv1 and
+hH
+ridHirEKΘ1 = uT
+1 diag{hH
+ridHirEK}, where v1 = [u1; 1]
+and Hsir
+=
+[Hirdiag{hsi}, hsr]. Substituting these for-
+mulas into (20a), and due to ∥uT
+1 diag{hH
+ridHirEK}∥2
+=
+∥diag{hH
+ridHirEK}u1∥2, the object function can be further
+rewritten as
+vH
+1 F1v1
+vH
+1 F2v1
+,
+(21)
+where F1 = γsHH
+sirhridhH
+ridHsir and F2 is written as (22) at
+the top of next page. The constraint (20b) for passive elements
+EL can be rewritten as
+|v1(i)|2 = 1,
+for i ∈ EL.
+(23)
+Obviously, ∥EKΘ1∥2
+F = ∥EKu1∥2, and the constraint (11c)
+can be translated to
+vH
+1 G1v1 ≤ γi,
+(24)
+where
+G1 =
+�
+γsdiag{hH
+si}EKdiag{hsi} + EK
+0N×1
+01×N
+0
+�
+.
+(25)
+Then for constraint (11d), according to the property of
+Hadamard product: tr(X(Y ⊙ Z)) = tr((X ⊙ YT )Z), where
+X ∈ Cm×n, Y ∈ Cn×m and Z ∈ Cn×m, we have
+∥AHirEKΘ1∥2
+F = ∥AHirEKdiag{u1}∥2
+F
+= tr{EKHH
+irAHAHirEKdiag{u1}diag{uH
+1 }}
+= tr{EKHH
+irAHAHir[EK ⊙ (u1uH
+1 )]}
+= uH
+1 [(EKHH
+irAHAHir) ⊙ EK]u1
+= uH
+1 [(HH
+irAHAHir) ⊙ EK]u1
+(26)
+Inserting hsr + HirΘ1hsi = Hsirv1 and (26) back into the
+constraint (11d), which can be rewritten as
+vH
+1 G2v1 ≤ γr,
+(27)
+where
+G2 =γsHH
+sirAHAHsir +
+�
+(HH
+irAHAHir) ⊙ EK
+0N×1
+01×N
+∥A∥2
+F
+�
+.
+(28)
+Similarly, (11e) can be written in the following form
+vH
+1 G3v1 ≤ γi,
+(29)
+where G3 is written as (30), as shown at the top of the page.
+Substituting the simplified objective function and constraints
+into (20), the optimization problem can be equivalently trans-
+formed into
+max
+v1
+(21)
+(31a)
+s.t.
+(23), (24), (27), (29), v1(N + 1) = 1.
+(31b)
+Aiming at further transforming the optimization problem, and
+defining V1 = v1vH
+1 , problem (31) can be equivalently given
+by
+max
+V1
+tr(F1V1)
+tr(F2V1)
+(32a)
+s.t.
+V1(i, i) = 1,
+for i ∈ EL,
+(32b)
+V1(N + 1, N + 1) = 1,
+(32c)
+tr(G1V1) ≤ γi, tr(G2V1) ≤ γr,
+(32d)
+tr(G3V1) ≤ γi, rank(V1) = 1, V1 ⪰ 0.
+(32e)
+
+6
+F2 =
+� diag{EKHH
+irhrid}diag{hH
+ridHirEK}
+0N×1
+01×N
+∥hH
+rid∥2 + ∥hH
+idEKΘ2∥2 + 1
+�
+,
+(22)
+G3 =γsHH
+sirAHHirΘH
+2 EKΘ2HH
+irAHsir
++
+�
+(HH
+irAHHirΘH
+2 EKΘ2HH
+irAHir) ⊙ EK
+0N×1
+01×N
+∥EKΘ2HH
+irA∥2
+F + ∥EKΘ2∥2
+F
+�
+.
+(30)
+Due to the fact that the object function is quasi-convex
+and constraint rank(V1) = 1 is non-convex, (32) is still
+a non-convex problem. Relaxing the rank-1 constraint, the
+problem is transformed into a SDR problem, which can also be
+solved by applying Charnes-Cooper transformation. Moreover,
+introducing a slack variable τ, then defining τ = tr(F2V1)−1
+and �V1 = τV1, the SDR problem of (32) can be translated to
+a SDP problem, i.e.,
+max
+�V1,τ
+tr(F1�V1)
+(33a)
+s.t.
+�V1(i, i) = τ,
+for i ∈ EL,
+(33b)
+�V1(N + 1, N + 1) = τ, τ > 0,
+(33c)
+tr(G1�V1) ≤ τγi, tr(G2�V1) ≤ τγr,
+(33d)
+tr(G3�V1) ≤ τγi, tr(F2�V1) = 1, �V1 ⪰ 0,
+(33e)
+which can be directly solved by CVX, thereby the solution V1
+of SDR problem of (32) is achieved, and Gaussian randomiza-
+tion method is used to recover a rank-one solution V1. Then
+the solution v1 is extracted from the eigenvalue decomposition
+of V1, subsequently, IRS reflecting coefficient matrix Θ1 can
+be obtained.
+C. Optimization of Θ2 Given A and Θ1
+In the subsection, defining u2 = [α21, · · · , α2N]H, we have
+hH
+rd +hH
+idΘ2HH
+ir = vH
+2 Hrid and Θ2HH
+irA(hsr +HirΘ1hsi) =
+diag{HH
+irA(hsr + HirΘ1hsi)}u∗
+2, where v2 = [u2; 1] and
+Hrid = [diag{hH
+id}HH
+ir; hH
+rd]. Similarly, when A and Θ1 are
+fixed, the SDP problem to optimize �V2 can be expressed as
+max
+�V2,ρ
+tr(H1�V2)
+(34a)
+s.t.
+�V2(i, i) = ρ,
+for i ∈ EL,
+(34b)
+�V2(N + 1, N + 1) = ρ, ρ > 0,
+(34c)
+tr(J�V2) ≤ ργi, tr(H2�V2) = 1, �V2 ⪰ 0,
+(34d)
+where ρ = tr(H2v2vH
+2 )−1 is a slack variable, �V2 = ρv2vH
+2 ,
+H1 = γsHridA(hsr + HirΘ1hsi)[HridA(hsr + HirΘ1hsi)]H,
+H2 =HridA(HirEKΘ1ΘH
+1 EKHH
+ir + IM)AHHH
+rid
++
+�
+diag{hH
+idEK}diag{EKhid}
+0N×1
+01×N
+1
+�
+,
+(35)
+and J is written as (36) at the top of next page, where
+H3
+=
+EKdiag{HT
+irA∗(hsr + HirΘ1hsi)∗} and H4
+=
+HH
+irAHirEKΘ1. It is observed that (34) is similar to (33),
+thus (34) can be solved in the same way as (33). Finally the
+solutions v2 and Θ2 are obtained, and the details are omitted
+here for brevity.
+D. Overall Algorithm and Complexity Analysis
+Since the objective function of problem (11) is non-
+decreasing and the transmit powers of S, AF relay and IRS
+active elements are limited, the objective function has an
+upper bound. Therefore, the convergence of the proposed HP-
+SDR-FP algorithm can be guaranteed. Our idea is alternative
+iteration, that is, the alternative iteration process are performed
+among A, Θ1 and Θ2 until the convergence criterion is
+satisfied, while the system rate is maximum. The proposed
+HP-SDR-FP method is summarized in Algorithm 1.
+Algorithm 1 Proposed HP-SDR-FP Method
+1. Initialize A0, Θ0
+1 and Θ0
+2. According to (9), R0 can be
+obtained.
+2.
+set the convergence error δ and the iteration number
+t = 0.
+3. repeat
+4. Given Θt
+1 and Θt
+2, solve problem (19) for �A
+t+1, recover
+rank-1 solution �A
+t+1 via Gaussian randomization, obtain
+At+1.
+5. Given At+1 and Θt
+2, solve problem (33) for �V
+t+1
+1
+,
+recover rank-1 solution Vt+1
+1
+via Gaussian randomization,
+obtain Θt+1
+1
+.
+6. Given At+1 and Θt+1
+1
+, solve problem (34) for �V
+t+1
+2
+,
+recover rank-1 solution Vt+1
+2
+via Gaussian randomization,
+obtain Θt+1
+2
+.
+7. Calculate Rt+1 by using At+1, Θt+1
+1
+and Θt+1
+2
+.
+8. Update t = t + 1.
+9. until
+��Rt+1 − Rt�� ≤ δ.
+After that, the complexity of Algorithm 1 is calculated
+and analyzed according to problems (19), (33) and (34).
+Problem (19) has 4 linear constraints with dimension 1, one
+linear matrix inequality (LMI) constraint of size M 2 and
+M 4+1 decision variables. Hence, the computational complex-
+ity of problem (19) is denoted as O{nA
+√
+M 2 + 4(M 6 + 4 +
+nA(M 4 + 4) + n2
+A)ln(1/ε)} float-point operations (FLOPs),
+where nA = M 4 + 1 and ε represents the computation
+accuracy. For problem (33), there exit L + 6 linear constraints
+with dimension 1, one LMI constraint of size N + 1 and
+(N +1)2+1 decision variables, so the computational complex-
+ity of problem (33) is written as O{nV1
+√
+N + L + 7((N +
+
+7
+J =
+�
+γsHH
+3 H3 + (H4HH
+4 + HH
+irAAHHir + IN) ⊙ EK
+0N×1
+01×N
+0
+�
+.
+(36)
+1)3 + L + 6 + nV1((N + 1)2 + L + 6) + n2
+V1)ln(1/ε)} FLOPs,
+where nV1 = (N + 1)2 + 1. For problem (34), there are L + 4
+linear constraints with dimension 1, one LMI constraint of
+size N + 1 and (N + 1)2 + 1 decision variables. Thus, the
+computational complexity of problem (34) is expressed as
+O{nV2
+√
+N + L + 5((N + 1)3 + L + 4 + nV2((N + 1)2 +
+L + 4) + n2
+V2)ln(1/ε)} FLOPs, where nV2 = (N + 1)2 + 1.
+Therefore, the total computational complexity of Algorithm 1
+is written by
+O{D1[nA
+�
+M 2 + 4(M 6 + 4 + nA(M 4 + 4) + n2
+A)
++ nV1
+√
+N + L + 7((N + 1)3 + L + 6 + nV1((N + 1)2
++ L + 6) + n2
+V1) + nV2
+√
+N + L + 5((N + 1)3 + L + 4
++ nV2((N + 1)2 + L + 4) + n2
+V2)]ln(1/ε)}
+(37)
+FLOPs, where D1 is the maximum number of alternating
+iterations needed for convergence in Algorithm 1. It is obvious
+that the highest order of computational complexity is M 13 and
+N 6.5 FLOPs.
+IV. PROPOSED A LOW-COMPLEXITY SCA-FP-BASED
+MAX-SNR METHOD
+In the previous section, HP-SDR-FP method is proposed
+to obtain AF relay beamforming matrix A, IRS reflecting
+coefficient matrices Θ1 and Θ2. However, its computational
+complexity is very high because of SDR algorithm with lots of
+FLOPs. To reduce the high computational complexity of HP-
+SDR-FP method, a low-complexity SCA-FP-based Max-SNR
+method is proposed in this section.
+A. Optimize A With Fixed Θ1 and Θ2
+For given Θ1 and Θ2, the optimization problem based on
+(13) is given by
+max
+a
+γsaHB1a
+aH(B2 + B3)a + ∥hH
+idEKΘ2∥2 + 1
+(38a)
+s.t.
+(15a), (15b).
+(38b)
+Observing the above objective function, it is seen that (38)
+is also a fractional optimization problem. In the subsection,
+Dinkelbachs transformation is introduced to solve problem
+(38). Here, introducing a slack variables µ, the problem (38)
+is reformulated as
+max
+a,µ
+γsaHB1a − µ[aH(B2 + B3)a + ∥hH
+idEKΘ2∥2 + 1]
+(39a)
+s.t.
+(15a), (15b),
+(39b)
+where µ is iteratively updated by
+µ(t + 1) =
+γsaH(t)B1a(t)
+aH(t)(B2 + B3)a(t) + ∥hH
+idEKΘ2∥2 + 1, (40)
+where t is the iteration number. µ is nondecreasing after each
+iteration, which guarantees the convergence of the objective
+function (39a). Note that the above problem is still non-
+convex due to the first term of the objective function is non-
+concave, which can be solved by using SCA method. We
+approximate the first term by using a linear function, i.e.,
+its first-order Taylor expansion at feasible vector �a, which
+is γsaHB1a ≥ 2γsℜ{aHB1�a} − γs�aHB1�a, where �a is the
+solution of previous iteration. Inserting the low bound of
+γsaHB1a back into problem (39) yields
+max
+a
+2γsℜ{aHB1�a} − γs�aHB1�a
+− µ[aH(B2 + B3)a + ∥hH
+idEKΘ2∥2 + 1]
+(41a)
+s.t.
+(15a), (15b).
+(41b)
+It is known that the above optimization problem consists of
+a concave objective function and several convex constraints.
+Therefore, problem (41) is a convex optimization problem.
+When �a and µ are fixed, a can be directly achieved by CVX.
+Correspondingly, A can be obtained.
+B. Optimize Θ1 With Fixed A and Θ2
+It is assumed that AF relay beamforming matrix A and Θ2
+are given. The constraint (11b) can be re-expressed as
+|u1(i)|2 = 1,
+for i ∈ EL,
+(42)
+which can be relaxed as
+uH
+1 (i)u1(i) ≤ 1,
+for i ∈ EL.
+(43)
+Problem (11) with respect to u1 can be rearranged as
+max
+u1
+γs|hH
+1 u1 + a|2
+∥diag{hH
+ridHirEK}u1∥2 + b
+(44a)
+s.t.
+uH
+1 (i)u1(i) ≤ 1,
+for i ∈ EL,
+(44b)
+γs∥EKdiag{hsi}u1∥2 + ∥EKu1∥2 ≤ γi,
+(44c)
+γs∥Ahsr + P1u1∥2 + ∥AHirEKdiag{u1}∥2
+F
++ ∥A∥2
+F ≤ γr,
+(44d)
+γs∥h2 + P2u1∥2 + ∥P3EKdiag{u1}∥2
+F ≤ �γi,
+(44e)
+where
+h1 = [(hH
+rd + hH
+idΘ2HH
+ir)AHirdiag{hsi}]H,
+(45a)
+h2 = EKΘ2HH
+irAhsr, P1 = AHirdiag{hsi},
+(45b)
+P2 = EKΘ2HH
+irAHirdiag{hsi},
+(45c)
+P3 = EKΘ2HH
+irAHir, a = (hH
+rd + hH
+idΘ2HH
+ir)Ahsr, (45d)
+b = ∥(hH
+rd + hH
+idΘ2HH
+ir)A∥2 + ∥hH
+idEKΘ2∥2 + 1,
+(45e)
+�γi = γi − ∥EKΘ2HH
+irA∥2
+F − ∥EKΘ2∥2
+F .
+(45f)
+
+8
+Problem (44) can be further converted to
+max
+u1
+γs|hH
+1 u1 + a|2
+∥diag{hH
+ridHirEK}u1∥2 + b
+(46a)
+s.t.
+uH
+1 (i)u1(i) ≤ 1,
+for i ∈ EL,
+(46b)
+uH
+1 (γsdiag{hH
+si}EKdiag{hsi} + EK)u1 ≤ γi,
+(46c)
+uH
+1 [γsPH
+1 P1 + (HH
+irAHAHir) ⊙ EK]u1 + ∥A∥2
+F
++ 2γsℜ{uH
+1 PH
+1 Ahsr} + γshH
+srAHAhsr ≤ γr,
+(46d)
+uH
+1 [γsPH
+2 P2 + (PH
+3 P3) ⊙ EK]u1
++ 2γsℜ{uH
+1 PH
+2 h2} + γshH
+2 h2 ≤ �γi.
+(46e)
+In the same manner, Dinkelbach’s transformation is also
+introduced to problem (46). Problem (46) can be rewritten
+as
+max
+u1
+γs|hH
+1 u1 + a|2 − ωb
+− ωuH
+1 diag{EKHH
+irhrid}diag{hH
+ridHirEK}u1 (47a)
+s.t.
+(46b), (46c), (46d), (46e),
+(47b)
+where ω is a variable scalar, which is defined as
+ω(t + 1) =
+γs|hH
+1 u1(t) + a|2
+∥diag{hH
+ridHirEK}u1(t)∥2 + b.
+(48)
+Similarly, aiming at converting the objective function (47a)
+to convex, the first-order Taylor expansion at the point �u1 is
+employed to γs|hH
+1 u1 + a|2 and transform it into the linear
+function, i.e., |hH
+1 u1 + a|2 ≥ 2ℜ{uH
+1 h1(hH
+1 �u1 + a)} + a∗a −
+�uH
+1 h1hH
+1 �u1. Inserting the low bound of
+��hH
+1 u1 + a
+��2 back into
+problem (47) yields the following optimization problem
+max
+u1
+2γsℜ{uH
+1 h1(hH
+1 �u1 + a)} + γsa∗a − γs�uH
+1 h1hH
+1 �u1−
+ω(b + uH
+1 diag{EKHH
+irhrid}diag{hH
+ridHirEK}u1) (49a)
+s.t.
+(46b), (46c), (46d), (46e),
+(49b)
+where the object function is concave and the constraints are
+convex, thus problem (49) is convex. For a given feasible
+vector �u1 and ω, problem (49) can be solved by CVX directly,
+thereby u1 is achieved.
+C. Optimize Θ2 With Fixed A and Θ1
+Similarly, given AF relay beamforming matrix A and Θ1,
+the optimization problem with respect to u2 is modeled as
+max
+u2
+2γsℜ{uH
+2 h3(hH
+3 �u2 + c∗)} + γscc∗ − γs�uH
+2 h3hH
+3 �u2
+− λ(uH
+2 Q1QH
+1 u2 + 2ℜ{uH
+2 Q1h4} + hH
+4 h4)
+− λ(uH
+2 Q2QH
+2 u2 + 2ℜ{uH
+2 Q2AHhrd})
+− λhH
+rdAAHhrd − λuH
+2 Q3QH
+3 u2 − λ
+(50a)
+s.t.
+uH
+2 (i)u2(i) ≤ 1, for i ∈ EL,
+(50b)
+uH
+2 [γsHH
+3 H3 + (H4HH
+4 + HH
+irAAHHir + IN) ⊙ EK]u2
+≤ γi,
+(50c)
+where λ is a variable, �u2 is a feasible vector, and
+h3 = diag{hH
+id}HH
+irA(hsr + HirΘ1hsi),
+(51a)
+h4 = (hH
+rdAHirEKΘ1)H,
+(51b)
+Q1 = diag{hH
+id}HH
+irAHirEKΘ1,
+(51c)
+Q2 = diag{hH
+id}HH
+irA, Q3 = diag{hH
+idEK},
+(51d)
+c = hH
+rdA(hsr + HirΘ1hsi),
+(51e)
+λ(t + 1) =
+γs|uH
+2 (t)h3 + c|2
+∥uH
+2 (t)Q1 + hH
+4 ∥2 + ∥uH
+2 (t)Q2 + hH
+rdA∥2 + ∥uH
+2 (t)Q3∥2 + 1.
+(51f)
+It is clear that problem (50) is a convex optimization problem
+with concave objective function and convex constraints. Given
+�u2 and λ, u2 can be effectively obtained by CVX.
+D. Overall Algorithm and Complexity Analysis
+In the same manner, the proposed LC-SCA-FP method
+is convergent with an upper bound. The alternate iteration
+idea is roughly as follows: for given Θ1 and Θ2, first-
+order Taylor expansion is applied in problem (39), AF relay
+beamforming vector a can be calculated by solving problem
+(41) iteratively; similarly, for given A and Θ2, reflecting
+coefficient vector u1 can be obtained by solving problem (49)
+iteratively; for given A and Θ1, reflecting coefficient vector
+u2 can be achieved by solving problem (50) iteratively. Then
+the alternative iteration process are operated among A, Θ1
+and Θ2 until the convergence criterion is satisfied, while the
+system rate is maximum. The proposed LC-SCA-FP method
+is summarized in Algorithm 2.
+Furthermore, we calculate and analyze the complexity of
+Algorithm 2 in accordance with problems (41), (49) and (50).
+It is observed that problem (41) consists of one SOC constraint
+of dimension M 2 and one SOC constraint of dimension
+M 2 + 1. The number of decision variables na = M 2. The
+computational complexity corresponding to problem (41) is
+represented as O{2na(M 4+(M 2+1)2+n2
+a)ln(1/ε)} FLOPs.
+Problem (49) includes L SOC constraints of dimension 1,
+one SOC constraint of dimension N, one SOC constraint of
+dimension N +1 and one SOC constraint of dimension N +2.
+The number of decision variables nu1 = N. The computa-
+tional complexity corresponding to problem (49) is written as
+O{nu1
+√
+2L + 6(L+N 2+(N +1)2+(N +2)2+n2
+u1)ln(1/ε)}
+FLOPs. Problem (50) is composed of L SOC constraints of
+dimension 1 and one SOC constraint of dimension N. The
+number of decision variables nu2 = N. The computational
+complexity corresponding to problem (50) is expressed as
+O{nu2
+√2L + 2(L+N 2+n2
+u2)ln(1/ε)} FLOPs. Consequently,
+the total computational complexity of Algorithm 2 is denoted
+as
+O{D2[2na(M 4 + (M 2 + 1)2 + n2
+a) + nu1
+√
+2L + 6
+· (L + N 2 + (N + 1)2 + (N + 2)2 + n2
+u1)
++ nu2
+√
+2L + 2(L + N 2 + n2
+u2)]ln(1/ε)}
+(52)
+FLOPs, where D2 is the maximum number of alternating
+iterations to obtain a, u1 and u2. For Algorithm 2, its highest
+
+9
+Algorithm 2 Proposed LC-SCA-FP Method
+1. Initialize A0, Θ0
+1 and Θ0
+2. According to (9), R0 can be
+obtained.
+2. set the convergence error δ and the iteration number
+t = 0.
+3. repeat
+4. Fix Θt
+1 and Θt
+2, initialize �a0, set δ and t1 = 0.
+5.
+repeat
+6.
+Update solution at1+1 with (�at1, µt1) by solving
+problem (41), t1 = t1 + 1.
+7.
+Set �at1+1 = at1+1 and update µt1+1.
+8.
+until (41a) converges, update at+1 = at1+1 and
+obtain At+1.
+9. Fix At+1 and Θt
+2, initialize �u0
+1, set δ and t2 = 0.
+10.
+repeat
+11.
+Update solution ut2+1
+1
+with (�ut2
+1 , wt2) by solving
+problem (49), t2 = t2 + 1.
+12.
+Set �ut2+1
+1
+= ut2+1
+1
+and update wt2+1.
+13.
+until (49a) converges, update ut+1
+1
+= ut2+1
+1
+and
+obtain Θt+1
+1
+.
+14. Fix At+1 and Θt+1
+1
+, initialize �u0
+2, set δ and t3 = 0.
+15.
+repeat
+16.
+Update solution ut3+1
+2
+with (�ut3
+2 , λt3) by solving
+problem (50), t3 = t3 + 1.
+17.
+Set �ut3+1
+2
+= ut3+1
+2
+and update λt3+1.
+18.
+until (50a) converges, update ut+1
+2
+= ut3+1
+2
+and
+obtain Θt+1
+2
+.
+19. Calculate R by using At+1, Θt+1
+1
+and Θt+1
+2
+, t = t+1.
+20. until
+��Rt+1 − Rt�� ≤ δ.
+order of computational complexity is M 6 and N 3 FLOPS,
+which is greatly reduced compared to the complexity of HP-
+SDR-FP method.
+V. PROPOSED A LOWER-COMPLEXITY
+WF-GPI-GRR-BASED MAX-SNR METHOD
+In what follows, to further reduce the computational com-
+plexity, a lower-complexity WF-GPI-GRR-based Max-SNR
+method is put forward. For gaining rate enhancement, we ap-
+ply WF operation to exploit the colored property of noise and
+present the related system model. Here, active IRS reflecting
+coefficient matrix is split into amplifying coefficient and IRS
+phase-shift matrix. The details of derivation on the amplifying
+coefficients, AF relay beamforming matrix and IRS phase-shift
+matrices are described as below.
+A. System Model
+For brevity, it is assumed that the amplifying coefficients of
+each IRS active element in the first time slot and the second
+time slot are |β1| and |β2|, respectively. Let us define
+Ψ1 = EK �Θ1, Φ1 = |β1|EK �Θ1,
+(53a)
+Ψ2 = EK �Θ2, Φ2 = |β2|EK �Θ2,
+(53b)
+where the phase-shift matrix �Θ1 = diag(ejθ1i, · · · , ejθ1N ),
+�Θ2 = diag(ejθ2i, · · · , ejθ2N), | �Θ1(i, i)| = 1 and | �Θ2(i, i)| =
+1. Thus we have
+Θ1 = (EK + |β1|EK) �Θ1, Θ2 = (EK + |β2|EK) �Θ2. (54)
+In the first time slot, the received signal at AF relay can be
+redescribed as
+yr =
+�
+Ps[hsr + Hir(EK + |β1|EK) �Θ1hsi]x
++ (|β1|HirEK �Θ1n1i + nr)
+�
+��
+�
+n1r
+.
+(55)
+As matter of fact, n1r is color, not white. It is necessary for
+us to whiten the color noise n1r by using covariance matrix
+Cn. The covariance W1r of n1r is given by
+W1r = β2
+1∥HirEK �Θ1∥2
+F σ2 + σ2.
+(56)
+While n1i and nr are the independent and identically dis-
+tributed random vectors, n1r has a mean vector of all-zeros
+and covariance matrix
+C1r = β2
+1σ2HirEK �Θ1 �ΘH
+1 EKHH
+ir + σ2IM,
+(57)
+where obviously C1r is a positive definite matrix. Defining the
+WF matrix W1r with W1rWH
+1r = C−1
+1r , which yields
+W1r = C
+− 1
+2
+1r = (Q1rΛ1rQH
+1r)− 1
+2 = Q1rΛ
+− 1
+2
+1r QH
+1r,
+(58)
+where Q1r is an unitary matrix, and Λ1r is a diagonal matrix
+consisting of eigenvalues. Performing the WF operation to (55)
+yields
+yr =
+�
+PsW1r[hsr + Hir(EK + |β1|EK) �Θ1hsi]x
++ W1r(|β1|HirEK �Θ1n1i + nr)
+�
+��
+�
+n1r
+,
+(59)
+where n1r is the standard white noise with covariance matrix
+IM. The transmit signal at AF relay is yt = Ayr. In the second
+time slot, the received signal at D is denoted as
+yd =
+�
+Ps[hH
+rd + hH
+id(EK + |β2|EK) �Θ2HH
+ir]AW1r
+· [hsr + Hir(EK + |β1|EK) �Θ1hsi]x
++ [hH
+rd + hH
+id(EK + |β2|EK) �Θ2HH
+ir]An1r
++ |β2|hH
+idEK �Θ2n2i + nd.
+(60)
+The corresponding SNR can be represented as (61), as shown
+at the top of next page. It is assumed that the power budgets
+Ps, Pr and Pi are respectively fully used to transmit signals
+at S, AF relay and IRS. Therefore, the optimization problem
+can be converted to
+max
+|β1|,|β2|, �
+Θ1, �
+Θ2,A
+(61)
+(62a)
+s.t.
+| �Θ1(i, i)| = 1,
+| �Θ2(i, i)| = 1.
+(62b)
+It is necessary to solve the above problem for optimal |β1|,
+|β2|, �Θ1, �Θ2 and A.
+
+10
+SNR =
+γs|[hH
+rd + hH
+id(EK + |β2|EK) �Θ2HH
+ir]AW1r[hsr + Hir(EK + |β1|EK) �Θ1hsi]|2
+β2
+1∥[hH
+rd + hH
+id(EK + |β2|EK) �Θ2HH
+ir]AW1rHirEK �Θ1∥2 + ∥[hH
+rd + hH
+id(EK + |β2|EK) �Θ2HH
+ir]AW1r∥2 + β2
+2∥hH
+idEK �Θ2∥2 + 1
+.
+(61)
+B. Solve |β1| and |β2|
+In the first time slot, the reflected signal at IRS is written
+by
+yt
+1i =
+�
+PsΘ1hsix + Φ1n1i,
+=
+�
+PsEK �Θ1hsix
+�
+��
+�
+ypt
+1i
++
+�
+Ps|β1|EK �Θ1hsix + |β1|EK �Θ1n1i
+�
+��
+�
+yat
+1i
+,
+(63)
+where ypt
+1i and yat
+1i are respectively the signals reflected by
+passive elements EL and active elements EK. Additionally, the
+power consumed by the active elements is Pi. We have
+Pi = Psβ2
+1∥EK �Θ1hsi∥2 + β2
+1∥EK �Θ1∥2
+F σ2
+1i
+= β2
+1Ps
+K
+�
+k=1
+|ejθ1khk
+si|2 + β2
+1
+K
+�
+k=1
+|ejθ1k|2σ2
+1i
+= β2
+1Ps
+K
+�
+k=1
+|hk
+si|2 + Kβ2
+1σ2,
+(64)
+where θ1k is the phase shift of the kth IRS active element
+in the first time slot, hk
+si is the channel between S and the
+kth IRS active element and follows Rayleigh distribution with
+the expression hk
+si =
+�
+PLk
+sigk
+sie−jϕsk, where PLk
+si, gk
+si and
+ϕsk denote the path loss, the channel gain and the channel
+phase from S to the kth IRS active element, respectively. |gk
+si|2
+follows Exponential distribution [41], and the corresponding
+probability density function is given by
+f|gk
+si|2(x) =
+
+
+
+1
+λsi
+e−
+x
+λsi
+x ∈ [0, +∞),
+(65a)
+0
+otherwise,
+(65b)
+where λsi is the Exponential distribution parameter. Let us
+define PLk
+si is equal to the path loss from S to IRS (i.e.,
+PLsi). Using the weak law of large numbers, (64) can be
+further written as
+Pi = β2
+1Ps
+K
+�
+k=1
+|
+�
+PLsigk
+sie−jϕsk|2 + Kβ2
+1σ2
+= β2
+1PsPLsi
+K
+�
+k=1
+|gk
+si|2 + Kβ2
+1σ2
+≈ Kβ2
+1PsPLsi · E(|gk
+si|2) + Kβ2
+1σ2
+= Kβ2
+1PsPLsiλsi + Kβ2
+1σ2,
+(66)
+β1 can be achieved as
+|β1| =
+�
+Pi
+KPsPLsiλsi + Kσ2 .
+(67)
+Similarly, the received signal of the kth active IRS element in
+the second time slot is
+yrk
+2i = hH
+rkAyr + n2i,k = hH
+rkyt + n2i,k,
+(68)
+where hH
+rk ∈ C1×M represents the channel between AF relay
+and the kth active IRS element,
+hH
+rk = [
+�
+PL1k
+ri g1k
+ri e−jϕ1k, · · · ,
+�
+PLMk
+ri gMk
+ri e−jϕMk], (69)
+where PLmk
+ri , gmk
+ri
+and ϕmk are the path loss, the channel gain
+and the channel phase between the mth antenna at AF relay
+and the kth active IRS element. Defining PLmk
+ri
+= PLri,
+PLri is the path loss from AF relay to IRS. The reflected
+signal of the kth active IRS element is
+ytk
+2i = |β2|ejθ2khH
+rkyt + |β2|ejθ2kn2i,k
+= |β2||hH
+rk||yt|ej(θ2k+ϕrkt) + |β2|ejθ2kn2i,k,
+(70)
+where θ2k is the phase shift of the kth IRS active element in
+the second time slot, ϕrkt is the phase of hH
+rkyt. It is assumed
+that the transmit power of AF relay is Pr, the corresponding
+power of the reflected signal of the kth active IRS element is
+P tk
+2i = β2
+2|hH
+rk|2|yt|2 + β2
+2σ2
+2i,k
+= β2
+2PrPLri
+M
+�
+m=1
+|gmk
+ri |2 + β2
+2σ2
+K
+= Mβ2
+2PrPLriλri + β2
+2σ2
+K ,
+(71)
+where σ2
+2i,k = σ2
+2i/K = σ2/K, λri is the Exponential
+distribution parameter of channel from AF relay to IRS. Thus
+the power of the reflected signal of K active IRS elements is
+Pi =
+K
+�
+k=1
+P tk
+2i = KMβ2
+2PrPLriλri + β2
+2σ2,
+(72)
+which yields
+|β2| =
+�
+Pi
+KMPrPLriλri + σ2 .
+(73)
+C. Optimize A Given �Θ1 and �Θ2
+Aiming at maximizing the received signal power, MRC-
+MRT method are applied to solve A as follows
+A = A [hrd + Hir �ΘH
+2 (EK + |β2|EK)hid]
+∥hH
+rd + hH
+id(EK + |β2|EK) �Θ2HH
+ir∥
+· [hsr + Hir(EK + |β1|EK) �Θ1hsi]HWH
+1r
+∥W1r[hsr + Hir(EK + |β1|EK) �Θ1hsi]∥
+= AΥ,
+(74)
+where A is the amplify factor of AF relay. Since the transmit
+power of AF relay is Pr, we have A as shown in (75) at the top
+of next page. Inserting A back into (74), A can be obtained.
+
+11
+A =
+�
+γr
+γs∥ΥW1r[hsr + Hir(EK + |β1|EK) �Θ1hsi]∥2 + β2
+1∥ΥW1rHirEK �Θ1∥2
+F + ∥ΥW1r∥2
+F
+,
+(75)
+�F2 =
+�
+β2
+1diag{EKHH
+ir�hrid}diag{�h
+H
+ridHirEK}
+0N×1
+01×N
+∥�h
+H
+rid∥2 + β2
+2∥hH
+idEK �Θ2∥2 + 1
+�
+.
+(77)
+D. Optimize �Θ1 Given A and �Θ2
+By defining �u1 = [ejθ1i, · · · , ejθ1N ]T , �v1 = [�u1; 1] and
+�Hsir = [Hir(EK + |β1|EK)diag{hsi}, hsr]. Given A and �Θ2,
+the optimization problem is equivalent to
+max
+�v1
+�vH
+1 �F1�v1
+�vH
+1 �F2�v1
+(76a)
+s.t.
+|�v1(i)| = 1, ∀i = 1, 2, · · · , N,
+(76b)
+�v1(N + 1) = 1,
+(76c)
+where �F1 and �F2 are Hermitian matrices, and �F2 is positive
+semi-definite. �F1 = γs �H
+H
+sir�hrid�h
+H
+rid �Hsir, �hrid = [(hH
+rd +
+hH
+id(EK +|β2|EK) �Θ2HH
+ir)AW1r]H, and �F2 is denoted as (77)
+at the top of next page. The above problem can be relaxed to
+max
+�v1
+�vH
+1 �F1�v1
+�vH
+1 �F2�v1
+s.t.
+∥�v1∥2 = N + 1,
+(78)
+which can be constructed as
+max
+�v1
+�vH
+1 �F1�v1
+�vH
+1 �F2�v1
+· �vH
+1 IN+1�v1
+�vH
+1 IN+1�v1
+s.t.
+∥�v1∥2 = N + 1. (79)
+�v1 can be solved by using GPI algorithm, the details of
+GPI procedure is presented in Algorithm 3, where we define
+Ω(�vt
+1) = (�vH
+1 �F1�v1)IN+1 + (�vH
+1 IN+1�v1)�F1 and Ξ1(�vt
+1) =
+(�vH
+1 �F2�v1)IN+1 + (�vH
+1 IN+1�v1)�F2.
+Algorithm 3 GPI Algorithm to Compute Phase-Shift
+Vector �v1 with Given A and �Θ2
+1. Given A and �Θ2, and initialize �v0
+1.
+2. Set the tolerance factor ξ and the iteration number t = 0.
+3. repeat
+4.
+Compute the function matrix Ω(�vt
+1) and Ξ1(�vt
+1).
+5.
+Calculate yt = Ξ1(�vt
+1)†Ω(�vt
+1)�vt
+1.
+6.
+Update �vt+1
+1
+=
+yt
+∥yt∥.
+7.
+Update t = t + 1.
+8. until
+∥�vt+1
+1
+− �vt
+1∥ ≤ ξ.
+E. Optimize �Θ2 Given A and �Θ1
+If
+A
+and
+�Θ1
+are
+fixed,
+let
+us
+define
+�u2
+=
+[ejθ2i, · · · , ejθ2N ]H, �v2 = [�u2; 1], �Hrid = [diag{hH
+id(EK +
+|β2|EK)}HH
+ir; hH
+rd]. Accordingly, the optimization problem is
+reduced to
+max
+�v2
+�vH
+2 �H1�v2
+�vH
+2 �H2�v2
+(80a)
+s.t.
+|�v2(i)| = 1, ∀i = 1, 2, · · · , N,
+(80b)
+�v2(N + 1) = 1,
+(80c)
+where �H1 and �H2 are Hermitian matrices, and �H2 is positive
+definite. �H1 = γs �HridAW1r[hsr +Hir(EK +|β1|EK) �Θ1hsi]·
+{�HridAW1r[hsr + Hir(EK + |β1|EK) �Θ1hsi]}H, and
+�H2 = �HridAW1r(β2
+1HirEK �Θ1 �ΘH
+1 EKHH
+ir + IM)WH
+1rAH·
+�H
+H
+rid +
+�
+β2
+2diag{hH
+idEK}diag{EKhid}
+0N×1
+01×N
+1
+�
+.
+(81)
+We have the following relaxed transformation
+max
+�v2
+�vH
+2 �H1�v2
+�vH
+2 �H2�v2
+s.t.
+�vH
+2 �v2 = 1
+(82)
+where �v2 =
+�v2
+√N+1. Moreover, in line with the GRR theorem,
+the optimal �v2 is obtained as the eigenvector corresponding
+to the largest eigenvalue of �H
+−1
+2 �H1. Thereby �v2 and �Θ2 is
+achieved.
+F. Overall Algorithm and Complexity Analysis
+The proposed lower-complexity WF-GPI-GRR method is
+summarized in Algorithm 4. The main idea consists of two
+parts: the amplifying coefficient of active element and the
+iterative idea. The analytic solutions of amplifying coefficients
+of IRS active elements in the first time slot and the second time
+slot, i.e., β1 and β2, are determined by the transmit power
+of S, AF relay, IRS. Furthermore, β1 and β2 are denoted as
+(67) and (73). The iterative idea can be described as follows:
+for given �Θ1 and �Θ2, the closed-form expression of A are
+represented as (74) by utilizing MRC-MRT; for given A and
+�Θ2, GPI is applied to achieve �Θ1; for given A and �Θ1, �Θ2 is
+obtained in a closed-form expression by using GRR theorem.
+The alternative iteration process are performed among A, �Θ1
+and �Θ2 until the stop criterion is satisfied, while the system
+rate is maximum.
+In the following, the total computational complexity of
+Algorithm 4 is calculated as
+O{D3(N 3 + 4M 3 + 4M 2N + 2MN 2 + 2M 2K+
+8M 2 + 6N 2 + 9MN + 5MK + 5M + 11N+
+3 + D4(7N 3 + 27N 2 + 43N + 18))}
+(83)
+
+12
+Algorithm 4 Proposed WF-GPI-GRR Method
+1.
+Calculate β1 and β2 through (67) and (73).
+2.
+Initialize A0, �Θ0
+1 and �Θ0
+2. According to (9) and (61),
+R0 can be obtained.
+3.
+set the convergence error δ and the iteration number
+t = 0.
+4. repeat
+5.
+Fix �Θt
+1 and �Θt
+2, compute At+1 through (74).
+6.
+Fix At+1 and �Θt
+2, solve problem (79) to achieve
+�vt+1
+1
+based on GPI presented in Algorithm 3, �Θt+1
+1
+=
+diag{�vt+1
+1
+(1 : N)}.
+7.
+Fix At+1 and �Θt+1
+1
+, solve problem (82) to achieve
+�vt+1
+2
+based on GRR theorem, �Θt+1
+2
+= diag{�vt+1
+2
+(1 : N)}.
+8.
+Update Rt+1 by using β1, β2, At+1, �Θt+1
+1
+and �Θt+1
+2
+.
+9.
+Update t = t + 1.
+10. until
+��Rt+1 − Rt�� ≤ δ.
+FLOPs, where D3 is the maximum number of alternating
+iterations for Algorithm 4 and D4 is the number of iteration in
+GPI algorithm. Its highest order of computational complexity
+is M 3 and N 3 FLOPs, which is lower than the complexity of
+Algorithm 1 and Algorithm 2.
+VI. SIMULATION AND NUMERICAL RESULTS
+In this section, in order to evaluate the rate performance
+among the proposed three methods, numerical simulations are
+performed. Moreover, it is assumed that S, D, hybrid IRS
+and AF relay are located in three-dimensional (3D) space, the
+related coordinate simulation setup is shown in Fig. 2, where
+S, D, hybrid IRS and AF relay are located at (0, 0, 0), (0, 100,
+0), (−10, 50, 20) and (10, 50, 10) in meter (m), respectively.
+The path loss is modeled as PL(d) = PL0 − 10αlog10( d
+d0 ),
+where PL0 = −30dB is the path loss at the reference distance
+d0 = 1m, d is the distance between transmitter and receiver,
+and α is the path loss exponent, respectively. Here, the path
+loss exponents of each channel link associated with IRS, i.e.,
+S-IRS, IRS-AF relay and IRS-D, are set as 2.0, and those of
+S-AF relay and AF relay-D links are considered as 3.0. The
+remaining system parameters are set as follow: σ2 = −80dBm
+and EK is randomly generated.
+Fig. 2. Simulation setup.
+Additionally, to demonstrate the proposed three methods,
+the following three benchmark schemes are taken into account.
+1) AF relay+passive IRS: A passive IRS-aided AF relay
+network is considered, where IRS only reflects the signal
+without amplifying the reflected signal, and the reflecting
+coefficient of each IRS element is set as 1;
+2) AF relay+passive IRS with random phase: With ran-
+dom phase of each reflection element uniformly and indepen-
+dently generated from the interval (0, 2π], the beamforming
+matrix A at AF relay is optimized.
+3) Only AF relay: A AF relay network without IRS
+is considered, while the AF relay beamforming matrix A
+can be achieved by MRC-MRT, which is given by A =
+�
+Pr
+Ps∥Γhsr∥2+σ2∥Γ∥2
+F Γ, where Γ =
+hrdhH
+sr
+∥hH
+rd∥∥hsr∥.
+Towards a fair comparison between a hybrid IRS-aided
+AF relay wireless network and the above three benchmark
+schemes, let us define that the total transmit power budgets
+of S and AF relay in the three benchmark schemes are the
+same as that of S, AF relay and IRS in the hybrid IRS-aided
+AF relay network. For instance, the AF relay transmit power
+budget PR in the three benchmark schemes is equal to Pi+Pr.
+4
+5
+6
+7
+8
+9
+10
+11
+12
+log2N
+105
+1010
+1015
+1020
+1025
+1030
+Computational Complexity (FLOPs)
+Proposed HP-SDR-FP
+Proposed LC-SCA-FP
+Proposed WF-GPI-GRR
+Fig. 3. Computational complexity versus N with (M, K, D1, D2, D3, D4) =
+(2, 4, 6, 10, 5, 2).
+Fig. 3 plots the computational complexity of the proposed
+three methods. By suppressing ln(1/ε) [42], the computational
+complexities of the proposed three methods, 1) HP-SDR-
+FP; 2) LC-SCA-FP; and 3) WF-GPI-GRR, increase as N
+increases. It is clear that the first method has the highest
+computational complexity, which is much higher than those
+of the other two methods. In addition, the third method has
+the lowest computational complexity.
+Fig. 4 demonstrates the proposed three methods are con-
+vergent under different Ps, respectively. Obviously, for Ps =
+10dBm, the proposed HP-SDR-FP, LC-SCA-FP and WF-GPI-
+GRR methods require about only four iterations to achieve the
+rate ceil. While for Ps = 30dBm, it takes ten iterations for the
+proposed three methods to converge to the rate ceil. From the
+above two cases, we conclude that the proposed three methods
+are feasible.
+Fig. 5 shows the achievable rate versus Ps with (M, N, K)
+= (2, 32, 4). It can be seen that the proposed HP-SDR-
+FP, LC-SCA-FP and WF-GPI-GRR methods with (Pi, Pr) =
+(30dBm, 30dBm) perform better than AF relay+passive IRS,
+
+Hybrid IRS (-10, 50, 20)
+y
+AF relay (10, 50, 10)
+S
+(0, 0, 0)
+(0, 100, 0)
+X13
+0
+3
+6
+9
+12
+15
+18
+Number of iterations
+4
+5
+6
+7
+8
+9
+10
+Achivable Rate(bits/s/Hz)
+Proposed HP-SDR-FP
+Proposed LC-SCA-FP
+Proposed WF-GPI-GRR
+30dBm
+10dBm
+Fig. 4. Convergence of proposed methods with (M, N, K, Pi, Pr) = (2, 32,
+4, 30dBm, 30dBm).
+0
+5
+10
+15
+20
+25
+30
+ Ps (dBm)
+0
+1
+2
+3
+4
+5
+6
+7
+8
+9
+Achivable Rate (bits/s/Hz)
+Proposed HP-SDR-FP
+Proposed LC-SCA-FP
+Proposed WF-GPI-GRR
+AF relay+passive IRS
+AF relay+passive IRS with random phase
+Only AF relay
+Fig. 5. Achievable rate versus Ps with (M, N, K) = (2, 32, 4).
+AF relay+passive IRS with random phase and only AF relay
+with PR = 33dBm. Furthermore, the rate performance of LC-
+SCA-FP method is the most closest to that of HP-SDR-FP
+method in the low and medium power Ps region. For instance,
+when power Ps is equal to 15dBm, the rate performance gaps
+between the method LC-SCA-FP, the worst method WF-GPI-
+GRR and the best method HP-SDR-FP method are respectively
+0.037bits/s/Hz and 0.245bits/s/Hz.
+Fig. 6 illustrates the achievable rate versus Pi
+with
+(M, N, K, Ps) = (2, 32, 4, 30dBm). It is particularly noted
+that the proposed three methods with Pr = 30dBm make
+a better rate performance improvement than that of AF re-
+lay+passive IRS, AF relay+passive IRS with random phase
+and only AF relay with PR = Pi + Pr. For example, when
+Pi equals 40dBm, the proposed worst method, WF-GPI-GRR
+method, can harvest up to 49.8% rate gain over AF re-
+lay+passive IRS. The best method HP-SDR-FP approximately
+has a 53.3% rate gain over AF relay+passive IRS. This shows
+that as Pi increases, significant rate gains are achieved for
+the proposed hybrid IRS-aided AF relay wireless network.
+Moreover, the rate performance of LC-SCA-FP is getting
+closer to that of HP-SDR-FP, and the gap between LC-SCA-
+FP and WF-GPI-GRR becomes smaller.
+10
+15
+20
+25
+30
+35
+40
+ Pi (dBm)
+3
+4
+5
+6
+7
+8
+9
+Achivable Rate (bits/s/Hz)
+Proposed HP-SDR-FP
+Proposed LC-SCA-FP
+Proposed WF-GPI-GRR
+AF relay+passive IRS
+AF relay+passive IRS with random phase
+Only AF relay
+Fig. 6. Achievable rate versus P i with (M, N, K, Ps) = (2, 32, 4, 30dBm).
+0
+1
+2
+3
+4
+5
+log2K
+4
+5
+6
+7
+8
+9
+10
+Achivable Rate(bits/s/Hz)
+Proposed HP-SDR-FP
+Proposed LC-SCA-FP
+Proposed WF-GPI-GRR
+AF relay+passive IRS
+AF relay+passive IRS with random phase
+Only AF relay
+Fig. 7. Achievable rate versus K with (M, N, Ps) = (2, 32, 30dBm).
+Fig. 7 presents the achievable rate versus the number of
+active IRS elements K with (M, N, Ps) = (2, 32, 30dBm)
+for the proposed three methods with (Pi, Pr) = (30dBm,
+30dBm) and the three benchmark schemes with PR = 33dBm.
+From Fig. 7, it can be observed that as the number of active
+IRS elements K increases, the rate gains of the proposed
+three methods over AF relay+passive IRS, AF relay+passive
+IRS with random phase and only AF relay increase gradually
+and become more significant. Meanwhile, the proposed three
+methods have the following increasing order on rate: HP-
+SDR-FP, LC-SCA-FP and WF-GPI-GRR. Compared with the
+benchmark scheme of AF relay+passive IRS, our proposed
+three methods perform much better, which shows that the
+optimization of beamforming is important and efficient.
+VII. CONCLUSIONS
+In this paper, we have made an investigation of beamform-
+ing methods of optimizing the beamforming matrix at AF
+relay and reflecting coefficient matrices at IRS in a hybrid
+IRS-aided AF relay network, where the hybrid IRS includes
+few active elements amplifying and reflecting the incident
+signal. By using the criterion of Max SNR, three schemes,
+namely HP-SDR-FP, LC-SCA-FP and WF-GPI-GRR, have
+
+14
+been proposed to improve the rate performance. Simulation
+results show that the proposed three methods can make a
+dramatic rate enhancement compared to AF relay+passive
+IRS, AF relay+passive IRS with random phase and only
+AF relay, which verifies the active IRS elements can break
+the “double fading” effect caused by conventional passive
+IRS. For instance, an approximate 50.0% rate gain over the
+three benchmark schemes can be achieved in the high power
+budget region of hybrid IRS. Therefore, a hybrid IRS-aided
+AF relay network can provide an enhancement in accordance
+with rate performance and extended coverage for the mobile
+communications.
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+

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+Science
+Education
+Do Inquiring Minds Have Positive
+Attitudes? The Science Education
+of Preservice Elementary Teachers
+CATHERINE RIEGLE-CRUMB,1 KARISMA MORTON,1 CHELSEA MOORE,2
+ANTONIA CHIMONIDOU,3 CYNTHIA LABRAKE,4 SACHA KOPP5
+1Department of Curriculum and Instruction, STEM Education, University of Texas at
+Austin; 2Department of Psychology, University of Massachusetts, Amherst; 3UTeach
+Primary, College of Natural Sciences, University of Texas at Austin; 4Department of
+Chemistry, University of Texas at Austin; and 5College of Arts and Sciences, SUNY
+Stonybrook
+Received 26 June 2014; revised 27 February 2015; accepted 30 March 2015
+DOI 10.1002/sce.21177
+Published online 14 July 2015 in Wiley Online Library (wileyonlinelibrary.com).
+ABSTRACT: Owing to their potential impact on students’ cognitive and noncognitive
+outcomes, the negative attitudes toward science held by many elementary teachers are a
+critical issue that needs to be addressed. This study focuses on the science education of
+preservice elementary teachers with the goal of improving their attitudes before they begin
+their professional lives as classroom teachers. Specifically, this study builds on a small
+body of research to examine whether exposure to inquiry-based science content courses
+that actively involve students in the collaborative process of learning and discovery can
+promote a positive change in attitudes toward science across several different dimensions.
+To examine this issue, surveys and administrative data were collected from over 200 students
+enrolled in the Hands on Science (HoS) program for preservice teachers at the University
+of Texas at Austin, as well as more than 200 students in a comparison group enrolled
+in traditional lecture-style classes. Quantitative analyses reveal that after participating in
+HoS courses, preservice teachers significantly increased their scores on scales measuring
+confidence, enjoyment, anxiety, and perceptions of relevance, while those in the comparison
+group experienced a decline in favorable attitudes to science. These patterns offer empirical
+support for the attitudinal benefits of inquiry-based instruction and have implications for
+the future learning opportunities available to students at all education levels.
+C⃝ 2015
+Wiley Periodicals, Inc. Sci Ed 99:819–836, 2015
+Correspondence to: Dr. Catherine Riegle-Crumb; e-mail: riegle@austin.utexas.edu
+C⃝ 2015 Wiley Periodicals, Inc.
+
+820
+RIEGLE-CRUMB ET AL.
+INTRODUCTION
+While the metaphor of science, technology, engineering, and mathematics (STEM) as a
+pipeline has been rightly criticized as too simplistic, it is nevertheless clear that students’
+early experiences in science classrooms shape their future achievement and interests (Xie &
+Shauman, 2003). With the goal of better understanding and ultimately improving elemen-
+tary science education in the United States, researchers and policymakers have increased
+their attention toward teachers. A growing body of research now focuses on the science
+content knowledge of elementary science teachers, as the subject matter expertise that
+they possess has clear implications for what students learn (Diamond, Maerten-Rivera,
+Rohrer, & Lee, 2014; Heller, Daehler, Wong, Shinohara, & Miratrix, 2012; Kanter & Kon-
+stantopoulus, 2010; Sadler, Sonnert, Coyle, Cook-Smith, & Miller, 2013). Compared to
+secondary teachers, elementary teachers are much more likely to be trained as generalists
+and consequently less likely to have extensive content knowledge, and this pattern appears
+particularly pronounced for science (Haefner & Zembal-Saul, 2004). Clearly there are con-
+tinued concerns about the need for teacher training programs and professional development
+to focus on increasing subject matter expertise.
+Yet content knowledge is a necessary but insufficient characteristic of a successful teacher.
+Teachers’ attitudes about the content they teach is another critical factor that has implications
+for classroom learning, and importantly, negative attitudes can exist independent of content-
+area expertise (Beilock, Gunderson, Ramirez, & Levine, 2010; Tosun, 2000). While there
+is comparatively less research on elementary teachers’ attitudes toward science than math
+(Bursal & Paznokas, 2006), there is nevertheless evidence that many elementary teachers
+are not favorably inclined toward science. Such negative attitudes on the part of teachers can
+impact their students’ attitudes toward science and can inhibit students’ learning (Beilock
+et al., 2010; Jarrett, 1999; Ramey-Gassert, Shroyer, & Staver, 1996), creating a vicious
+cycle that must be interrupted. However, effectively changing how teachers view science is
+a challenging task (Mulholland & Wallace, 1996; Palmer, 2002).
+The goal of this study is to examine whether inquiry-based science content classes might
+function to help break this cycle by improving preservice teachers’ attitudes at a critical
+juncture before they begin their professional lives as classroom teachers. While there is
+much research on the positive impact of inquiry on outcomes for K–12 students (Borman,
+Gamoran, & Bowdon, 2008; Diamond et al., 2014; National Research Council, 2012b), we
+build on a smaller body of qualitative research regarding the benefits of inquiry instruction
+in college for preservice elementary teachers (Mulholland & Wallace, 1996; Palmer, 2002).
+Specifically, we suggest that students can become empowered and enthusiastic about the
+domain of science through active involvement in the process of inquiry, defined as engaging
+in the pursuit of scientific questions via data collection, experimentation, exploration, and
+discussion (National Research Council, 2000).
+To examine this issue, we collected data from the hands-on science (HoS) undergraduate
+program at the University of Texas at Austin (Ludwig et al., 2013) to determine whether
+exposure to these inquiry-based science content courses promoted a change in the science
+attitudes of a sample of over 200 preservice elementary teachers. In exploring preservice
+teachers’ attitudes, we go beyond the typical singular focus of much research on self-
+efficacy to instead consider personal attitudes toward science across several dimensions
+(van Aalderen-Smeets, Walma van der Molen, & Asma, 2012). Additionally, to ensure
+the robustness of our results, our design utilizes a comparison group of noneducation and
+nonscience majors enrolled in more traditional lecture-based science courses; our analyses
+also account for students’ social and academic background. Our study offers promising
+evidence that science content classes in college can be a positive vehicle for changing the
+Science Education, Vol. 99, No. 5, pp. 819–836 (2015)
+
+DO INQUIRING MINDS HAVE POSITIVE ATTITUDES?
+821
+attitudes of future elementary teachers, and subsequently has potential implications for
+the opportunities to learn both cognitive and noncognitive skills that are offered to future
+generations of elementary science students.
+LITERATURE REVIEW
+Framework: Considering Attitudes Across Multiple Dimensions
+When exploring attitudes toward science, it is critical to recognize multiple relevant
+dimensions. Based on a comprehensive review of prior research on teacher attitudes, and
+motivated by the lack of substantive clarity and empirical transparency of most prior
+research, van Aalderen-Smeets et al. (2012) recently advanced a new theoretical framework
+to provide a cohesive model that captures primary teachers’ attitudes toward science.
+Specifically, they developed a tripartite model of primary teachers’ attitudes toward science
+that distinguishes between three overarching dimensions, each of which is composed of
+different elements: (1) perceived control (which includes elements such as self-efficacy), (2)
+affective states (which includes enjoyment and anxiety), and (3) cognitive beliefs (such as
+perceived relevance). This framework is informed by earlier theoretical models of attitudes
+(Eagly & Chaiken, 1993), but departs from prior models by considering perceived control
+(e.g., self-efficacy) as a core dimension, and furthermore defining behavioral intentions
+as a consequence rather than a component of science attitudes. Their model also calls for
+researchers to make a clear distinction regarding whether the focus is on teachers’ personal
+attitudes toward science or their professional attitudes toward teaching science, as studies
+that combine teachers’ views of science as a domain with their views on science instruction
+in their own classroom into one empirical scale blur the object of teachers’ attitudes, making
+substantive interpretation difficult.
+The three dimensions of attitudes (whether personal or professional) advanced by van
+Aalderen-Smeets et al. (2012) are logically related to one another. For example, Bursal and
+Paznokas(2006) found that teachers with higher levels of self-efficacy had lower anxiety,
+a finding supported by several other studies (Bleicher, 2007; Palmer, 2002). Yet while
+related, they nevertheless capture somewhat distinct thoughts and beliefs. For instance, an
+individual might expect to master an activity or believe that it is useful, but nevertheless
+find it is unappealing (e.g., flossing their teeth) or even anxietyproducing (e.g., running 10
+miles). Therefore, considering elements of all three dimensions is critical to developing a
+comprehensive picture of teachers’ attitudes toward science.
+Yet most of the literature about the attitudes of elementary science teachers (either
+preservice or in-service) focuses on their perceived control in the form of self-efficacy,
+as there is relatively scant research on either the cognitive beliefs or affective attitudes
+of teachers (van Aalderen-Smeets et al., 2012). Thus, a key contribution of our study is
+our consideration of elements of all three dimensions of attitude to more fully capture the
+complexity of preservice teachers’ attitudes toward science.1 Below we discuss the prior
+literature on the science attitudes of preservice elementary teachers in more detail, including
+how such attitudes may influence the outcomes of future students. Because our research
+questions and subsequent empirical analyses focus on preservice teachers well before they
+1In this paper, we address all three dimensions of van Aalderen-Smeets et al.’s (2012) theoretical model,
+but do not discuss (or model) every element within each dimension. For example, while the authors include
+context dependency as an element that falls under the dimension of perceived control, we do not address this
+here as it refers to the support that practicing teachers receive from their administrators and therefore is not
+particularly relevant for a study concerning the personal (rather than professional) attitudes of preservice
+teachers.
+Science Education, Vol. 99, No. 5, pp. 819–836 (2015)
+
+822
+RIEGLE-CRUMB ET AL.
+actually enter the elementary classroom, we concentrate on literature on personal attitudes
+toward science. We then turn to a discussion of why inquiry-based science content classes
+for preservice teachers have the potential to increase individuals’ attitudes across all three
+dimensions.
+Perceived Control: Considering Self-Efficacy.
+A key element of the attitudinal dimen-
+sion of perceived control is self-efficacy, which according to Bandura’s (1977, 1982)
+foundational work is defined as an individual’s belief that she can successfully master a
+situation or deal with an obstacle that arises. An individual’s self-efficacy has logical im-
+plications for her subsequent behaviors and choices, as she is likely to attempt to avoid
+those situations or activities where she does not feel she can be efficacious, and persist
+where she feels confident that she can be successful. While research has demonstrated
+that in-service elementary teachers exhibit low levels of efficacy in their science teaching
+(see, for example,Atwater, Gardner, & Kight, 1991; Harlen, 1997; Ramey-Gassert et al.,
+1996), not surprisingly this pattern is also evident among preservice teachers, who feel less
+efficacious about their own ability to learn science. For example, Skamp (1991) determined
+that less than half of the preservice elementary teachers in his study reported having even
+a fair amount of confidence in their science ability. Similarly, Bleicher’s (2007) study of
+preservice elementary teachers found that participants in a science methods class exhibited
+low scores on science self-efficacy scales. Low efficacy is often attributed to prior negative
+educational experiences in science. For example, in a study of five preservice teachers,
+Mulholland and Wallace (1996) found that their respondents reported very little confidence
+in their own science abilities and attributed this to negative experiences in their own science
+schooling as children.
+Affective States: Considering Enjoyment and Anxiety.
+Individuals’ affect toward a
+domain represents another attitudinal dimension. As articulated by van Aalderen-Smeets
+et al. (2012), the dimension of affect can be further categorized into the positive element
+of enjoyment and the negative element of anxiety. Beginning with the former, Liang and
+Gabel (2005) found that most preservice elementary teachers in their study reported that
+science had never been enjoyable for them. These teachers also indicated that they took
+science courses only because it was required for their degree program. Smith (2000) and
+Howes (2002) found similar results. Additionally, the preservice participants in all of these
+studies attributed their low levels of enjoyment to negative experiences in either (or both)
+their high school and college science courses, a point to which we will return to later.
+Anxiety captures the negative aspect of the affective dimension. Early research by Mallow
+(1981; also Mallow & Greenburg, 1983) as well as Westerback (1984) define science anxiety
+as the fear, worry, or apprehension that some individuals experience when presented with
+the task of learning science. While research on the math anxiety of preservice teachers is
+extensive (Udo, Ramsey, & Mallow, 2004), and research considering both math anxiety and
+science anxiety find a strong association between the two (Cady & Rearden, 2007), there is
+comparatively little research focusing specifically on science anxiety (Bursal & Paznokas,
+2006). Yet several studies do provide evidence that preservice elementary teachers report
+relatively high levels of science anxiety about teaching science, as well as more generalized
+anxiety about learning science themselves (Cady & Rearden, 2007; Udo et al., 2004;
+Westerback & Long, 1990).
+Cognitive Beliefs: Considering the Relevance of Science.
+Finally, we discuss research
+on preservice elementary teachers’ beliefs in the relevance of science, a critical element
+Science Education, Vol. 99, No. 5, pp. 819–836 (2015)
+
+DO INQUIRING MINDS HAVE POSITIVE ATTITUDES?
+823
+of the attitudinal dimension of cognitive beliefs. The limited extant research on this topic
+focuses on perceptions of science as useful or relevant for society and for them personally,
+and finds that preservice elementary teachers report generally favorable views. Specifically,
+in a study of 200 preservice teachers, Coulson (1992) used a survey instrument that in-
+cluded a personal usefulness science scale and found that on average, respondents indicated
+moderate levels of agreement. Cobern and Loving’s (2002) study at a large Midwestern
+university also found that on Likert scales measuring the perceived importance of science
+to all citizens, preservice elementary teachers on average agreed with this sentiment. These
+findings echo the sentiments of the general population, which generally regard science and
+technology as useful for making their lives better (Evans & Durant, 1995; Kohut, Keeter,
+Doherty, & Dimock, 2009). Therefore among the three attitudinal dimensions discussed,
+promoting preservice teachers’ views of the relevance of science is perhaps somewhat less
+of a pressing problem than promoting both their perceived control in the form of science
+efficacy and their affective attitudes of enjoyment and anxiety.
+Examining the Impact of Teacher Attitudes
+There is a logical connection between teachers’ attitudes toward a subject and student
+outcomes, both in terms of impacting students’ opportunities to learn science and their own
+developing attitudes. First, research indicates that the negative attitudes toward science held
+by many elementary teachers are likely to result in less coverage of science content and
+less engaging and effective instruction. Teachers who are not confident in their own science
+knowledge are likely to worry that they cannot effectively answer students’ questions nor
+keep them engaged with interesting activities (Jarrett, 1999). Consequently, elementary
+teachers who feel less efficacious and more anxious are likely to try to avoid teaching
+science and spend less time teaching when they cannot avoid it altogether (Brownlow,
+Jacobi, & Rogers, 2000; Pine et al., 2006; Ramey-Gassert et al., 1996). Appleton and Kindt
+(1999) also found evidence that avoidance occurred when teachers did not find science to
+be as relevant as other subjects such as English or math. One such teacher reported that “If
+you’re running out of time in the week, you think ‘Oh I just won’t worry about that science
+activity’” (p. 162).
+When teachers do teach science, their negative attitudes can impact their pedagogical
+practices and subsequently their capacity to reach and engage students. Appleton and
+Kindt’s (1999) study of in-service elementary teachers found that those exhibiting low
+confidence were less likely to engage their students in hands-on learning. Ramey-Gassert
+et al. (1996) found that elementary teachers with low science self-efficacy had a minimal
+desire to engage in professional development activities that could improve their teaching of
+science. Lack of belief in the usefulness or relevance of science could also logically result
+in a reduced effort to provide engaging instruction to students.
+Additionally, the negative attitudes toward science held by many elementary teachers can
+result in the socialization of students toward a negative stance toward science. As students
+look to their teachers as role models and authorities, they begin to mimic and potentially
+internalize such attitudes as their own (Jussim & Eccles, 1992; McKown & Weinstein,
+2002). This can lead to a vicious cycle where negative attitudes such as anxiety and low
+efficacy are passed from teachers to their students, and therefore from one generation
+to the next. For example, Beilock et al. (2010) found that elementary teachers’ math
+anxiety influenced students’ own attitudes toward math, with girls in particular more likely
+to evidence declining math attitudes and increasingly gender-stereotyped views of math
+over the course of the year. This particular study highlights additional concerns regarding
+gender rolemodeling; as most elementary teachers are female, and a substantial number
+Science Education, Vol. 99, No. 5, pp. 819–836 (2015)
+
+824
+RIEGLE-CRUMB ET AL.
+of them exhibit negative attitudes toward science (as well as math), this could be a strong
+conduit through which young girls begin to believe that these subjects are less interesting,
+less important, and overall less-suited for them. Furthermore, as students’ own attitudes
+decline, so does their engagement and motivation to learn, which further impacts their
+achievement (Eccles, 1994).
+In sum, the research literature provides compelling evidence that teachers’ negative
+attitudes can impact students’ learning and attitudinal outcomes, and as such are a critical
+issue that needs to be addressed. To effectively break this cycle, necessitates intervening
+to change teachers’ attitudes before they enter the classroom. In short, the education of
+preservice teachers is an ideal place to focus.
+Educating Preservice Teachers: Inquiry as a Tool for Improving
+Attitudes
+As mentioned earlier, the negative attitudes of preservice elementary teachers can at
+least in part be traced back to their own negative experiences with science classes in high
+school and in college. For example, Liang and Gabel (2005) found that preservice teachers
+attributed their lack of enjoyment of science to their prior experiences in classes dominated
+by lectures that necessitated copious amounts of note-taking and memorization. Similarly,
+Smith (2000) reports that preservice elementary teachers in his study often complained
+about the boredom of their procedure-based high school and college science courses.
+Simply put, many preservice teachers have had little exposure to inquiry-based science
+instruction in their lives as students. We posit that exposure while in college to pedagogy
+that actively involves them in the process of scientific discovery can ultimately help them to
+see that science is meaningful, interesting, and accessible, thereby changing their attitudes
+toward science.
+Several studies support the supposition that inquiry classes can promote such changes,
+in spite of the possibility of some initial student frustration with an approach that deviates
+from teachers’ didactic presentations of the “right” answer (Volkmann, Abell, & Zgagacz,
+2005). For example, in a qualitative study of preservice teachers at a large university in
+the southwest, Kelly (2000) found that after completing an active, inquiry-based science
+methods course, most participants reported that their interest in science had increased.
+Kelly (2000) attributes this shift to the use of hands-on explorations and discussions where
+students came to embody the process of scientific inquiry by formulating and exploring
+ideas. Similarly, in a study of preservice elementary teachers, Palmer (2002) interviewed
+four participants who reported that their attitudes toward science had changed from negative
+to positive due to the excitement of inquiry-based lessons. A study of five teachers by
+Mulholland and Wallace (1996) reported similar results. Finally, in a quantitative study
+of 112 preservice elementary teachers, Jarrett (1999) found that an inquiry-based science
+methods class increased the participants’ personal interest in science. The author attributed
+this change to preservice teachers becoming active agents in the classroom and learning to
+view science as a process of discovery. Thus, a small body of research finds that inquiry-
+based pedagogy in science educational methods courses can have a positive impact on
+preservice elementary teachers’ attitudes.
+CURRENT STUDY
+Building on the insights of this prior research, we posit that required science content
+courses could also represent a powerful venue for change for college students at the
+beginning stages of preparing for their careers as future teachers. Specifically, the purpose
+Science Education, Vol. 99, No. 5, pp. 819–836 (2015)
+
+DO INQUIRING MINDS HAVE POSITIVE ATTITUDES?
+825
+of this study is to examine whether inquiry-based science content courses promote a change
+in attitudes toward science among preservice elementary teachers. The courses are part of
+the HoS undergraduate program, developed at The University of Texas at Austin in the
+College of Natural Sciences, with cooperation from the College of Education. HoS is
+required of all students in the elementary education program and covers four semesters of
+science courses with a curriculum that is composed heavily of the topics that preservice
+teachers will be expected to teach their students once they become teachers (Ludwig et al.,
+2013). The design of the HoS program is based on the Physics and Everyday Thinking
+(Goldberg, 2008) framework that centers around the development of students’ physical
+science understandings through experimentation and follows the example of work from
+Western Washington University extending this framework to other disciplines (Nelson,
+2008). The curriculum is based upon big ideas in science, with specific emphasis on the
+themes of Matter and Energy, which are integrated across different science disciplines. The
+course sequence focuses on physics in Semester 1, chemistry and geology in Semester 2,
+biological systems in Semester 3, and astronomy and earth science in Semester 4.
+HoS classes were designed to utilize the essential elements of inquiry-based learning
+as defined by the influential National Research Council report on inquiry (Forbes, 2011;
+National Research Council, 2000); specifically, students engage in scientifically oriented
+questions by collecting, organizing, and analyzing data. From that data they formulate
+explanations, connect them to scientific knowledge, and subsequently evaluate their expla-
+nations in contrast to alternative explanations. Additionally, students share and justify those
+explanations with others.
+The HoS program is best categorized as teacher-directed or guided inquiry (Cuevas, Lee,
+Hart, & Deaktor, 2005; National Research Council, 2000; Volkmann et al., 2005). Initial
+questions are posed by the instructor and all activities are carefully designed to present
+opportunities for students to confront misconceptions, with both topics and skills developed
+in a structured progression. The instructor does not lecture but probes student knowledge and
+offers helpful nudges in the right direction, working with students both in small groups and
+via whole class discussions to construct knowledge and develop explanations (Volkmann
+et al., 2005). Finally, consistent with inquiry approaches to learning, the course emphasizes
+big ideas in science while engaging students in hands-on learning and constant exploration
+and discussion while in groups with their peers (National Research Council, 2000, 2012b).
+An example of a typical 2-hour class session is a lesson on sound that begins with the
+following teacher-generated questions: How does sound travel through a room? How does
+sound travel from your classmate to you? Students then work in groups of three or four to
+answer these questions and discuss their initial ideas or preconceptions; they subsequently
+share their ideas with the entire class, so that there is a collective knowledge of alternative
+ways of thinking about how sound travels. Working in their small groups, students then
+engage in a series of experiments, using an “airzooka” and candles, a string telephone, and
+a loudspeaker, that require that they gather data and then generate models based on these
+data. They are regularly prompted by the instructor to make predictions and connect trends
+to other previously seen concepts. Finally, students reflect and summarize key ideas and
+connect them to other contexts by using evidence-based reasoning from the data collected
+in their experiments. These questions are the basis for whole-class discussions, where
+students present and justify their ideas to their peers. Students typically end lessons by
+writing a scientific narrative summarizing how their thinking was revised from their initial
+ideas, drawing on evidence gained through the experiments and whole class sharing and
+discussion. For example, a student might share how her initial idea that “sound travels as a
+wave” has become more sophisticated, discussing the role of vibrations and the transfer of
+mechanical energy from the source to the receiver.
+Science Education, Vol. 99, No. 5, pp. 819–836 (2015)
+
+826
+RIEGLE-CRUMB ET AL.
+Our examination of whether the inquiry-based science content courses of the HoS pro-
+gram promote a change in attitudes toward science among preservice elementary teachers is
+described in detail below. Here, we briefly note that our study contributes to the literature a
+rigorous quantitative examination of change in attitudes across multiple dimensions among
+preservice elementary teachers, utilizing a comparison group of students in more traditional
+science content courses, as well as accounting for differences among students in academic
+and social background characteristics that may have implications for their attitudes.
+Data and Method
+Analytic Sample.
+Our analytic sample includes 238 preservice elementary teachers who
+enrolled in the HoS program between Fall 2010 and Spring 2012. Students completed
+presurveys prior to the start of the first course and postsurveys at the end of the second
+course in the sequence.2 Surveys were done online, and were administered during class
+by members of the research team. Additionally, our sample includes a comparison group
+composed of 263 nonscience and noneducation majors enrolled in traditional lecture-style
+undergraduate science courses in Spring 2012. While our design falls shorts of a pure
+treatment versus control comparison, we chose courses that represent what our sample of
+preservice elementary teachers would have been required to take in the absence of the
+HoS program. Therefore, our comparison group includes students who were enrolled in
+either an introductory-level chemistry or biology class. These students were surveyed at
+the beginning and end of the semester. As the time period between pre- and postsurveys
+is longer for HoS students than for non-HoS students, we discuss the implications and
+limitations of this comparison later.
+Administrative records were linked to student surveys, allowing us to examine the demo-
+graphic and academic background information of students in our sample and to consider
+how HoS students differed from those in the comparison group. Not surprisingly given the
+gender composition of the elementary teacher population nationwide, HoS students were
+overwhelmingly female (95%), compared to 65% of our comparison sample of nonedu-
+cation and nonscience majors. There were no statistically significant differences between
+the two groups of students by mother’s education; for race/ethnicity, there were signifi-
+cantly fewer students who identified as American Indian among HoS students compared
+to non-HoS students. Regarding academic background prior to college entry, HoS students
+had lower SAT math scores than their fellow students taking typical science classes by
+more than one-third of a standard deviation (see Table 1). To ensure that any differences
+observed between the two groups in their changes in attitudes over time are not confounded
+by differences in their background characteristics, our subsequent multivariate models will
+control for all of these factors.3
+Student Attitude Surveys.
+The pre- and postsurveys administered to both groups con-
+sisted of 21 items geared toward assessing student attitudes toward science. To examine
+multiple dimensions of attitudes, we selected items from preexisting surveys to measure
+2At the time of this study, students were only required to take the first two semesters of the total four
+semester sequence.
+3Mother’s education level is an ordinal variable with the following categories: (1) did not attend high
+school, (2) attended high school but did not graduate, (3) high school diploma or GED, (4) some college, (5)
+earned associate’s degree, (6) bachelor’s degree, (7) graduate or professional degree. Math SAT score and
+mother’s education level were imputed for those students who had missing values. We utilized information
+on gender, race, high school rank, family income, and father’s education to conduct single imputation using
+Stata. Analyses using list-wise deletion produced similar results.
+Science Education, Vol. 99, No. 5, pp. 819–836 (2015)
+
+DO INQUIRING MINDS HAVE POSITIVE ATTITUDES?
+827
+TABLE 1
+Descriptive Statistics
+Hands-on Science
+(HoS) Students
+Non-HoS Students
+N = 238
+N = 263
+Gender
+Female***
+94.5%
+65.4%
+Race/ethnicity
+White (non-Hispanic)
+63.4%
+60.1%
+Black
+2.9%
+3.4%
+Hispanic
+21%
+19%
+Asian
+11.3%
+14.4%
+American Indian**
+0.4%
+2.7%
+Native Hawaiian/Pacific Islander
+0.8%
+0.4%
+Other background characteristics
+Mean
+SD
+Mean
+SD
+SAT math score***
+574.87
+75.60
+607.62
+73.39
+Mother’s educational level
+5.10
+1.60
+5.07
+1.69
+***p < .001, **p < .01, *p < .05, p < .10.
+confidence (an element of perceived control)4, enjoyment and anxiety (positive and neg-
+ative elements of affective states), and relevance (a key element of cognitive beliefs). To
+measure anxiety, we used the Math Anxiety Rating Scale (Hopko, 2003) and substituted
+the word science for all references to math. To measure all other attitudes, we selected
+items developed by the National Center for Education Statistics (www.nces.ed.gov), and
+used in national surveys, including the Educational Longitudinal Study, the High School
+Longitudinal Study, and the U.S. component of the Trends in International Mathematics
+and Science Study. Principal component analyses with promax rotation using the complete
+set of 21 survey items revealed four factors with Eigenvalues greater than one. We sub-
+sequently created separate scales (described below) composed of the individual items that
+loaded onto each of the four factors. The scales clearly corresponded to the elements of
+confidence, enjoyment, anxiety, and relevance; using the full sample, the Cronbach’s alpha
+for each scale was .8 or higher.5
+The confidence scale is composed of four items that gauge students’ level of confidence
+when engaging in scientific activities. The items were as follows: “I have always done well
+in science,” “Science is not one of my strengths” (reverse coded), “I am confident that I can
+understand the most difficult material presented in my science textbooks,” and “Science
+4Because the questions ask students to reflect more on their assessment of their current success in science,
+we refer to this scale as measuring confidence rather than self-efficacy for future success. However, prior
+research has consistently noted a very strong correspondence between indicators of self-confidence and
+self-efficacy (Wigfield & Eccles, 2000).
+5To further test the reliability of the four scales, we calculated Cronbach’s alphas separately for (a)
+treatment and reference groups; (b) first year cohort and second year cohorts (for treatment group), and
+(c) survey responses at time 1 and time 2 (for different cohorts and for different treatment groups). In all,
+we calculated alpha reliabilities for each of our four scales for 15 different subsamples with remarkably
+consistent results ranging from a low of .77 to a high of .88. (exploratory factor analyses using each of
+these groups also yielded the same four factor model).
+Science Education, Vol. 99, No. 5, pp. 819–836 (2015)
+
+828
+RIEGLE-CRUMB ET AL.
+has always been one of my best subjects.” The five items in the enjoyment scale capture
+students’ level of positive affect for science and included “I enjoy learning science,” “I look
+forward to going to science courses,” “Science is fun,” “I like science,” and “Science is
+boring” (reversecoded for inclusion in the scale). Categories of response were the same as
+those for the confidence scale.
+For each of the eight items in the anxiety scale, students were asked to indicate how
+much the situation made them feel anxious or worried. Similar to the other measures, the
+responses ranged from (1) not at all anxious to (5) very much anxious or worried. The items
+included “Looking through the pages in a science text,” “Thinking about an upcoming
+science test one day before the test,” “Reading and interpreting a scientific graph, chart, or
+illustration,” “Taking an exam in a science course,” “Watching and listening to a teacher
+explaining a scientific concept or phenomena,” “Waiting to get a science test returned in
+which you expected to do well,” “Walking on campus and thinking about a science course,”
+and “Being given a ‘pop’ quiz in science class.”
+Finally, the four items making up the relevance scale include questions that focused
+on students’ perception of the meaning or usefulness of science in their daily lives. The
+items are as follows: “The subject of science is not very relevant to most people,” “It is not
+important for most people to understand science,” “I think learning science will help me in
+my daily life,” and “Science is important to me personally.” Students indicated the extent to
+which they agreed or disagreed with the statements with possible responses ranging from
+strongly agree (1) to strongly disagree (5).
+ANALYSES AND RESULTS
+To examine whether inquiry-based science content courses promote a positive change
+in attitudes toward science for preservice elementary teachers, we begin by comparing the
+means on the pre- and postmeasures of our four attitudinal scales for HoS students. Results
+of paired t-tests indicate a statistically significant improvement over time for each scale.
+Specifically, for confidence, the mean increased from 2.61 to 2.98 (p < .001), reflecting
+a 0.37 point increase or almost half of a standard deviation change from pre to post. HoS
+students’ science enjoyment increased by about a fourth of point (and about a third of a
+standard deviation), from the presurvey mean of 3.24 to a postsurvey mean of 3.51 (p <
+.001). The largest change was observed for the anxiety scale, where the average decreased
+(meaning that students became less anxious over time) from a presurvey mean of 3.11 to a
+postsurvey mean of 2.63 (p < .001), a difference of almost half of a point and approximately
+two-thirds of a standard deviation. Finally, as discussed earlier, students view science as a
+relevant domain, as evidenced by a relatively high presurvey mean of 3.66 on the utility
+scale. Nevertheless, they slightly increased their views of the relevance of science after
+taking HoS courses (p < .05) to a postsurvey mean of 3.74, a change of about a tenth of a
+standard deviation.
+Subsequently, we turn to an examination of how the changes we observe for HoS students
+compare to changes in attitudes toward science among students taking more traditional
+lecture-based science content courses. Here, we utilize multilevel mixed-effects models,
+an extension of regression analysis that is similar to a mixed-design analysis of variance
+and appropriate when data are nested. For this study, repeated measures of attitudes are
+nested within individuals with time treated as a random effect (Rabe-Hesketh & Skrondal,
+2008).The goal of this analysis was to determine whether the change in different dimensions
+of science attitudes observed for HoS students was similar or different than that observed
+for non-HoS students while controlling for students’ background characteristics (which
+Science Education, Vol. 99, No. 5, pp. 819–836 (2015)
+
+DO INQUIRING MINDS HAVE POSITIVE ATTITUDES?
+829
+TABLE 2
+Regression Analyses Predicting Attitudes to Sciencea
+Model 1
+Model 2
+Model 3
+Model 4
+Confidence
+Enjoyment
+Anxiety
+Relevance
+Hands-on science (HoS) students
+(ref = non-HoS students)
+–0.391***
+–0.163*
+0.106
+–0.051
+(0.071)
+(0.072)
+(0.066)
+(0.059)
+Time (change from pre- to postsurvey)
+–0.127**
+–0.131***
+0.074�
+–0.116**
+(0.039)
+(0.039)
+(0.041)
+(0.036)
+Time ×HoS
+0.500***
+0.391***
+–0.553***
+0.200***
+(0.056)
+(0.056)
+(0.060)
+(0.052)
+Female
+–0.402***
+–0.395***
+0.311***
+0.046
+(0.085)
+(0.084)
+(0.076)
+(0.069)
+Race/ethnicity (ref = white)
+Black
+0.078
+–0.069
+0.018
+–0.159
+(0.188)
+(0.185)
+(0.168)
+(0.149)
+Hispanic
+–0.136
+0.054
+0.055
+–0.023
+(0.093)
+(0.091)
+(0.083)
+(0.073)
+Asian
+–0.314**
+–0.110
+0.232**
+–0.002
+(0.098)
+(0.097)
+(0.088)
+(0.078)
+American Indian/Alaska native
+–0.591*
+–0.181
+0.287
+0.112
+(0.251)
+(0.249)
+(0.224)
+(0.204)
+Native Hawaiian/other Pacific
+Islander
+0.635
+0.380
+–0.524
+0.537�
+(0.418)
+(0.407)
+(0.374)
+(0.322)
+SAT math score
+0.002***
+–0.000
+–0.002***
+0.000
+(0.000)
+(0.000)
+(0.000)
+(0.000)
+Mother’s education level
+–0.038�
+–0.013
+0.029
+–0.003
+(0.022)
+(0.022)
+(0.020)
+(0.017)
+Constant
+2.680***
+4.179***
+3.748***
+3.762***
+(0.320)
+(0.315)
+(0.288)
+(0.255)
+aCoefficients calculated from a multilevel regression model in Stata where observations of
+attitudes are nested within individuals.
+***p < .001, **p < .01, *p < .05, �p < .1; standard errors in parentheses; n = 501.
+is particularly important as our two groups of students differed by gender and math SAT
+scores, both factors which likely predict attitudes).
+Table 2 displays the results of separate analyses for each dependent variable. The first
+row displays the coefficient comparing HoS and non-HoS students on the presurvey (or
+time 1) for each attitudinal dimension. The second row displays the average change over
+time between the pre- and postsurvey, while the third row displays the interaction between
+student group (HoS or non-HoS) and time. The change in attitudes from pre- to postsurvey
+for HoS students is calculated as the sum of the main effect of time and the interaction term,
+while change for non-HoS students is captured by the main effect of time only. To simplify
+the presentation of results, we include figures for each attitudinal outcome (Figures 1–4)
+that display the changes over time for each group, adjusted for the social and academic
+characteristics discussed above.6
+6Figures 1–4 display the adjusted pre and post means for each group. These are calculated using a
+postestimation command in Stata where all other variables in the model(other than student group, time, and
+the interaction) are set to the mean (or alternatively to the mode for categorical variables).
+Science Education, Vol. 99, No. 5, pp. 819–836 (2015)
+
+830
+RIEGLE-CRUMB ET AL.
+Figure 1. Confidence.
+Figure 2. Enjoyment.
+Figure 3. Anxiety.
+Science Education, Vol. 99, No. 5, pp. 819–836 (2015)
+
+3.8
+3.6
+3.4
+score
+3.2
+Adjusted
+HoSstudents
+3
+...Non-Hosstudents
+2.8
+2.6
+2.4
+Pre
+Post3.8
+3.6
+3.4
+score
+3.2
+Adjusted :
+HoSstudents
+3
+...Non-Hos students
+2.8
+2.6
+2.4
+Pre
+Post3.8
+3.6
+3.4
+Adjusted score
+3.2
+HoSstudents
+3
+.Non-HoSstudents
+2.8
+2.6
+2.4
+Pre
+PostDO INQUIRING MINDS HAVE POSITIVE ATTITUDES?
+831
+Figure 4. Relevance.
+Beginning with the first column predicting confidence, the results reveal that compared to
+their peers in traditional science classes, HoS students reported significantly lower science
+confidence on the presurvey (–.391***). The coefficient for time is negative and significant,
+yet the interaction between time and student group is positive and significant, indicating
+that HoS students increased their confidence over time relative to non-HoS students. To help
+clarify the patterns for the two groups, Figure 1 displays the trends for each group. Here, we
+see clearly that changes in attitudes occurred for both groups in opposite directions. While
+HoS students initially had lower confidence than their non-HoS peers, they significantly
+increased their confidence over time. In contrast, non-HoS students significantly decreased
+their science confidence (–.127***), such that their confidence was slightly lower than
+non-HoS students on the postsurvey.
+Returning to Table 2, we see a similar pattern when predicting change in science en-
+joyment. HoS students initially reported significantly lower levels of enjoyment than their
+peers in more traditional classes (–.163**). However, once again we see a negative main
+effect of time but a positive and significant interaction between student group and time, in-
+dicating opposite directions of change for each group. As displayed graphically in Figure 2,
+HoS students significantly increase their enjoyment over time, while on average non-HoS
+students report a decrease in their affect toward science (–.131***), and end their course
+reporting lower enjoyment than HoS students.
+Regarding changes in anxiety, we note that HoS and non-HoS students do not differ
+significantly on the presurvey. The main effect of time is positive and borderline significant,
+yet the interaction term reveals a statistically significant difference between the two groups’
+average change in anxiety. Figure 3 clearly shows the marked decrease in anxiety from the
+pre- to the postsurvey for HoS students. For their non-HoS peers, however, the figure shows
+a slight increase in anxiety (�.074).
+Finally, Table 2 displays the results for models predicting changes in attitudes toward
+the relevance of science. HoS and non-HoS students do not differ significantly on the
+presurvey. But once again the interaction term reveals a statistically significant difference
+between groups in change over time, and the sign of the coefficient is positive in contrast
+to negative main effect of time. Figure 4 displays the disordinal patterns for the groups.
+HoS students’ views of relevance increase a small amount from the pre- to the postsurvey,
+whereas their peers’ perceptions of the relevance of science decreases (–.116**).
+Science Education, Vol. 99, No. 5, pp. 819–836 (2015)
+
+3.8
+3.6
+3.4
+score
+3.2
+Adjusted
+HoSstudents
+3
+... Non-HoS students
+2.8
+2.6
+2.4
+Pre
+Post832
+RIEGLE-CRUMB ET AL.
+Finally, while the main focus of our analyses was to assess differences between HoS and
+non-HoS students regarding changes in their science attitudes, our multivariate analyses
+revealed patterns consistent with prior research, namely that females are significantly less
+confident in their science ability and report significantly less enjoyment and more science
+anxiety than their male peers (Correll, 2001; Eccles, 1994). Our models also indicate some
+evidence of racial/ethnic differences in attitudes (see the models predicting confidence and
+anxiety), as well as differences by prior math achievement, as those with higher scores on
+the math portion of the SAT report significantly higher levels of confidence in their science
+ability and less anxiety. Given the differential distribution of HoS and non-HoS students on
+several of these covariates (see Table 1), including them in our models generally decreased
+the size of the difference between groups on the presurvey (e.g., the more male composition
+of the non-HoS group helped to account for their initially stronger math confidence), and
+also revealed larger differences between the groups on the postsurvey than could be detected
+in simpler models that did not account for these factors.
+DISCUSSION AND CONCLUSION
+The goal of this study was to focus on future elementary teachers’ views toward science
+at a point when they are still explicitly occupying the role of a learner, before they are tasked
+with assuming the role of a science teacher. Specifically, we investigated whether enrollment
+in HoS, a program of inquiry-based science content courses, promoted a favorable change
+in the science attitudes of a sample of over 200 preservice elementary teachers. Building
+on the theoretical framework advanced by van Aalderen-Smeets and colleagues (2012), our
+study examined changes in attitudes on multiple dimensions, and also utilized a comparison
+group of noneducation/nonscience majors to help contextualize the changes observed for
+our focal sample of elementary preservice teachers.
+Data analyses reveal a remarkably consistent and positive story for HoS students; students
+significantly changed their views toward science from the pre- to the postsurvey, such that
+after participating in inquiry-based content courses they reported more confidence in their
+skills as science learners, more enjoyment and less anxiety toward science, and perceived it
+as more relevant. Conversely, patterns for those in the comparison group revealed a decline
+in favorable attitudes toward science after enrolling in a traditional, lecture-based content
+course. Importantly, these results are independent of differences between HoS and non-HoS
+students (e.g., gender and SAT math score) that are associated with attitudes; therefore,
+our analyses indicates that it was differences in the courses that the two groups of students
+enrolled in, rather than characteristics of the individual students themselves, that led to
+changes in attitudes.
+Our study has several likely implications for the future classrooms of preservice teach-
+ers, as prior research suggests that teachers with negative views toward science may both
+socialize their young students to develop similar views, as well as offer less science instruc-
+tion in class due to a desire to avoid the subject (Bursal & Paznokas, 2006; Jarrett, 1999).
+Thus, to the extent that HoS students have more positive attitudes toward science as a result
+of inquiry-based instruction, we have positively intervened to disrupt the vicious cycle of
+elementary school teachers passing their negative views of science onto the next generation.
+Instead, future elementary students could ultimately be the beneficiaries, having teachers
+who are favorably inclined and excited about teaching science and therefore spend more
+time and focus on it.
+Additionally, we suggest that our results have potential implications for gender equity in
+the classroom, particularly because, while all observed changes for HoS students were in
+a favorable direction, the largest changes were for decreasing anxiety. Recent research by
+Science Education, Vol. 99, No. 5, pp. 819–836 (2015)
+
+DO INQUIRING MINDS HAVE POSITIVE ATTITUDES?
+833
+Beilock et al. (2010) offered evidence that the math anxiety of female elementary teachers
+had a negative impact on their female students in particular. The authors argue that due
+to the inclination for children to more strongly connect to adults of the same gender as
+role models, young girls in the classroom were more susceptible to teachers’ math anxiety,
+and consequently exhibited more negative attitudes of their own as well as lower math
+achievement. Their study provides powerful evidence that teacher role-modeling can be a
+key factor that leads to the development of the gender gap in math as early as elementary
+school (Beilock et al., 2010). Such a pattern can be logically extended to science anxiety;
+thus by intervening to decrease the anxiety of (predominantly female) preservice elementary
+teachers, more young girls could have the opportunity to interact with a positive female role
+model, thereby thwarting emerging gender disparities in children’s views and performance
+(Eccles, 1994).
+Our study also has potential ramifications for the type of instruction and classrooms
+experienced by future elementary students. As research suggests that preservice teachers
+tend to teach their students in ways similar to how they were taught (Gess-Newsome &
+Lederman, 1999), programs such as HoS can serve as a powerful model for implement-
+ing inquiry in their own classrooms, and thus ultimately contribute to greater uptake of
+recommended science reform. The Next Generation Science Standards call for elementary
+teachers to engage their students in inquiry-based practices, as well as emphasize disci-
+plinary core ideas in the classroom (National Research Council, 2012b). The HoS classes
+offer preservice teachers the critical opportunity to develop an understanding of and real
+experience with inquiry-based teaching and learning that is consistent with these standards.
+We concur with Volkmann et al. (2005), who argue that “if learning through inquiry is to
+become a reality in today’s schools, then university science courses must model inquiry so
+that pre-service teachers may experience it” (p. 867). Programs such as HoS have the power
+to do exactly this, and thereby help break the cycle of teacher-centered didactic instruction.
+While the primary focus of this study is the educational experiences of preservice teachers
+during college and the subsequent implications for future elementary classrooms, our study
+also speaks to the need to improve undergraduate science education more broadly. A recent
+meta-analysis of student academic performance in STEM undergraduate courses provides
+evidence that traditional lecture formats lead to higher failure rates and lower achievement
+when compared to classes that are more constructivist based (Freeman et al., 2014 ). Our
+study focuses on attitudinal rather than performance outcomes, and in doing so heeds recent
+calls by the National Research Council to examine a more comprehensive range of student
+outcomes at the postsecondary level (National Research Council, 2012). Specifically, we
+find that, even after adjusting for differences in social and academic background between
+our two groups (preservice education students and noneducation/nonscience students),
+enrollment in traditional lecture classes has the opposite effect of enrolling in inquiry-
+based content classes. We suggest that the decline in confidence, affect, and utility, as well
+as a slight increase in anxiety that we observed for students in traditional lecture-based
+science content classes, are important consequences of their relatively low engagement in
+the classroom, and perhaps linked to patterns of lower performance documented elsewhere
+(Freeman et al., 2014; Seymour & Hewitt, 1997).
+As with any study, ours has limitations. First, we note a lack of parallel time frames for
+our focal HoS students (two semesters between pre- and postsurveys) and our comparison
+group (one semester between pre- and postsurveys). Therefore, it is possible that part of the
+positive change in attitudes observed for HoS students is due to the longer exposure period;
+while we cannot dismiss this possibility entirely we are nevertheless skeptical that a shorter
+survey window for HoS students would have substantively changed our findings regarding
+opposite directions of change for the two groups. Indeed, we did collect postsurveys for a
+Science Education, Vol. 99, No. 5, pp. 819–836 (2015)
+
+834
+RIEGLE-CRUMB ET AL.
+small subsample of HoS students at the end of the first semester, and the results, although
+slightly weaker in magnitude, were statistically significant and in the same positive direction
+as the full sample of HoS students included here.
+Additionally, we did not collect data to assess what features of these inquiry-based
+classes students found most favorable; for instance, it could be that the significant amount
+of time spent on group work was a particularly influential factor leading to their change
+in attitudes (Park Rogers & Abell, 2008). We think this is an important area for future
+research to address, to better ascertain which aspects of inquiry-based classrooms are most
+effective at promoting favorable shifts on different dimensions of science attitudes. More
+long-term studies are also needed to assess whether and how the potential implications we
+discuss above come to fruition and make a difference for elementary students in the science
+classroom.
+Finally, it is important to point out that designing and implementing inquiry-based science
+content courses that depart from the typical lecture-based format of most postsecondary
+instruction is certainly not without its challenges and difficulties (Allen & Tanner, 2005;
+Armbruster, Johnson, & Weiss, 2009). One such obstacle is convincing science instructors
+and university administrators of the benefits of inquiry-based instruction for their students.
+Our study contributes to the small number of studies that offer robust empirical evidence
+on this topic (Seymour, 2002), and in doing so offers additional support to the call to
+implement inquiry-based science instruction at all levels and for all students.
+This research was supported by a grant from the National Science Foundation (NSF DUE 0942943,
+PI: Sacha Kopp), and a grant from the Eunice Kennedy Shriver National Institute of Health and Child
+Development (5 R24 HD042849) awarded to the Population Research Center at The University of
+Texas at Austin. Opinions reflect those of the authors and do not necessarily reflect those of the
+granting agencies.
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+

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+Addressing Cosmological Tensions by Non-Local Gravity
+Filippo Bouchè,1, a Salvatore Capozziello,1, 2, 3, b and V. Salzano4, c
+1Dipartimento di Fisica “E. Pancini”, Università di Napoli “Federico II", Via Cinthia 21, I-80126, Napoli, Italy
+2DipaScuola Superiore Meridionale, Largo S. Marcellino 10, I-80138, Napoli, Italy
+3Istituto Nazionale di Fisica Nucleare, Sez. di Napoli, Via Cinthia 21, I-80126, Napoli, Italy
+4Institute of Physics, University of Szczecin, Wielkopolska 15, 70-451 Szczecin, Poland
+(Dated: January 5, 2023)
+Alternative cosmological models have been under deep scrutiny in recent years, aiming to address
+the main shortcomings of the ΛCDM model. Moreover, as the accuracy of cosmological surveys
+improved, new tensions have risen between the model-dependent analysis of the Cosmic Microwave
+Background and lower redshift probes. Within this framework, we review two quantum-inspired
+non-locally extended theories of gravity, whose main cosmological feature is a geometrically driven
+accelerated expansion. The models are especially investigated in light of the Hubble and growth
+tension, and promising features emerge for the Deser–Woodard one. On the one hand, the cosmolog-
+ical analysis of the phenomenological formulation of the model shows a lowered growth of structures
+but an equivalent background with respect to ΛCDM. On the other hand, the study of the lensing
+features at the galaxy cluster scale of a new formulation of non-local cosmology, based on Noether
+symmetries, makes room for the possibility of alleviating both the H0 and σ8 tension. However, the
+urgent need for a screening mechanism arises for this non-local theory of gravity.
+I.
+INTRODUCTION
+Recent astrophysical and cosmological surveys, both from ground-based and space experiments, have provided
+extremely high-quality data. The observations point towards a Universe in which the cosmological principle holds
+on large scales, namely the Universe appears homogeneous and isotropic if averaged over scales of ∼100 h−1Mpc or
+more [1, 2]. Moreover, the Universe is undergoing an accelerated expansion phase [3, 4], subsequent to a decelerated
+era in which the structure formation occurred. All these features, together with the nuclei abundances produced in
+the Big Bang Nucleosynthesis (BBN), the Baryon Acoustic Oscillations (BAO) and many others, are well predicted
+by the Lambda Cold Dark Matter (ΛCDM) model, which has been adopted as the standard cosmological model
+accordingly. Such a model provides an effective description of the Universe, which relies on the assumption that
+General Relativity (GR) is the final theory of gravitation that governs the cosmic dynamics. As a consequence, two
+more fluids other than baryonic matter and radiation must be inserted in the matter–energy content of the Universe
+in order to adjust GR predictions to data: the Dark Energy (DE), responsible for the late time cosmic acceleration,
+and the Dark Matter (DM), which accounts for the structure formation. Together, they should represent ∼95% of
+the matter–energy budget of the Universe [5–7], thus dominating the cosmic dynamics at all scales.
+Building on its capability to fit the whole cosmological and astrophysical dataset with a relatively small number of
+parameters, the ΛCDM model stands as the pillar of our comprehension of the Universe. However, several shortcomings
+[8] and recently risen tensions [9–11] affect its reliability. On the one hand, we have a huge assortment of candidates
+but no final solution for DM [12, 13] and DE [14–16]. On the other hand, the presence of singularities, as well as the
+inconsistency at quantum level, undermine the credibility of GR as the final theory of gravity. Even more puzzling
+are the cosmological tensions, which have emerged in recent years as the result of the growing availability of a wide
+range of extremely precise data. A multitude of independent observations appears indeed to be in a ≳2σ tension
+with the reference ΛCDM estimates by the Planck collaboration [5]. Even though systematic experimental errors may
+account for part of these tensions, their statistical significance and their persistence after several check analyses have
+thrown up some serious red flags. Since the Planck constraints for the cosmological parameters relies on a strongly
+ΛCDM-model-dependent analysis of the Cosmic Microwave Background (CMB), such tensions may be the signature
+of the brakedown of the concordance model, hence of new physics. In this paper, we pay particular attention to the
+two most well-known tensions: the H0 tension [10], which emerges from the comparison between early-time [5] and
+late-time measurements [17] of the Hubble constant, and the growth tension [11] between the CMB value [5] of the
+aElectronic address: filippo.bouche-ssm@unina.it
+bElectronic address: capozziello@na.infn.it
+cElectronic address: vincenzo.salzano@usz.edu.pl
+arXiv:2301.01503v1  [astro-ph.CO]  4 Jan 2023
+
+2
+cosmological parameters ΩM and σ8 and those from lower redshift probes, such as Weak Lensing (WL) [18], Cluster
+Counts (CC) [19] and Redshift Space Distortion (RSD) [20].
+In order to meet the challenges posed by ΛCDM theoretical shortcomings, as well as by its observational tensions, a
+zoo of alternative cosmological models has been formulated in recent years. The common feature of any proposed model
+is the introduction of additional degrees of freedom, whether in the gravitational or the matter–energy Lagrangian.
+Several approaches have been adopted: from the simple generalization of the Hilbert–Einstein action to functions
+of the curvature scalar, namely f(R) theories [21–27], to the addition of further geometric invariants such as the
+torsion scalar T [27–32] or the Gauss–Bonnet scalar G [33–36]. The introduction of scalar/vector fields minimally or
+non-minimally coupled to gravity [37–41], as well as the emergence of non-trivial dynamics in the dark sector [42–46],
+also represent intriguing possibilities in the extremely wide framework of the alternatives to GR/ΛCDM (see [47, 48]
+for the state of the art). In this paper, we want to inquire into a specific class of alternative cosmological models
+ruled by non-local gravitational interactions [49]. Among others, we investigate the cosmological implications of two
+non-locally extended theories of gravity: the Deser–Woodard (DW) model [50] and the Ricci-Transverse (RT) model
+[51]. These theories have drawn increasing attention in recent years due to their capability to account for late-time
+cosmic acceleration, thus avoiding the introduction of any form of unknown dark energy. Moreover, the non-local
+corrections may provide a viable mechanism to alleviate some of the main cosmological tensions.
+The paper is organized as follow: in Section II, we outline the main motives for formulating non-local theories
+of gravity, and we present the two ways in which dynamical non-locality can be implemented. Then, we introduce
+the two chosen models and their theoretical features. In Section III, we investigate the mechanisms through which
+the DW and the RT model account for the accelerated expansion of the Universe. In Section IV, we present the
+non-locally driven evolution of cosmological perturbations for the two models and the resulting impact on the σ8
+tension. Moreover, in Section V, we assess the H0 tension in light of the non-local theories. Finally, in Section VI,
+we present the main astrophysical tests of the non-local gravity models. The conclusions are drawn in Section VII.
+II.
+NON-LOCAL GRAVITY
+Non-locality naturally emerges in Quantum Physics, both as a kinematical and a dynamical feature. On the other
+hand, locality is a key property of classical field theories, and thus represents one of the main obstacles to overcome
+in order to merge gravitational interaction formalism with that of Quantum Field Theory (QFT). As a consequence,
+the introduction of non-locality in our theory of gravitation seems to be an unavoidable step towards the unification
+of the fundamental interactions.
+There exist at least two ways to achieve non-locality: at fundamental level, in
+which kinematical non-locality can be implemented by discretizing spacetime and introducing a minimal length scale
+(usually the Planck length); as an effective approach, in which non-local geometrical operators can be added to the
+gravitational Lagrangian to obtain a non-local dynamics in a continuum background spacetime [52]. Here, we want
+to focus on the latter scenario, which is of great interest for cosmological applications.
+Two main classes of non-locally extended theories of gravity have been developed in recent years [49]: Infinite
+Derivative theories of Gravity (IDGs), involving entire analytic transcendental functions of a differential operator,
+and Integral Kernel theories of Gravity (IKGs) based on integral kernels of differential operators, such as
+□−1R(x) =
+�
+d4x′G(x, x′)R(x′) ,
+(2.1)
+where G(x′x′) is the Green function associated to the inverse d’Alembertian. IDGs usually address the ultraviolet (UV)
+problems of the ΛCDM model by ensuring classical asymptotic freedom. The gravitational interaction is weakened
+on small scales and the singularities disappear accordingly. Non-singular black holes [53], as well as inflationary [54]
+and bouncing cosmologies [55], are indeed forecast in the IDG framework. On the other hand, IKGs are introduced
+to account for the infrared (IR) shortcomings of the concordance model of cosmology. The phenomenology of both
+dark fluids can be actually reproduced by non-local corrections that switch on at large scales [50, 56, 57].
+In this paper, we focus on two specific curvature-based IKGs [50, 51] and their cosmological features. IKGs indeed
+have special relevance due to the fact that they combine suitable cosmological behavior with well-justified Lagrangians
+at the fundamental level. GR is actually plagued by quantum IR divergences that already appear for pure gravity in
+flat space [58]. This pathological behavior implies that the long-range dynamics of the gravitational interaction may
+be non-trivial, and non-perturbative techniques are thus required. Applying such non-perturbative methods to the
+renormalization of the quantum effective action of the gravity theory, non-local terms emerge both associated [59, 60]
+or not associated [61, 62] to a dynamical mass scale. Analogous results can be recovered through the trace anomaly
+[63].
+
+3
+A.
+The Deser–Woodard Model
+The first model that we want to highlight is an IKG initially proposed in [50]. The non-locally extended gravitational
+action of the Deser–Woodard model reads
+S =
+1
+16πG
+�
+d4x√−g
+�
+R
+�
+1 + f(□−1R)
+��
+,
+(2.2)
+where the non-local correction is given by the so-called distortion function, namely a general function of the inverse
+box of the Ricci scalar, as in Equation (2.1).It is worth noticing that the non-local theory reduces to GR as soon as
+f(□−1R) vanishes. The modified field equations descending from Equation (2.2) are
+Gµν + ∆Gµν = κT (m)
+µν
+,
+(2.3)
+where the non-local correction reads
+∆Gµν =
+�
+Gµν + gµν□ − ∇µ∇ν
+��
+f
+�
+□−1R
+�
++ □−1�
+Rf ′�
+□−1R
+���
++
+�
+1
+2
+�
+δα
+µδβ
+ν + δβ
+µδα
+ν
+�
+− 1
+2gµνgαβ
+�
+∂α
+�
+□−1R
+�
+∂β
+�
+□−1�
+Rf ′�
+□−1R
+���
+.
+(2.4)
+Furthermore, the non-local gravitational action in Equation (2.2) can be easily rewritten under the standard of
+local scalar–tensor theories by introducing an auxiliary scalar field
+R(x) = □η(x) ,
+(2.5)
+which does not carry any independent degree of freedom.
+The local canonical form of the scalar–tensor action,
+equivalent to the non-local theory, thus reads [64]
+S = 1
+2κ
+�
+d4x√−g
+�
+R
+�
+1 + f
+�
+η
+��
+− ∂µξ∂µη − ξR
+�
+,
+(2.6)
+where ξ(x) is a Lagrangian multiplier which has been promoted to a position- and time-dependent scalar field. In this
+formulation, the gravitational field equation is
+Gµν =
+1
+1 + f(η) − ξ
+�
+κT (m)
+µν
+− 1
+2gµν∂αξ∂αη + 1
+2
+�
+∂µξ∂νη + ∂µη∂νξ
+�
+−
+�
+gµν□ − ∇µ∇ν
+��
+f(η) − ξ
+�
+�
+,
+(2.7)
+while the Klein–Gordon equations for the two auxiliary scalar fields are
+□η = R ,
+(2.8)
+□ξ = −R∂f(η)
+∂η
+.
+(2.9)
+B.
+The Ricci-Transverse Model
+The second non-local model that we investigate through this paper is a metric IKG proposed in [51]. This is a
+quantum-inspired model, whose quantum effective action is
+Γ =
+1
+64πG
+�
+d4x
+�
+hµνEµν,αβhαβ − 2
+3m2�
+P µνhµν
+�2
+�
+,
+(2.10)
+
+4
+where gµν = ηµν+κhµν is the linearized metric tensor, Eµν,αβ is the Lichnerowicz operator1, P µν = ηµν−(∂µ∂ν/□) is a
+projector operator and m is the mass of the conformal mode of the gravitational field. Performing the covariantization
+of Equation (2.10), the modified gravitational field equation reads
+Gµν − 1
+3m2(gµν□−1R)T = κTµν ,
+(2.11)
+where we take the transverse part of the symmetric non-local tensor Sµν = gµν□−1R
+Sµν = ST
+µν + 1
+2(∇µSν + ∇νSµ) ,
+(2.12)
+and Sµ is an associated four-vector. The Bianchi identities are guaranteed accordingly, i.e., ∇µST
+µν = 0.
+In the same way as the DW model, the Ricci-Transverse model can be localized through a scalar–tensor–vector
+formulation [51, 65]. Here, we introduce two auxiliary objects, namely
+U(x) = −□−1R(x) ,
+Sµν(x) = −U(x)gµν(x) = gµν(x)□−1R(x) .
+(2.13)
+An auxiliary four-vector field Sµ(x) therefore enters the localized equations because of Equation (2.12).
+The
+gravitational field equation, Equation (2.11), turns into
+Gµν + m2
+6
+�
+2Ugµν + ∇µSν + ∇νSµ
+�
+= κTµν
+(2.14)
+plus the two equations of motions of the two auxiliary fields
+□U = −R ,
+(2.15)
+�
+δµ
+ν □ + ∇µ∇ν
+�
+Sµ = −2∂νU .
+(2.16)
+III.
+THE LATE-TIME COSMIC ACCELERATION
+The Universe is currently undergoing an accelerated expansion. The first evidence of this peculiar behavior dates
+back to the end of the twentieth century, when the observation of several Type Ia Supernovae (SNIa) [3, 4] pointed out
+the unavoidable necessity of a cosmological constant to fit the cosmic expansion history. On the one hand, these results
+have been corroborated by the observations of all the recent surveys [5, 66–68]. On the other hand, the theoretical
+explanation of this issue has two main drawbacks: the fine tuning problem [69] and the coincidence problem [70]. The
+former is related to the huge discrepancy (∼120 orders of magnitude) between the observed value of the cosmological
+constant and the vacuum energy density calculated via QFT. The latter is linked with the similar current values of
+ΩΛ and ΩM, despite their radically different evolution laws.
+The next generation of cosmological surveys should boost the investigation of the nature of the so-called cosmological
+constant, providing powerful data to discriminate between DE solutions and extended theories of gravity. Within this
+framework, non-local gravity provides viable mechanisms that could account for the observed accelerated expansion
+of the Universe.
+1 Eµν,αβ = 1
+2 (ηµρηνσ + ηµσηνρ − 2ηµνηρσ)□ + (ηρσ∂µ∂ν + ηµν∂ρ∂σ) − 1
+2 (ηµρ∂σ∂ν + ηνρ∂σ∂µ + ηµσ∂ρ∂ν + ηµσ∂ρ∂µ)
+
+5
+A.
+The DW Case: Delayed Response to Cosmic Events
+The main reason why the DW model has been in the spotlight since its formulation is its effective way to explain
+late-time cosmic acceleration without the introduction of any form of dark energy. Computing the non-local correction
+of Equation (2.2) in the Friedmann–Lemaitre–Robertson–Walker (FLRW) metric,
+ds2 = −dt2 + a2(t)d⃗x · d⃗x ,
+(3.1)
+one obtains a non-negligible geometrical contribution,
+�
+□−1R
+�
+(t) =
+� t
+0
+dt′
+1
+a3(t′)
+� t′
+0
+dt′′a3(t′′)R(t′′) = −6s(2s − 1)
+3s − 1
+�
+�ln
+�
+t
+teq
+�
+−
+1
+3s − 1 +
+1
+3s − 1
+�
+teq
+t
+�3s−1�
+� ,
+(3.2)
+where a(t) ∼ t s, and the integration constant is set to make the non-local correction vanish before the radiation–
+matter equivalence time, teq. Then, for t > teq the non-local correction starts to grow, becoming non-negligible at
+late time and driving the accelerated cosmic expansion. Non-locality thus emerges in the cosmological framework as
+a delayed response to the radiation-to-matter dominance transition, i.e., as a time-like non-local effect.
+The introduction of non-locality may therefore have beneficial effects on cosmological scales, both at background
+level and perturbations level. Two different approaches can be adopted for the DW model: on the one hand, one can
+exploit the freedom guaranteed by the undetermined form of the distortion function to fit the observed expansion
+history of the universe. In such a scenario, the DW cosmology is made equivalent to ΛCDM at the background
+level. However, different features emerge when perturbations are taken into account, and the physics of structure
+formation is affected accordingly. On the other hand, one can select the form of the distortion function building on
+some fundamental principles, such as the Noether symmetries of the system. In this case, the non-local cosmology is
+also modified at background level and stronger deviations from the concordance model should rise.
+In [71], the first method has been applied, and the ΛCDM expansion history of the Universe has been accurately
+reproduced by matching the data through a non-trivial form of the distortion function
+f(□−1R) = 0.245
+�
+tanh
+�
+0.350X + 0.032X2 + 0.003X3�
+− 1
+�
+,
+(3.3)
+where X = □−1R + 16.5.
+B.
+The RT Case: Dynamical Dark Energy
+Considering a spatially flat FLRW metric, Equation (3.1), the equations of motion of the RT model, Equa-
+tions (2.14)–(2.16), become [72]
+H2 − m2
+9 (U − ˙S0) = 8πG
+3
+ρ ,
+(3.4)
+¨U + 3H ˙U = 6 ˙H + 12H2 ,
+(3.5)
+¨S0 + 3H ˙S0 − 3H2S0 = ˙U ,
+(3.6)
+where the spatial components of the vector field Sµ vanish to preserve the rotational invariance of the FLRW metric,
+and the stress–energy tensor is taken to be T µ
+ν
+=diag(−ρ, p, p, p).
+Defining Y = U − ˙S0, ˜h = H/H0 and the
+dimensionless variable x ≡ ln a(t), then the modified Friedmann equation, Equation (3.4), reads
+˜h2(x) = ΩMe−3x + γY (x) ,
+(3.7)
+
+6
+with γ ≡ m2/(9H2
+0). An effective dark energy thus appears
+ρDE(t) = ρ0γY (t) ,
+(3.8)
+where ρ0 = 3H2
+0/(8πG). Once the initial conditions for the auxiliary fields are set (see [56] for details), the evolution
+of ρDE(t)/ρT OT (t) can be studied: the non-local effective dark energy is actually negligible until recent time and then
+starts to dominate the cosmic expansion. Moreover, it is possible to study the DE equation of state
+˙ρDE + 3H(1 + ωDE)ρDE = 0 ,
+(3.9)
+and different evolutions for ρDE(z) follow from different choices for the initial conditions of the auxiliary fields. For
+small values of the initial conditions, one obtains a fully phantom DE, namely ωDE(z) is always less than −1. For
+large values of the initial conditions, ρDE(z) has a "phantom crossing" behavior, i.e., there is a transition from the
+phantom regime to −1 < ωDE < 0 of about z ≃ 0.3. Regardless of the initial conditions, therefore, the non-local
+model provides a dynamical DE density which drives the accelerated expansion of the Universe.
+IV.
+THE GROWTH OF PERTURBATIONS AND THE σ8 TENSION
+Building on the primordial density fluctuations emerged from the inflation, cosmic structures have formed due
+to gravitational instability. Studying the large-scale structure of the Universe and its evolution through the cosmic
+epochs, it is possible to trace the growth of the so-called cosmological perturbations.
+Associated to this observable, one of the main cosmological tensions has risen: the growth tension. It has come
+about as the result of the discrepancy between the Planck value of the cosmological parameters ΩM and σ8 and those
+from WL measurements, CC and RSD data. The former dynamical probes point towards lower values of the amplitude
+(σ8) or the rate (fσ8 = [ΩM(z = 0)]0.55σ8) of growth of structures with respect to the CMB experiments, giving rise to
+a 2−3σ tension [9, 11]. Moreover, the Planck 2018 value of the joint parameter S8 = σ8
+�
+ΩM/0.3 (S8 = 0.834±0.016
+[5]) is confirmed by another recent CMB analysis by the ACT + WMAP collaboration (S8 = 0.840 ± 0.030 [66]),
+thus erasing the possibility of a systematic error related to the excess of lensing amplitude measured by Planck [73].
+A 2.3σ tension emerges accordingly with both the original WL analysis of KiDS-450 [74] and KiDS-450 + VIKING
+data [75], while updated constraints from the same datasets [76, 77] show greater discrepancies.
+The same 2.3σ
+tension also occurs with the data from DES’s first year release (DESY1) [78], while the combination of KiDS-450 +
+VIKING + DESY1 weak lensing datasets results in a 2.5σ [79] or 3.2σ [80] tension depending on the analysis. The
+most recent cosmic shear data release from both KiDS-1000 and DESY3 confirms the previous estimates [18, 81–84]
+(S8 = 0.759+0.024
+−0.021 from KiDS-1000 [18]). Analogous results have been obtained with the 3 × 2 pt correlation function
+analysis (cosmic shear correlation function, galaxy clustering angular auto-correlation function and galaxy–galaxy
+lensing cross-correlation function) of KiDS-1000 + BOSS + 2dFLenS datset [85]. Additional results, in agreement
+with those from WL surveys, have been achieved by number counting of galaxy clusters, using multiwavelength
+datasets [19, 86–91]. Supplementary observational evidence for the weaker growth of structures is also given by the
+exploitation of RSD data [20, 92–95].
+A.
+The Deser–Woodard Evolution of Scalar Perturbations
+To investigate the growth of structures in the non-local DW model, we select the phenomenological form of the
+distortion function given by Equation (3.3). As a consequence, the non-local background evolution is made equivalent
+to that of ΛCDM, and any deviation is enclosed in the cosmological perturbations.
+Consider the field equations of the scalar–tensor equivalent of the DW model, Equations (2.7)–(2.9). Specializing
+to the cosmological case by assuming the FLRW metric, Equation (3.1), the field equations now read
+H2�
+1 + f − ξ
+�
++ H
+�
+f ′ ˙η − ˙ξ
+�
+− 1
+6 ˙η ˙ξ = 8πG
+3
+ρ ,
+(4.1)
+˙H
+�
+1 + f − ξ
+�
+− H
+2
+�
+f ′ ˙η − ˙ξ
+�
++ 1
+2 ˙η ˙ξ + 1
+2
+�
+f ′′ ˙η2 + f ′¨η − ¨ξ
+�
+= −4πG(p + ρ) ,
+(4.2)
+
+7
+¨η + 3H ˙η = −6
+� ˙H + 2H2�
+,
+(4.3)
+¨ξ + 3H ˙ξ = 6f ′� ˙H + 2H2�
+,
+(4.4)
+where the former two are the (0,0) and the (1,1) component of Equation (2.7), while the latter two are the cosmological
+formulation of Equations (2.8) and (2.9).
+The linear perturbation equations have been derived in [96] for the scalar–tensor equivalent of the non-local theory,
+and then analogous results have been found in [97] for the original formulation. Using the perturbed FLRW metric
+in the Newtonian gauge,
+ds2 = −(1 + 2Ψ)dt2 + a2(t)(1 + 2Φ)δijdxidxj ,
+(4.5)
+the growth equation reads
+¨δM + (2 − ξ) ˙δM =
+3H2
+0
+�
+1 − ξ − 8f ′(η) + f(η)
+�
+Ω0
+M
+2a3H2�
+1 − ξ − 6f ′(η) + f(η)
+��
+1 + f(η) − ξ
+� δM ,
+(4.6)
+where δM = δρM/ρM is the matter density perturbation in the sub-horizon limit.
+Numerical results for the growth rate fσ8 ≡ σ8δ′
+M/δM have been obtained in [96] and [97] for both the formulations
+of the Deser–Woodard model, and good agreement with the Redshift Space Distortion (RSD) data has emerged
+(σNL
+8
+= 0.78). Moreover, when the DW cosmological parameters are inferred by matching the CMB data [98], a lower
+growth amplitude with respect to that of ΛCDM turns out. The non-local model thus alleviates the growth tension,
+predicting compatible values of σ8 both from Planck–CMB and the other dynamical probes, as shown in Figure 1.
+However, even though the non-local clustering of linear structures is weakened with respect to ΛCDM, and the DW
+prediction for the matter power spectrum is about 10% lower [98], the non-local lensing response is counterintuitively
+enhanced due to a severe increase in the lensing potential. This peculiar behavior results in a slight tension between
+CMB and RSD [98]: performing the joint fit, the RSD dataset tends to push the DW predictions for the CMB lensing
+potential Cφφ
+ℓ
+out of the 1σ error bars at low-ℓ. Applying the Bayesian tools for the model selection, a “weak evidence”
+[103] for the ΛCDM model consequently emerges.
+B.
+The Ricci-Transverse Evolution of Scalar Perturbations
+The scalar perturbations of the RT model have been investigated in [56, 104], using the FLRW metric in the
+Newtonian gauge, Equation (4.5), and perturbing the auxiliary fields as
+U(t, x) = ¯U(t) + δU(t, x) ,
+(4.7)
+Sµ(t, x) = ¯S0(t) + δSµ(t, x) = ¯S0(t) + δS0(t, x) + ∂i
+�
+δS(t, x)
+�
+,
+(4.8)
+where the spatial part of the vector perturbation does not vanish and, for scalar perturbations, only depends on
+δS. Building on the RT cosmological equations Equations (3.4)–(3.6), the growth equation for the matter density
+perturbation in the sub-horizon limit reads [104]
+¨δM + 2H ˙δM = 3
+2
+Geff
+G
+H2
+0ΩMδM ,
+(4.9)
+where in Geff
+�
+Ψ, Φ, δU, δS0, S
+�
+is encoded the deviation of the non-local theory from GR. In the sub-horizon modes,
+namely ˆk ≫ 1, such deviation is
+
+8
+0.70
+0.75
+0.80
+0.85
+S8
+CMB Planck (DW non-local)
+CMB Planck + RSD + JLA (DW non-local)
+CMB Planck + BAO + Pantheon (RT-minimal non-local)
+CMB Planck + BAO + Pantheon (RT non-local, ΔN=64)
+CMB Planck TT,TE,EE + lowE
+CMB ACT + WMAP
+WL KiDS-1000
+WL DES-Y3
+WL CFHTLenS
+WL KiDS + VIKING + DES-Y1
+WL + CMB lensing DES-Y3 + SPT + Planck
+WL + GC KiDS-1000 3×2pt
+WL + GC DES-Y3 3×2pt
+WL + GC KiDS + VIKING-450 + BOSS
+GC BOSS + eBOSS
+GC BOSS power spectra
+GC + CMB lensing DESI + Plank
+CC AMICO KiDS-DR3
+CC SDSS-DR8
+CC Planck tSZ
+RSD + BAO + Pantheon
+RSD
+RSD
+0.70
+0.75
+0.80
+0.85
+S8
+FIG. 1: Estimates of S8 provided by the two non-local cosmological analyses [56, 98], and the ΛCDM fit of the CMB [5, 66],
+the WL data [18, 20, 80, 83, 84, 99], the combination of WL and galaxy clustering observations [100–102], cluster counting
+[19, 87, 88] and RSD surveys [92, 94]. The colored band corresponds to the S8 value derived by the analysis of the Planck–CMB
+data in the ΛCDM framework [5].
+1 − Geff
+G
+= O
+� 1
+ˆk2
+�
+,
+(4.10)
+and the RT model is thus safe regarding the time variation of the effective Newton’s constant. Geff indeed reduces
+to G at the Solar System scale, while a deviation of ∼1% rises at cosmological scales.
+In [56], the growth rate f(z, k) ≡ d ln δM/d ln a is also derived. The results do not differ from those of ΛCDM
+cosmology: f(z, k) can be fitted with a k-independent function f(z) = [ΩM(z)]γ, where γ ≃ 0.55 is roughly constant.
+Accordingly, any possible deviation in the growth of perturbations should be due to the amplitude σ8. In order to find
+any signature of the non-local model at perturbation level, which could account for the growth tension, the theory was
+compared with cosmological observations: Planck–CMB, Pantheon SNIa and SDSS-BAO. The Bayesian parameter
+estimation shows a full equivalence between the RT non-local cosmology and the ΛCDM one. No statistically signifi-
+cant deviation in the σ8 parameter emerges for any of the tested versions of the Ricci-Transverse model. Eventually,
+this theory cannot alleviate the growth tension, as shown in Figure 1.
+V.
+HUBBLE TENSION IN LIGHT OF THE NON-LOCAL MODELS
+Hubble tension is certainly the most renowned and significant tension of the ΛCDM model. It emerges from the
+comparison between early-time and late-time measurements of the Hubble constant. From one side, CMB analysis
+[5, 66, 105–108], BAO surveys [6, 101, 109, 110] with standard BBN constraints [111] and combinations of CMB,
+BAO, SNIa [112], RSD and cosmic shear data [78, 113, 114] point towards lower values of H0 (H0 = 67.4 ± 0.5 km
+s−1Mpc−1 from Planck 2018 [5]). On the other side, the local measurements based on standard candles prefer higher
+values for the Hubble constant [115] (H0 = 73.04 ± 1.04 km s−1Mpc−1 from SH0ES 2022 [17]). The main results are
+achieved by the SH0ES collaboration using Hubble Space Telescope observations: on the one hand, they analyzed
+
+9
+65
+70
+75
+80
+H0 ( km s-1 Mpc-1 )
+CMB Planck (DW non-local)
+CMB Planck + RSD + JLA (DW non-local)
+CMB Planck + BAO + Pantheon (RT-minimal non-local)
+CMB Planck + BAO + Pantheon (RT non-local, ΔN=64)
+CMB Planck
+CMB Planck + lensing
+SPT-3G CMB
+ACT + WMAP CMB
+Planck + SPT + ACT CMB
+BOSS correlation function + BAO + BBN
+BOSS power spectrum + BAO + BBN
+BOSS DR12 + BBN
+BAO + RSD
+LSS equivalence-time ruler
+LSS equivalence-time ruler + lensing
+SNIa-Cepheid
+SNIa-TRGB
+SNIa-Miras
+SneII
+SnIa-Cepheid + TD lensing
+Time-delay lensing
+Time-delay lensing
+Time-delay lensing + SLACS
+GW Standard Sirens
+GW Standard Sirens
+GW Standard Sirens
+Masers
+Masers
+Tully Fisher
+Surface Brightness Fluctuations
+0.70
+0.72
+0.74
+0.76
+0.78
+0.80
+S8
+FIG. 2: Estimates of H0 provided by the two non-local cosmological analyses [56, 98], and the ΛCDM fit of the CMB [5, 66,
+105, 107], the matter power spectrum combined with BAO [20, 110, 126] and RSD [127], the Large Scale Structure teq standard
+ruler [128, 129], the supernovae [17, 121, 124, 130, 131], the time-delay lensing [121, 132, 133], the gravitational waves [134–136],
+the water megamasers [125, 137], the Tully–Fisher relation [138] and the SBF [139]. The colored bands correspond to the H0
+estimates derived by the Planck–CMB analysis in the ΛCDM framework (purple) [5] and the SNIa–Cepheids analysis by SH0ES
+(orange) [17].
+SNIa data with distance calibration by Cepheid variables in the host galaxies [116, 117]; on the other hand, they
+targeted long-period pulsating Cepheid variables [17, 118], calibrating the geometric distance to the Large Magellanic
+Cloud, both from eclipsing binaries and parallaxes from the Gaia satellite [119, 120]. Moreover, other independent
+local measurement of H0 have been performed by using time delays between multiple images of strong lensed quasars
+[121, 122], the tip of the Red Giant Branch [123] and Miras (variable red giant stars) [124] with water megamaser as
+distance indicator [125]. All these measurements agree on higher values of the Hubble constant, thus generating a
+4 − 5σ tension with early-time model-dependent estimates.
+A.
+The Deser–Woodard Expansion History
+The stat-of-the-art investigation of the DW non-local model does not allow any attempt to address H0 tension. To
+make the model predictive, the form of the distortion function needs to be specified, and most of the analyses have
+been focused on the phenomenological ΛCDM form of Equation (3.3), until now (see [98] for the latest results). This
+choice implies that the DW cosmology is made equivalent to that of the concordance model at background level, and
+the same expansion history, as well as the same H0, are thus predicted (see Figure 2).
+However, another option is also available for the selection of the distortion function. In [57, 140, 141], a specific
+form of f(η) has been derived by exploiting the Noether symmetries [142] of a spherically symmetric background
+spacetime
+f(η) = 1 + e η .
+(5.1)
+The accurate cosmological analysis of this form of the DW model has yet to be carried out, but some results have
+already been achieved, such as exact solutions [49] and a phase-space view of solutions [143]. Furthermore, several
+
+10
+astrophysical tests have been performed on very different scales, and viable results turned out. In [141], the lensing
+properties of the galaxy clusters have been investigated in light of the exponential DW model, and a fully non-local
+regime with enhanced lensing strength has been highlighted. This feature clearly resembles the improved lensing
+response of the phenomenological form of the DW model, which is co-responsible for the lowered estimate of the σ8
+parameter. A promising insight upon the cosmological behavior of the non-local theory based on Equation (5.1) thus
+emerges. Therefore, this model may alleviate not only the growth tension but also the Hubble tension, since it also
+deviates from GR at the background level [144]. Further analysis of the exponential DW model should be carried out
+accordingly.
+B.
+The Ricci-Transverse Expansion History
+For what concerns the RT model, the most updated cosmological analysis is due to Belgacem et al. [56]. As we
+saw in Section IV, different versions of the non-local model, relying on different choices for initial conditions of the
+auxiliary fields, have been compared against Planck–CMB, Pantheon SNIa and SDSS-BAO data. Since the RT model
+has no freedom with regard to the functional form of the action, the theory cannot be adjusted to data, and no
+background equivalence to the ΛCDM cosmology can be established a priori. In such a scenario, the solution to the
+Hubble tension may thus be possible. However, the estimator tool defined in Equation (4.10), which accounts for
+the deviation from GR of the non-local theory, only shows little discrepancies. This manifests in the values of the
+cosmological parameters inferred via Markov Chain Monte Carlo (MCMC), which are almost equivalent to those of
+the concordance model. The only model that exhibits some slight discrepancies is the so-called "RT-minimal", which
+relies on the assumption that the auxiliary scalar field U(x) starts its evolution during the radiation dominance era.
+On the one hand, this model predict a non-vanishing value for the sum of neutrino masses, while the ΛCDM model
+and the other versions of the RT model show a marginalized posterior which is peaked in zero. On the other hand,
+the RT-minimal model provides a barely higher estimate of the Hubble constant, i.e., H0 = 68.74+0.59
+−0.51 km s−1Mpc−1.
+Accordingly, the Hubble tension is just reduced to ∼4σ, as shown in Figure 2.
+The RT model in its minimal setup thus provides a viable mechanism to account for the late-time cosmic acceleration,
+as well as for the inclusion of non-zero neutrino masses. However, the predictions for both the background evolution
+and the linear perturbations are too similar to that of the ΛCDM model, hence the cosmological tensions cannot be
+alleviated.
+VI.
+ASTROPHYSICAL TESTS OF NON-LOCAL GRAVITY
+As we saw, good cosmological behavior emerges for both of the non-local models, thus enabling the possibility to
+avoid the introduction of any form of unknown dark energy. In order to further investigate the viability of such models
+as alternatives to GR, it is then necessary to test the non-local predictions down to astrophysical scales. Moreover,
+an accurate investigation of the possible screening mechanisms should be performed, if necessary.
+A.
+Testing the Deser–Woodard Model by Galaxy Clusters, Elliptical Galaxies and the S2 Star
+The DW model has been tested on a wide range of astrophysical scales, from the galaxy clusters [141] to the stellar
+orbits around Sagittarius A* [140]. Most of the tests have been carried out for the exponential form of the non-local
+model, namely
+S = 1
+2κ
+�
+d4x√−g
+�
+R
+�
+2 + e η�
+− ∂µξ∂µη − ξR
+�
+,
+(6.1)
+where the distortion function has been picked out by exploiting the Noether Symmetry Approach [145]. The analyses
+presented are all performed in the post-Newtonian limit, hence the non-local gravitational and metric potential are
+φ(r) = − GMηc
+r
++ G2M 2
+2c2r2
+�
+14
+9 η2
+c + 18rξ − 11rη
+6rηrξ
+r
+�
+− G3M 3
+2c4r3
+�
+50rξ − 7rη
+12rηrξ
+ηcr + 16
+27η3
+c −
+2r2
+ξ − r2
+η
+r2ηr2
+ξ
+r2
+�
+,
+(6.2)
+ψ(r) = −GMηc
+3r
+− G2M 2
+2c2r2
+�
+2
+9η2
+c + 3rη − 2rξ
+2rηrξ
+r
+�
+,
+(6.3)
+
+11
+where ηc is set to 1 so as to recover GR in the limit of φ(r). The two length scales rη and rξ are the characteristic
+non-local parameters that define the scale at which the non-local gravity corrections become effective.
+The first test of this form of the DW model has been carried out in [140], where the weak field non-local predictions
+have been compared against the NTT/VLT observations of S2 star orbit [146]. Exploiting a modified Marquardt–
+Levenberg algorithm, the fit between the simulated orbit and the observed one has shown a slightly better agreement
+for the non-local model with respect to the Keplerian orbit. Moreover, some constraints have been set on the non-local
+length scales.
+A further test has been subsequently performed in [141], exploiting the CLASH lensing data from 19 massive clusters
+[147, 148]. The point-mass potentials of Equations (6.2) and (6.3) were extended to a spherically symmetric mass
+distribution, i.e., the Navarro–Frenk–White density profile, and the non-local predictions for the lensing convergence
+were achieved. Therefore, the MCMC analysis has highlighted two effective regimes in which the non-local model
+is able to match the observations at the same level of statistical significance as GR. In the high-value limit of the
+non-local parameters, the non-local model reduces to a GR-like theory, whose lensing strength is 2/3 of the standard
+one. In this scenario, the DW theory is thus able to fit the data at the cost of increased cluster mass estimates.
+On the other hand, approaching the low-value limit of the non-local length scales, the non-local corrections to the
+lensing potential become larger and comparable to the zeroth-order terms. In this regime, the non-local model is
+able to reproduce GR phenomenology, neither affecting the mass estimates nor the statistical viability of the model.
+Furthermore, when the non-local contributions becomes completely dominant, the non-local theory seems to be able
+to fit the lensing observations with extremely low cluster masses. Accordingly, an intriguing possibility to fit data
+with no dark matter emerges. Additional constraints on the non-local parameters were also derived.
+The most recent astrophysical test of the exponential-DW model was carried out in [57], using the velocity distri-
+bution of elliptical galaxies [149]. Computing the non-local velocity dispersion as a function of the galaxy effective
+radius, the empirical relation of the so-called Fundamental Plane has been recovered so as to constrain the non-local
+gravity parameters. The results of the fit highlight the possibility to recover the fundamental plane without the dark
+matter hypothesis, setting new constraints for rη and rξ.
+It is worth noticing, however, that the non-local Deser–Woodard model exhibits worrisome features at the scale of
+the Solar System. Indeed, in [150], it was demonstrated that the screening mechanism proposed by the same authors
+of the non-local model does not work. As a consequence, the DW model would show a time dependence of the effective
+Newton constant in the small-scale limit, and it would be ruled out by Lunar Laser Ranging (LLR) observations. This
+conclusion, however, seems to be too strong, since it is still not clear how an FLRW background quantity behaves
+when evaluated from cosmological scales down to Solar System ones, where the system decouples from the Hubble
+flow. In fact, a full non-linear time- and scale-dependent solution around a non-linear structure would be necessary.
+A number of proposals go in this direction, and the so-called Vainshtein mechanism [151] can be regarded as the
+paradigm to realize the screening. Basically, any screening mechanism requires a scalar field coupled to matter and
+mediating a fifth force which might span from Solar System up to cosmological scales. Since non-local terms can be
+“localized”, thus resulting in effective scalar fields depending on the scale, some screening mechanism could naturally
+emerge so as to solve the above reported problems.
+B.
+Testing the Ricci-Transverse Model by Solar System Observations and Gravitational Waves Detection
+The main astrophysical tests of the RT model are related to Solar System observations. As we saw in the previous
+sub-section, any theory that extends GR has to reduce to Einstein’s theory at small scales. However, this is highly
+non-trivial when additional degrees of freedom are included, and screening mechanisms involving non-linear features
+are required. The RT model, instead, smoothly reduces to GR already at linear level, and no vDVZ discontinuity
+arises when m → 0 [56]. Note that such results are valid both in the flat and the Schwarzschild spacetime. Moreover,
+the non-local model passes the LLR test about the time variation of the effective Newton constant [150]. As stated
+in Equation (4.10), indeed, the deviation parameter Geff reduces to GN as soon as the system’s characteristic scale
+decreases.
+Another non-local feature of the RT model that has been investigated is the deviation from GR of the predicted
+gravitational radiation [152, 153]. The RT model, similarly to some other extended theories of gravity such as f(R)
+gravity and DHOST theories, has survived the GW170817 event, which set a stringent constraint on the Gravitational
+Waves (GW) speed [154]. Indeed, this non-local model only modifies the friction term in the GW propagation equation,
+thus predicting a massless graviton. Moreover, neither the coupling with matter nor the gravitational interaction
+between the coalescing binaries are affected (the RT model reduces to GR at short distances), and the only difference
+will therefore be due to the free propagation of the GW from the source to the observer. The GW amplitude indeed
+undergoes a modified dampening in the non-local model
+
+12
+˜hA(η, k) ∼
+1
+dgw
+L (z) =
+�
+dem
+L (z) exp
+�
+−
+� z
+0
+dz′
+1 + z′ δ(z′)
+��−1
+,
+(6.4)
+where
+δ(η) = m2 ¯S0(η)
+6H(η) ,
+(6.5)
+and ˜hA(η, k) are the Fourier modes of the GW amplitude, with A = ×, + labeling the polarization. Computing
+the ratio between the non-local behavior given by Equation (6.4) and the GR behavior, ˜hA(η, k) ∼ 1/dem
+L (z), little
+deviation emerges for the RT-minimal model, while a 20−80% deviation manifests at large z for the RT formulations
+in which the auxiliary fields start their evolution during the de Sitter inflation. The more e-folds we consider between
+the onset of the auxiliary fields and the end of the inflation, the greater the deviation from GR. We can use a simple
+parametrization for the considered ratio
+dgw
+L (z)
+dem
+L (z) = Ξ0 + 1 − Ξ0
+(1 + z)n ,
+(6.6)
+where Ξ0 is the asymptotic value reached by the ratio and n is the rate at which Ξ0 is approached. Then,
+δ(z) =
+δ(0)
+1 − Ξ0 + Ξ0(1 + z)n ,
+(6.7)
+with δ(0) = n(1 − Ξ0), and the event GW170817 provided the following constraint for such a parameter [153]:
+δ(0) = −7.8+9.7
+−18.4.
+More stringent constraints will certainly be set with the next generation of GW detectors by
+exploiting the observations of GW events with electromagnetic counterparts.
+VII.
+CONCLUSIONS AND PERSPECTIVES
+In this paper, we reviewed the cosmological properties of two of the main proposals in the framework of the non-
+locally extended theories of gravity. In particular, we considered two metric IKGs that are inspired by quantum
+corrections and manifest a suitable cosmological behavior as well. Both the DW and RT models are able to reproduce
+the expansion history of the Universe, exhibiting a late-time accelerated expansion driven by the onset of the non-local
+corrections. The non-local extensions of the Hilbert–Einstein Lagrangian thus provide a viable mechanism to avoid
+the introduction of any form of unknown dark energy. Building on these appealing properties, we inquired into the
+chance of addressing the two main cosmological tensions, namely the σ8 and H0 tensions.
+On the one hand, the non-local DW model has shown suitable features towards this aim. The phenomenological
+formulation of the model indeed predicts a lowered amplitude of growth of perturbations, therefore solving the σ8
+tension. However, this model is made equivalent to the ΛCDM cosmology at the background level, hence no chance
+to account for the Hubble tension arises. Another formulation of the DW theory, based on the Noether symmetries
+of the system, may address both the tensions. This model lacks a proper cosmological analysis, but the investigation
+of its lensing properties at the galaxy clusters scale has shown the same features that, in the phenomenological DW
+model, allow the weakening of the growth of structures. Moreover, this formulation of the non-local theory deviates
+from GR also at background level, thus enabling the possibility to alleviate the Hubble tension as well. The model has
+been also tested on astrophysical scales, and substantial statistical equivalence to GR has emerged in very different
+systems, such as the S2 star, the elliptical galaxies and the galaxy clusters. The main drawback of the DW model,
+however, is the absence of an effective screening mechanism on small scales, which has to be further investigated.
+On the other hand, the non-local RT model perfectly reduces to GR at the Solar System scale, thus avoiding the
+necessity of non-trivial screening mechanisms. Accordingly, the model is not ruled out by the LLR test. Moreover, the
+non-minimal formulations of the RT model show a strong deviation from GR for what concerns the GW propagation
+at large redshift. A powerful tool to test the model with the next generation of GW detectors thus emerges. However,
+this non-local model is not able to address any of the cosmological tensions, as it mimics the ΛCDM evolution both
+at the background and linear perturbations level.
+
+13
+In view of the fact that the next generation of cosmological surveys are expected to provide sufficiently accurate
+data to reach a turning point in our comprehension of the Universe, it is of great interest to further investigate the
+main alternatives to GR. A complete cosmological analysis should especially be carried out for the non-local DW
+model in its formulation based on the Noether Symmetry Approach. This model indeed provides one of the most
+promising windows towards the solution of both the cosmological tensions and the dark energy problem. The large-
+scale structure especially appears as a privileged environment for testing the non-local models, since one of their main
+features is the emergence of characteristic length scales. However, it must be stressed that as long as no screening
+mechanism will be found for the DW model, its reliability will be compromised.
+Acknowledgments
+This article is based upon work from COST Action CA21136 Addressing observational tensions in cosmology
+with systematic and fundamental physics (CosmoVerse) supported by COST (European Cooperation in Science and
+Technology). FB and SC acknowledge the support of Istituto Nazionale di Fisica Nucleare (INFN), iniziative specifiche
+QGSKY and MOONLIGHT2.
+Appendix A: Abbreviations
+The following abbreviations are used in this manuscript:
+BBN
+Big Bang Nucleosynthesis
+BAO
+Baryon Acoustic Oscillations
+ΛCDM Lambda Cold Dark Matter
+GR
+General Relativity
+DE
+Dark Energy
+DM
+Dark Matter
+CMB
+Cosmic Microwave Background
+WL
+Weak Lensing
+CC
+Cluster Counts
+RSD
+Redshift Space Distortion
+DW
+Deser–Woodard
+RT
+Ricci-Transverse
+QFT
+Quantum Field Theory
+IDG
+Infinite Derivative Theory of Gravity
+IKG
+Integral Kernel Theory of Gravity
+UV
+UltraViolet
+IR
+InfraRed
+SNIa
+Type Ia Supernovae
+FLRW Friedmann–Lemaitre–Robertson–Walker
+MCMC Markov Chain Monte Carlo
+LLR
+Lunar Laser Ranging
+GW
+Gravitational Waves
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+

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+A NEW SYMMETRIC HOMOMORPHIC FUNCTIONAL
+ENCRYPTION OVER A HIDDEN RING FOR POLYNOMIAL PUBLIC
+KEY ENCAPSULATIONS
+Randy Kuang, Maria Perepechaenko, Ryan Toth
+Quantropi Inc.
+Ottawa, Canada
+{randy.kuang, maria.perepechaenko, ryan.toth}@quantropi.com
+ABSTRACT
+This paper proposes a new homomorphic functional encryption using modular multiplications over
+a hidden ring. Unlike traditional homomorphic encryption where users can only passively perform
+ciphertext addition or multiplication, the homomorphic functional encryption retains homomorphic
+addition and scalar multiplication properties, but also allows for the user’s inputs through polynomial
+variables. The homomorphic encryption key consists of a pair of values, one used to create the hidden
+ring and the other taken from this ring to form an encryption operator for modular multiplication en-
+cryption. The proposed homomorphic encryption can be applied to any polynomials over a finite field,
+with their coefficients considered as their privacy. We denote the polynomials before homomorphic
+encryption as plain polynomials and after homomorphic encryption as cipher polynomials. A cipher
+polynomial can be evaluated with variables from the finite field, GF(p), by calculating the monomials
+of variables modulo a prime p. These properties allow functional homomorphic encryption to be
+used for public key encryption of certain asymmetric cryptosystems, such as Multivariate Public Key
+Cryptography schemes or MPKC to hide the structure of its central map construction. We propose
+a new variant of MPKC with homomorphic encryption of its public key. This variant simplifies
+MPKC central map to two multivariate polynomials constructed from polynomial multiplications,
+applying homomorphic encryption to the map, and changing its decryption from employing inverse
+maps to a polynomial division. We propose to use a single plaintext vector and a noise vector of
+multiple variables to be associated with the central map, in place of the secret plaintext vector to be
+encrypted in MPKC. We call this variant of encrypted MPKC, a Homomorphic Polynomial Public
+Key algorithm or HPPK algorithm. The HPPK algorithm holds the property of indistinguishability
+under the chosen-plaintext attacks or IND-CPA. The overall classical complexity to crack the HPPK
+algorithm is exponential in the size of the prime field GF(p). We briefly report on benchmarking
+performance results using the SUPERCOP toolkit. Benchmarking results demonstrate that HPPK
+offers rather fast performance, which is comparable and in some cases outperforms the NIST PQC
+finalists for key generation, encryption, and decryption.
+Keywords Homomorphic Functional Encryption · Post-Quantum Cryptography · Public-Key Cryptography · PQC ·
+Key Encapsulation Mechanism · KEM · Multivariate Public Key Cryptosystem · MPKC · PQC Performance.
+1
+Introduction
+Homomorphic encryption was first proposed by Rivest et al. in 1978 [1], one year after filing the patent for the
+RSA public key cryptography [2]. Homomorphic encryption commonly refers to privacy encryption for computation
+in an encrypted mode, without knowing the homomorphic key and the decryption procedure. This is noticeably
+different from the cryptographic algorithms used to encrypt data for secure communications or storage with public key
+mechanisms such as RSA [2] and Diffie-Hellman [3], and Elliptic Curve Cryptography [4, 5] to establish the shared key
+for symmetric encryption using algorithms as Advanced Encryption Standard or AES.
+arXiv:2301.11995v1  [cs.CR]  27 Jan 2023
+
+Novel Homomorphic Functional Encryption over a Hidden Ring
+Homomorphic encryption can be classified into partially homomorphic and fully homomorphic. The partially homomor-
+phic encryption supports either multiplicative or additive homomorphic operations, RSA [6] and ElGamal cryptosystems
+[7] are multiplicatively homomorphic; Goldwasser–Micali [8], Benaloh [9], and Paillier [10] are additively homo-
+morphic. The first milestone for fully homomorphic encryption was achieved by Gentry in 2009 using lattice-based
+cryptography [11] to support both addition and multiplication operators in the encrypted mode. Meanwhile, Chan in
+2009 proposed a symmetric homomorphic scheme based on improved Hill Cipher [12]. Kipnis and Hibshoosh proposed
+their symmetric homomorphic scheme in 2012 [13] with a randomization function for non-deterministic encryption.
+Gupta and Sharma proposed their symmetric homomorphic scheme based on linear algebraic computation in 2013 [14].
+Very recently, Li et al. in 2016 proposed a new symmetric homomorphic scheme, called Li-Scheme for outsourcing
+databases [15]. Their scheme, at large, is associated with two finite fields: a secret small field Fq and a big public field
+Fp, with modular exponentiation with its secret base s followed by modular multiplication with plaintext message m.
+Li-Scheme supports both additive and multiplicative operations so it is a full homomorphic encryption. Wang et al.
+performed a cryptoanalysis of the Li-Scheme in 2018 [16] and broke the scheme with certain known plaintext-ciphertext
+pairs. Wang et al. further improved their cryptoanalysis in 2019 and successfully recovered the secret key with the
+ciphertext-only attack using lattice reduction algorithm [17].
+Homomorphic encryption solely focuses on addition and multiplication operations on the encrypted data for plain data
+privacy. However, it would be very interesting to see an extension of homomorphic encryption from data privacy to
+functional privacy with variables to take the user’s inputs in a framework of a public key scheme. It is rarely seen
+that a public key cryptosystem, more precisely quantum-safe public key cryptosystem, is purposely designed with
+careful considerations not only to leverage homomorphic properties of ciphertext addition and scalar multiplication
+but also to take user’s secrets into ciphertext computation. This served as a motivation for our paper. We introduce a
+new Homomorphic Polynomial Public Key encapsulation or HPPK, which is an asymmetric key encapsulation scheme,
+with public keys encrypted using functional homomorphic encryption. HPPK uses multivariate polynomials to not
+only leverage homomorphic properties of addition and scalar multiplication but also allows for encrypting party’s
+input during the ciphertext creation. That is, the public key polynomial coefficients are encrypted using homomorphic
+function to ensure they are never truly public and hide the structure of the public key, at the same time, treating variables
+of the said public key polynomials as user input allows for freedom during the encryption process.
+HPPK cryptosystem has two distinct features, namely, the homomorphic encryption of the public key that allows
+for the user’s input during ciphertext creation, and the use of a hidden ring. HPPK is not the first cryptosystem to
+use hidden structure. For instance, the work of Li et al. describes a cryptosystem with a hidden prime ring [15].
+Another important example is Hidden Field Equations (HFE) cryptosystems. The examples of asymmetric multivariate
+encryption schemes that are based on HFE include [18, 19, 20, 21]. Various signature schemes based on HFE were
+also proposed [22, 23, 24, 25, 26]. In the framework of HFE, the private polynomials as well as the structure they are
+defined over, a field extension, are both hidden using affine transformations.
+The algorithms based on HFE, mentioned above fall in the category of quantum-safe algorithms. More precisely,
+multivariate quantum-safe algorithms. Quantum computing developments have been receiving a lot of focus from the
+academic community as well as industry leaders since Google announced its first quantum advantage in 2019 [27].
+But it was the National Institution of Standards and Technology (NIST) that opened the arena for quantum-resistant
+cryptography, when they started the post-quantum cryptography (PQC) standardization process in November 2017.
+Recently, they have announced third-round finalists which include four key exchange mechanism schemes (KEM) and
+three finalists for digital signatures [28]. Four KEM finalists include code-based Classic McEliece [29], lattice-based
+CRYSTALS-KYBER [30], NTRU [31, 32], and Saber [33] algorithms. At the latest announcement, NIST selected
+CRYSTALS-KYBER to be standardized algorithm for KEM. In addition to the aforementioned finalists for KEM, the
+Multivariate Public Key Cryptosystems or MPKC is worth a special discussion. Algorithms based on multivariate
+polynomial problems are considered to be quantum-safe, but they also make an excellent candidate for homomorphic
+encryption due to the use of multivariate polynomials.
+The framework of MPKC is built on a system of quadratic polynomials. The public key is represented by a central map
+P : Fm
+p → Fℓ
+p with m variables and ℓ polynomials [34]. Many variants of MPKC central map constructions have been
+proposed since Matsumoto and Imai first introduced this cryptosystem in 1988 [35], including single field systems and
+mixed field systems [36]. Single field MPKC includes several Triangular systems and the Oil and Vinegar system since
+Patarin and Goubin in 1997 [37] and unbalanced Oil and Vinegar scheme by Kipnis et. al. in 1999 [18]. The mixed
+field MPKC refers to Matsumoto-Imai system [35] and Hidden Field Equation [38]. In addition, Wang et al. in 2006
+proposed a Medium-Field MPKC scheme [39] and an improved scheme in 2008 [40]. Ding and Schmidt proposed
+Rainbow as a MPKC digital signature scheme in 2005 [25].
+Attacks on MPKC cryptosystems are mainly classified into two categories: algebraic solving attacks and linear algebra
+attacks. Algebra solving attacks attempt to solve the MPKC multivariate equation system from the public key with
+2
+
+Novel Homomorphic Functional Encryption over a Hidden Ring
+ciphertext (z1, z2, . . . , zℓ) to recover the pre-image (x1, x2, . . . , xm). Faugére reported his first attack on MPKC in
+1999[41] and in 2002[42] using Gröber bases (F4), later in 2003 Faugére and Antoine reported their attack on HFE
+Gröber bases (F5). Ding et. al. proposed their new Zhuang-Zi algorithm to solve the multivariate system in 2006 [43].
+In linear algebra attacks, Courtois et al. reported their attack on MPKC using the relinearization technique, aclled XL
+in 2000 [44]. The Minrank attack has been successfully applied by Goubin and Courtois on the single field system in
+2000 [45] and by Kipnis and Patarin on the mixed field system in 1999 [46].
+A new type of polynomial public key has been recently proposed by Kuang in 2021 [47], based on univariate polynomial
+multiplications, by Kuang and Barbeau in 2021 [48, 49, 50], based on multivariate polynomial multiplications with two
+noise functions to increase the public key security against possible public key attacks. The digital signature scheme of
+the multivariate polynomial public key or MPPK has been prosoed by Kuang, Perepechaenko and Barbeau in 2022 [51].
+This paper explores the possibility to combine a new homomorphic encryption to key construction to further enhance
+the security of the MPPK cryptography for key encapsulation mechanism or KEM.
+The proposed HPPK scheme can be also considered as a new variant of MPKC scheme with public keys being
+encrypted using homomorphic functional encryption. We begin by introducing the proposed symmetric homomorphic
+encryption scheme in A New Symmetric Homomorphic Functional Encryption over a Hidden Ring. The proposed
+HPPK algorithm is then discussed in Homomorphic Polynomial Public Key Cryptosystem. We present the reader with
+thorough security analysis of HPPK in HPPK Security Analysis, and report on benchmarking the performance of HPPK
+in Brief Benchmarking Performance. We conclude with Conclusion.
+2
+A New Symmetric Homomorphic Functional Encryption over a Hidden Ring
+In contrast to conventional homomorphic cryptography used for data privacy, in this paper we propose homomorphic
+functional encryption to be applied to the public key in the framework of multivariate asymmetric cryptography. This
+will allow for an asymmetric scheme with encrypted public keys. Moreover, by construction, functional homomorphic
+encryption allows for user’s input during the ciphertext generation procedure. That is, the ciphertext can be created
+with the input of the encrypting party, however, the public key used for encryption is itself encrypted using functional
+homomorphic operator. The decrypting party is the only party that has knowledge of the private key associated
+with the functional homomorphic encryption operator as well as the asymmetric scheme private key. Essentially, the
+homomorphic functional encryption defined in this paper provides a round-trip envelope for a public key encryption.
+In a way, such approach combines three main areas of cryptography, namely, asymmetric cryptography, homomorphic
+encryption, and symmetric cryptography with self-shared key. This phenomenon is illustrated in Fig. 1. In the figure, the
+traditional public key derived from a given assymetric algorithm is called plain public key or PPK, the homomorphically
+encrypted PPK is called cipher public key or CPK. The cipher is produced by evaluating the public key polynomial
+values using a user-selected secret. The decryption would perform in two stages: homomorphic decryption and then
+secret extraction.
+Figure 1: Illustration of a cryptosystem combining asymmetric cryptography, symmetric cryptography with a single
+self-shared key, and homomorphic encryption.
+We begin by introducing the Homomorphic Functional Encryption Operator. In order to allow for the user’s input during
+ciphertext creation, and leverage additive and scalar multiplicative homomorphic features, the functional homomorphic
+3
+
+Symmetric
+Encrypt
+0
+Enc
+Crypto
+PPK
+Dec
+Dec
+HPPK
+997
+CPK
+Cipher
+Homomorphic CryptoNovel Homomorphic Functional Encryption over a Hidden Ring
+encryption is applied to polynomials. We discuss the reason for this further. Hence, when introducing the said operator
+we assume that it will be applied to polynomials.
+2.1
+Homomorphic Encryption Operator
+Let S be a positive integer, and R be a randomly chosen value such that R ∈ ZS and gcd(R, S) = 1. We propose a
+Homomorphic Functional Encryption Operator ˆE(R,S), with a secret homomorphic key being a tuple (R, S). The values
+S and R are never shared.
+In its general form, the encryption operator is defined as a multiplicative operation modulo a hidden value S as
+ˆE(R,S)(f) = (R ◦ f) mod S,
+(1)
+where f denotes any univariate or multivariate polynomial f = �k
+i=0 fiXi, over Fp with Xi being its monomials. The
+encryption operator acts on the coefficients of f, which we refer to as plain coefficients. This produces what we call
+cipher coefficients, denoted hi, as
+ˆE(R,S)fi = Rfi mod S = hi
+for every i. Similarly, we can define a Homomorphic Functional Decryption Operator as
+ˆE(R−1,S)h = (R−1 ◦ h) mod S.
+(2)
+Such operator decrypts the coefficients of the polynomial h. That is, it successfully decrypts the cipher coefficients
+back to the plain coefficients. More precisely,
+ˆE(R−1,S)hi = R−1(Rfi) mod S = (R−1R)fi mod S = fi
+for any i.
+The above defined homomorphic operator ˆE(R,S) holds following homomorphic properties:
+• ˆE(R,S) is additively homomorphic: if a and b are two plain constants, then ˆE(R,S)(a+b) = R(a+b) mod S =
+Ra + Rb mod S = ˆE(R,S)a + ˆE(R,S)b;
+• ˆE(R,S) is scalar multiplicatively homomorphic: if a is a plain constant and x is a variable, then ˆE(R,S)(ax) =
+R(ax) = (Ra)x = [ ˆE(R,S)a]x.
+Thus, the operator ˆE(R,S) offers partially homomorphic encryption. We leave it to the reader to verify that the
+same properties hold true for the proposed homomorphic functional decryption operator ˆE(R−1,S). Note that these
+homomorphic properties come from linearity, and thus are natural to polynomials. Indeed, polynomials hold additive
+and scalar multiplicative properties through their coefficients. Moreover, polynomials can be defined and evaluated with
+coefficients in a field or a ring, different from a field or a ring for variables. We leverage this property, and thus, apply
+the functional homomorphic encryption to public key cryptosystems with polynomial public keys.
+2.2
+Homomorphically Encrypted Polynomials
+As we have previously stated, the proposed homomorphic encryption is applicable to all polynomials over a ring Zp
+or finite field Fp characterized by a prime p. In this work, when we refer to polynomials, we imply that the plain
+polynomials, to be encrypted, are considered modulo p, unless stated otherwise. A generic multivariate polynomial has
+the following form
+p(x1, . . . , xm) =
+ℓ1
+�
+j1=1
+· · ·
+ℓm
+�
+jm=1
+pij1...jmxj1
+1 · · · xjm
+m .
+Alternatively, let Xj = xj1
+1 · · · xjm
+m denote the monomials of such polynomial, then
+p(x1, . . . , xm) =
+L
+�
+j=1
+pjXj,
+(3)
+where L denotes the total number of terms.
+To successfully encrypt and decrypt any polynomial p(x1, . . . , xm) using the functional homomorphic encryption and
+decryption operators defined in Eq. (1) and (2) respectively, the following conditions must be met:
+4
+
+Novel Homomorphic Functional Encryption over a Hidden Ring
+• The monomials Xj are to be computed as Xj = (Xj mod p). The values of monomials reduced modulo p are
+used to compute the value of the polynomial p(x1, . . . , xm).
+• The homomorphic secret key value S should satisfy the bit length condition: |S|2 > 2|p|2 + |L|2.
+The first condition ensures that polynomial p(x1, . . . , xm) is evaluated as if the mononials Xj are new variables over
+Fp, Indeed, the operator ˆE(R,S) is applied to the polynomial p(x1, . . . , xm) in the following way
+ˆE(R,S)p(x1, . . . , xm) =
+L
+�
+j=1
+[Rpj mod S]Xj.
+(4)
+Such encrypted polynomial can be computed as
+L
+�
+j=1
+[Rpj mod S](Xj mod p) = ¯p.
+Note that the computed value was not reduced modulo any integer, nor is the arithmetic performed modulo any integer.
+Thus, the user’s input through monomials Xj remains intact and can be decrypted correctly. Let the plain value of the
+polynomial with user’s input, that is, if the polynomial was not encrypted with ˆE(R,S), be
+ˆp =
+L
+�
+j=1
+pjXj mod p.
+To ensure successful decryption, the second condition must be met. If the size of S is sufficiently large, the values of
+coefficients and variables remains the same after decryption, and it is possible to recover ˆp. Indeed,
+ˆE(R−1,S)¯p =
+L
+�
+j=1
+[R−1Rpj mod S](Xj mod p) =
+L
+�
+j=1
+pj(Xj mod p),
+(5)
+and then the value �L
+j=1 pj(Xj mod p) can be reduced modulo p to yield �L
+j=1 pjXj mod p = ˆp.
+To elaborate more on this, we present the reader with two examples of functional homomorphic encryption of linear
+and quadratic polynomials.
+2.2.1
+Linear Polynomials
+Recall, that we encrypt the coefficients of the polynomials defined over Fp, which successfully maps polynomials from
+Fp[x1, . . . , xm] to ZS[x1, . . . , xm], leaving x1, . . . , xm ∈ Fp. A generic linear multivariate polynomial over a finite
+field Fp has form
+p(x1, x2, . . . , xm) =
+m
+�
+j=1
+pjxj mod p.
+(6)
+Conventionally, in the asymmetric encryption schemes, the public key inherits mathematical logic from the private
+key, making it vulnerable. Hence, if public key consists of polynomials, we wish to encrypt the coefficients of the said
+polynomials using functional homomorphic operator, to hide the mathematical logic. In this case we share the cipher
+public key, encrypted using functional homomorphic operator. To ensure that the ciphertext can be still created in the
+framework of asymmetric public key scheme, the variables in the public key polynomials are used for user’s input.
+They are not encrypted using homomorphic encryption, but only using the encryption procedure from the asymmetric
+scheme. Such variable values can consist of the plaintext only, or plaintext and noise used for obscurity.
+Applying homomorphic encryption operator to the above linear polynomial, defined in Eq.(6), produces a cipher linear
+polynomial with coefficients in a hidden ring ZS, and variables in Fp :
+P(x1, x2, . . . , xm) = ˆE(R,S)p(x1, x2, . . . , xm)
+=
+m
+�
+j=1
+(Rpj mod S)xj =
+m
+�
+j=1
+Pjxj.
+(7)
+5
+
+Novel Homomorphic Functional Encryption over a Hidden Ring
+While the plain coefficients, pj, are encrypted into cipher coefficients, Pj, the cipher polynomial P(x1, x2, . . . , xm)
+can still be evaluated with a set of chosen values r1, . . . , rm ∈ Fp to produce value ¯P
+¯P = P(r1, r2, . . . , rm) =
+m
+�
+j=1
+Pj(rj mod p).
+(8)
+Let the value ¯p = �m
+i=1 pjrj mod p, be the original ciphertext of the asymmetric scheme. However, it is encrypted
+into the value ¯P using homomorphic encryption. To recover the plain polynomial value, that is, decrypt the cipher
+coefficients, into the plain coefficients and evaluate polynomial modulo p, we first apply the functional homomorphic
+decryption operator ˆE(R−1,S) to get ˆE(R−1,S) ¯P, and then reduce this value modulo p. More precisely,
+ˆE(R−1,S) ¯P =
+m
+�
+j=1
+[R−1Pj mod S](rj mod p) =
+m
+�
+i=1
+pj(rj mod p),
+which reduced modulo p is
+m
+�
+i=1
+pjrj mod p = ¯p.
+In a framework of asymmetric scheme with functional homomorphic encryption element, polynomials such as in Eq. (6)
+are associated with plain coefficients, that is, the original public keys. The cipher polynomials have form as in Eq. (7),
+with coefficients being encrypted from the plain public keys, using homomorphic encryption. Such cipher public keys
+are shared, and the plain public keys are stored securely and never shared. The ciphertext in this combined algorithm
+is of the form as in Eq. (8). The decrypting party first needs to decrypt the ciphertext to nullify the homomorphic
+encryption of the public key, as shown in Eq. (8). Afterwards, the decryption party can perform decryption procedure
+that corresponds to the given asymmetric scheme.
+2.2.2
+Quadratic Polynomials
+Multivariate quadratic polynomials serve as the foundation of Multivariate Public Key Cryptosystem or MPKC[34,
+52, 53]. Thus, we want to pay special attention on applications of functional homomorphic encryption on multivariate
+quadratic polynomials. A general quadratic multivariate polynomial p(x1, x2, . . . , xn) over a finite field Fp has the
+following form
+p(x1, x2, . . . , xm) =
+m
+�
+1≤i≤j
+pijxixj mod p,
+(9)
+where the coefficients pij are considered as the privacy constants for this polynomial function so they must be hidden
+from public. This is done by applying the functional homomorphic encryption operator to this polynomial as follows
+P(x1, x2, . . . , xm) = ˆE(R,S)p(x1, x2, . . . , xm)
+(10)
+=
+m
+�
+1≤i≤j
+(Rpij mod S)xixj =
+m
+�
+1≤i≤j
+Pijxixj.
+(11)
+(12)
+Here, the encrypted coefficients are defined over the hidden ring ZS, however, all the variables x1, . . . , xm are still
+elements of the field Fp. As we have previously mentioned, we refer to the coefficients pij as plain coefficients, and Pij
+are referred to as cipher coefficients. Similarly, P(x1, x2, . . . , xm) and p(x1, x2, . . . , xm) are referred to as cipher and
+plain polynomials respectively.
+While coefficients are encrypted with homomorphic encryption operator, the polynomial P(x1, x2, . . . , xm) still accepts
+user’s input. That is, the cipher polynomial value ¯P can be still calculated with a chosen set of r1, . . . , rm from the
+field Fp as follows
+¯P = P(r1, r2, . . . , rm) =
+m
+�
+1≤i≤j
+Pij(xixj mod p).
+(13)
+Note that the computed value ¯P is an integer. The arithmetic to compute such value was not performned modulo any
+integer. The plain polynomial values are securely hidden through the hidden ring ZS. To recover the plain polynomial
+6
+
+Novel Homomorphic Functional Encryption over a Hidden Ring
+equation, decryption opertor ˆE(R−1,S) can be applied to the cipher polynomial value ¯P, followed by reduction mod p:
+ˆE(R−1,S) ¯P = R−1 ¯P mod S, then
+(14)
+(R−1 ¯P mod S) mod p = p(r1, r2, . . . , rm) = ¯p.
+(15)
+The value ¯p is the plain polynomial value for the chosen values of variables x1, . . . , xm by the encrypting party.
+Similar to the linear case, the public key of the asymmetric scheme consist of quadratic polynomials of the form (9), to
+be encrypted using homomorphic functional encryption operators. The cipher public keys are of the form (10). Such
+cipher public keys are the ones shared, while the plain public keys are not. The ciphertext in the combined scheme is of
+the form (13), which needs to be decrypted back to the plain value. For that a homomorphic decryption operator is
+applied, as in Eq (14), and the plain ciphertext value is recovered.
+3
+Homomorphic Polynomial Public Key Cryptosystem
+3.1
+Brief Summary of MPKC
+An interested reader can find the detail description of MPKC schemes by Ding and Yang [34]. In this section, we
+briefly outline the basic mechanism of MPKC algorithms. The framework mainly consists of ℓ quadratic multivariate
+polynomials
+p1(x1, . . . , xm), p2(x1, . . . , xm), . . . , pℓ(x1, . . . , xm)
+in m variables over finite field Fp. Each polynomial pk(x1, . . . , xm) can be written in its expanded form as
+pk(x1, . . . , xm) =
+m
+�
+i<j=1
+pijkxixj
+(16)
+for k = 1, 2, . . . , l. In the literature Eq.(16) is generally written in a matrix form as
+pk(x1, . . . , xm) = (x1, . . . , xm)
+�
+�
+�
+�
+�
+p11k
+p12k
+. . .
+p1mk
+. . .
+. . .
+. . .
+. . .
+pi1k
+pi2k
+. . .
+pimk
+. . .
+. . .
+. . .
+. . .
+pm1k
+pm2k
+. . .
+pmmk
+�
+�
+�
+�
+� (x1, . . . , xm)T
+(17)
+briefly expressed as,
+pk(x1, . . . , xm) = ⃗x · Pk · ⃗xT ,
+where Pk is an m × m square matrix. Considering all ℓ polynomials, we can write the MPKC map from Fm
+p to Fℓ
+p as a
+3-dimensional matrix P[ℓ][m][m], which is a trapdoor called the central map. The central map is selected to be easily
+invertible. In order to protect this central map and its structure, two affine linear invertible maps T and S are chosen to
+construct the MPKC public key:
+• Public Key: ¯P = T ◦ P ◦ S.
+• Private Key: (T, P, S).
+The MPKC encryption procedure simply to evaluates ℓ polynomials over the field Fp as
+⃗z = ¯P(⃗x) = {z1 = p1(x1, . . . , xm), z2 = p2(x1, . . . , xm), . . . , zℓ = pℓ(x1, . . . , xm)}
+(18)
+and decryption works as follows
+⃗u = T −1(⃗z),⃗v = P−1(⃗u), ⃗x = S−1(⃗v).
+The major step to use MPKC is to construct the invertible central map P over a finite field Fp to perform a map:
+Fm
+p → Fℓ
+p.
+There may be a potential way to enhance the security of MPKC cryptosystem by applying the proposed homomorphic
+encryption on its map: Fm
+p → Fℓ
+p. The homomorphic encryption effectively hides the public key construction logic
+over a hidden ring ZS. In this case, an encryption key Rk is required for each quadratic polynomial pk(x1, . . . , xm),
+with value Rk chosen over the hidden ring ZS for all k. Hence, there are a total of ℓ encryption keys for MPKC. The
+MPKC encryption in this case is almost the same as the original MPKC encryption. The ciphertext (z1, z2, . . . , zℓ) is to
+be homomorphically decrypted to create original multivariate equation system, as illustrated in Eq.(18). This means,
+7
+
+Novel Homomorphic Functional Encryption over a Hidden Ring
+Eq.(18) is hidden under the hidden ring ZS. On one hand, applying the homomorphic encryption would increase the
+public key size for MPKC, however, the number of variables can be reduced due to the homomorphic encryption.
+In this paper, we are not going to further explore this variant of MPKC schemes but we will focus on another variant of
+MPKC, called HPPK which we propose in the new section.
+3.2
+HPPK Encapsulation
+We propose a new variant of an MPKC scheme, called the Homomorphic Polynomial Public Key or HPPK, with the
+following considerations:
+• The vector on the left hand side of the map P is treated as ⃗xl and the vector on the right hand side as ⃗xr;
+• The vector ⃗xl is replaced with ⃗xl = (x0, x1, x2, . . . , xn), considering ⃗xl as a message vector in a polynomial
+vector space represented by a basis {x0, x1, x2, . . . , xn} for a message variable x and ⃗xr = (x1, . . . , xm) as a
+noise vector for noise variables x1, . . . , xm;
+• The proposed homomorphic encryption is applied to the central map P, mapping the elements from Fp → ZS:
+¯P = ˆE(R,S)P
+and the decryption is de-mapping from ZS → Fp:
+P = ˆE(R−1,S) ¯P mod p
+• The number of polynomials is reduced to ℓ = 2;
+• The decryption mechanism is changed from inverting maps to modular division, which automatically cancels
+the noise used for obscurity.
+3.2.1
+Key Construction
+Without loss of generality, we change the notation of the unencrypted central map to P. Under the above considerations,
+the central map P consists of two multivariate polynomials
+p1(x, x1, x2, . . . , xm) = (1, x1, . . . , xn)
+�
+�
+�
+�
+�
+�
+�
+p011
+p021
+. . .
+p0m1
+p111
+p121
+. . .
+p1m1
+. . .
+. . .
+. . .
+. . .
+pi11
+pi21
+. . .
+pim1
+. . .
+. . .
+. . .
+. . .
+pn11
+pn21
+. . .
+pnm1
+�
+�
+�
+�
+�
+�
+�
+(x1, x2, . . . , xm)T ,
+(19)
+and
+p2(x, x1, x2, . . . , xm) = (1, x1, . . . , xn)
+�
+�
+�
+�
+�
+�
+�
+p012
+p122
+. . .
+p0m2
+p112
+p122
+. . .
+p1m2
+. . .
+. . .
+. . .
+. . .
+pi12
+pi22
+. . .
+pim2
+. . .
+. . .
+. . .
+. . .
+pn12
+pn22
+. . .
+pnm2
+�
+�
+�
+�
+�
+�
+�
+(x1, x2, . . . , xm)T .
+(20)
+Note that the matrix maps P1 and P2 are of size (n + 1) × m, thus, no longer square.
+The construction of
+p1(x, x1, . . . , xm) and p2(x, x1, . . . , xm) can alternatively be achieved with polynomial multiplications
+p1(x, x1, . . . , xm) = b(x, x1, . . . , xm)f1(x)
+(21)
+p2(x, x1, . . . , xm) = b(x, x1, . . . , xm)f2(x),
+where the base multivariate polynomial b(x, x1, x2, . . . , xm) and univariate polynomials f1(x) and f2(x) have the
+following generic forms
+b(x, x1, . . . , xm) =
+nb
+�
+i=0
+m
+�
+j=1
+bijxixj
+(22)
+f1(x) =
+λ
+�
+i=0
+f1ixi
+f2(x) =
+λ
+�
+i=0
+f2ixi.
+8
+
+Novel Homomorphic Functional Encryption over a Hidden Ring
+Here, nb and λ are orders of base multivariate polynomial and univariate polynomials with respect to message variable
+x respectively. Without loss of generality, we assume that the univariate polynomials f1(x) and f2(x) are solvable, in
+other words λ < 5. Using Eq.(21) and (22), we can express
+p1(x, x1, . . . , xm) =
+n
+�
+i=0
+m
+�
+j=1
+pij1xixj = ⃗xl · P1 · ⃗xr
+(23)
+p2(x, x1, . . . , xm) =
+n
+�
+i=0
+m
+�
+j=1
+pij2xixj = ⃗xl · P2 · ⃗xr,
+with pij1 = �
+s+t=i bsjf1t and pij2 = �
+s+t=i bsjf2t being the coefficients, and n = nb + λ. It is apparent that
+the plain central map P as shown in Eq.(19) and Eq.(20) expanded as Eq.(23) inherits a lot of structure. Given
+that the components of the central map are private key elements, the map in its unaltered form is not secure against
+potential attacks such as polynomial factorization, root finding, etc. To secure the central map, we apply functional
+homomorphic encryption operator to the plain central map by acting with ˆE(R1,S) on p1(x, x1, . . . , xm) and ˆE(R2,S) on
+p2(x, x1, . . . , xm). To be more precise, the cipher central map consists of two polynomials
+P1(x, x1, . . . , xm) =
+n
+�
+i=0
+m
+�
+j=1
+(R1pij1 mod S)xixj = ⃗xl · P1 · ⃗xr
+(24)
+P2(x, x1, . . . , xm) =
+n
+�
+i=0
+m
+�
+j=1
+(R2pij2 mod S)xixj = ⃗xl · P2 · ⃗xr.
+We set public key to be the cipher central map P, while private key consists of the homomorphic operators, the hidden
+ring, together with univariate polynomials:
+• Security parameter: the prime finite field Fp which is agreed on before the key generation procedure.
+• Private Key:
+◦ hidden ring ZS with a randomly selected S for the required bit length;
+◦ homomorphic encryption key values R1 and R2 chosen from ZS;
+◦ univariate polynomials f1(x) and f2(x) with coefficients randomly selected from FS;
+• Public Key: the map P, consisting of
+◦P1(⃗xl, ⃗xr) = ⃗xl · P1 · ⃗xr
+◦P2(⃗xl, ⃗xr) = ⃗xl · P2 · ⃗xr
+3.2.2
+Encryption
+Encryption is straightforward by determining the value for the secret x and randomly choosing values for the noise
+variables x1, . . . , xm over the field Fp and evaluating ciphertext integer values ¯P1 and ¯P2. That is, the ciphertext
+consists of two integer values C = ( ¯P1, ¯P2), where
+¯P1 =
+n
+�
+i=0
+m
+�
+j=1
+Pij1(xjxi mod p)
+(25)
+¯P2 =
+n
+�
+i=0
+m
+�
+j=1
+Pij2(xjxi mod p).
+Here, Pij1 and Pij2 denote the cipher coefficients encrypted with the homomorphic encryption operators. Note that the
+cipher polynomials have coefficients in the hidden ring ZS, and all monomial calculations are performed mod p, the rest
+of the arithmetic is performed over integers. The values ¯P1, and ¯P2 are integers forming the ciphertext C = {P1, P2}.
+3.2.3
+Decryption
+It is easy to verify that the HPPK map as in Eq.(19) and Eq.(20), under construction as shown in Eq.(21), holds a
+division invariant property on the multiplicand or the base multivariate polynomial b(x, x1, x2, . . . , xm). Indeed,
+p1(x, x1, . . . , xm)
+p2(x, x1, . . . , xm) = b(x, x1, . . . , xm)f1(x)
+b(x, x1, . . . , xm)f2(x) = f1(x)
+f2(x).
+9
+
+Novel Homomorphic Functional Encryption over a Hidden Ring
+The first step in the decryption process is to apply the functional homomorphic decryption operator to the cipher-
+text to recover plain polynomial values ¯p1 and ¯p2, which are evaluation results of plain multivariate polynomials
+p1(x, x1, . . . , xm) and p2(x, x1, . . . , xm) at the chosen message and noises respectively. This can be done as
+ˆE(R−1
+1
+,S) ¯P1 = p1(x, x1, . . . , xm) = ¯p1
+ˆE(R−1
+2
+,S) ¯P2 = p2(x, x1, . . . , xm) = ¯p2.
+These values are used to compute the ratio K modulo p of the form
+K = ¯p1
+¯p2
+= p1(x, x1, . . . , xm)
+p2(x, x1, . . . , xm) = f1(x)
+f2(x) mod p
+(26)
+Note that the noise vector ⃗xr is automatically eliminated through the division. The secret x can then be found from
+Eq.(26) by radicals if f1(x) and f2(x) are solvable such as linear or quadratic polynomials.
+Note that when λ > 1, an extra 8-bit flag σ should be added to the plaintext to distinguish the correct plaintext during
+the decryption procedure. We propose a formatted plaintext X = (σ|x), with σ to be a one byte flag. That is, σ is
+concatenated with x, such that the most significant 8 bits of X are set as σ and the remaining bits as x. The flag σ
+can be generated by a cyclic redundancy check or CRC with the secret x. There are different CRC algorithms such as
+CRC-8, CRC-32, etc. 8 bits of CRC codes should be sufficient to make a right decision from the roots obtained during
+decryption. After decryption with successful flag verification, the shared secret would be established by removing the
+most significant 8 bits of the obtained value X. Note that the field size should account for the flag σ. In this work,
+however, we focus mainly on the case λ = 1.
+This division invariant property is the foundation for the HPPK encapsulation to be indistinguishable under chosen
+plaintext attacks.
+3.3
+A Toy Example
+We demonstrate how HPPK works with a toy example.
+3.3.1
+Key Pair Generation
+Considering a prime field F13 with the prime p = 13 and two noise variables x1, x2 for the simplicity of the
+demonstration purpose only, we can choose the hidden ring characterized by an integer of length > 12 bits. The private
+key consists of the following values:
+• S = 6798, R1 = 4267, R2 = 6475
+• f1(x) = 4 + 9x
+• f2(x) = 10 + 7x
+• B(x, x1, x2) = (8 + 7x)x1 + (5 + 11x)x2 (note: just for key pair construction procedure; this polynomial is
+not stored in the memory)
+The plain public key or PPK is simply constructed as
+• P1(x, x1, x2) = f1(x)B(x, x1, x2) mod 13 = x1(6 + 9x + 11x2) + x2(7 + 11x + 8x2),
+• P2(x, x1, x2) = f2(x)B(x, x1, x2) mod 13 = x1(2 + 9x + 10x2) + x2(11 + 2x + 12x2).
+The PPK polynomials are encrypted with the self-shared key R1, R2 over the ring ZS
+• P1(x, x1, x2) = E(R−1
+1
+,S)P1(x, x1, x2) = x1(5208 + 4413x + 6149x2) + x2(2677 + 6149x + 146x2)
+=⇒ P1 =
+�5208
+2677
+4413
+6149
+6149
+146
+�
+• P2(x, x1, x2) = E(R−1
+1
+,S)P2(x, x1, x2) = x1(6152 + 3891x + 3568x2) + x2(3245 + 6152x + 2922x2)
+=⇒ P2 =
+�6152
+3245
+3891
+6152
+3568
+2922
+�
+to create the so-called CPK P1 and P2.
+10
+
+Novel Homomorphic Functional Encryption over a Hidden Ring
+3.3.2
+Encryption
+We randomly choose variables from F13: x = 8, x1 = 3, x2 = 6. We, then, pre-calculate values
+x11 = xx1 mod 13 = 8 × 3 mod 13 = 11,
+x12 = x2x1 mod 13 = 82 × 3 mod 13 = 10,
+x21 = xx2 mod 13 = 8 × 6 mod 13 = 9,
+x22 = x2x2 mod 13 = 82 × 6 mod 13 = 7.
+Now we can calculate the ciphertext C = {198082, 192229} as follows
+• ¯P1 = x1(5208 + 4413x + 6149x2) + x2(2677 + 6149x + 146x2) = 198082
+• ¯P2 = x1(6152 + 3891x + 3568x2) + x2(3245 + 6152x + 2922x2) = 192229
+3.3.3
+Decryption
+We first perform the homomorphic decryption to rebuild the plain polynomial equations
+• P1(x, x1, x2) = f1(x)B(x, x1, x2) = [E(R−1
+1
+,S)19808] mod 13 = [ 19808
+4267 mod 6798] mod 13 = 8
+• P2(x, x1, x2) = f2(x)B(x, x1, x2) = [E(R−1
+2
+,S)192229] mod 13 = [ 192229
+6475 mod 6798] mod 13 = 9
+then we can eliminate the noise introduced by the base multivariate polynomial
+P1(x, x1, x2)
+P2(x, x1, x2) = 4 + 9x
+10 + 7x = 8
+9 mod 13 = 11
+where the secret x can be easily extracted as x = 8. The encryption can be done with any possible values for x1 and x2
+at a given secret x, which would produce different ciphertext C, but the decryption would reveal the same secret. This
+simple toy example demonstrates its capability of randomized encryption.
+4
+HPPK Security Analysis
+In this section, we analyze the security of the proposed HPPK algorithm. The security of HPPK relies on computational
+hardness of Modular Diophantine Equation, introduced in Definition 4.1, and Hilbert’s tenth Problem, introduced in
+Definition 4.7. We begin by proving that HPPK satisfies the IND-CPA indistinguishability property. These results are
+then extended to prove that task of recovering plaintext from ciphertext in the framework of HPPK is NP-complete, and
+state its classical and quantum complexity. Afterwards, we focus on the private key attack and prove that the problem of
+obtaining the private key from the public key is NP-complete. Here we also provide classical and quantum complexities
+of obtaining privates key from public key.
+4.1
+Plaintext attack
+An attentive reader will notice that the evaluated ciphertext as illustrated in Eq. (25) has not been reduced modulo
+any integer. Thus, an adversary looking to perpetrate an attack to recover the plaintext can treat the coefficients of the
+polynomials in Eq. (25) and evaluated ciphertext as integers. The plaintext values, sought after by the adversary, are
+elements of the field Fp, thus the malicious party can reduce the public values of the ciphertext modulo p to solve for
+plaintext variables in the Eq (25). We formally phrase it in the following remark.
+Remark 4.0.1. For the purpose of obtaining the plaintext, the ciphertext and cipher coefficients as illustrated in the
+Eq. (25) can be considered modulo p as follows
+C =
+��n
+i=0
+�m
+j=1 Pij1xjxi − ¯P1 = 0 (mod p)
+�n
+i=0
+�m
+j=1 Pij2xjxi − ¯P2 = 0 (mod p)
+(27)
+Definition 4.1 (Modular Diophantine Equation). The Modular Diophantine Equation asks whether an integer solution
+exists to the equation
+P(y1, . . . , yk) − 1 = 0
+mod p,
+given as an input of a polynomial P(y1, . . . , yk) and a prime p.
+Remark 4.0.2. A positive answer to this question would include a solution.
+11
+
+Novel Homomorphic Functional Encryption over a Hidden Ring
+Let m + 1 > 2. Note that the system in the Eq. (27) can be normalized as
+C =
+��n
+i=0
+�m
+j=1 P′
+ij1xjxi − 1 = 0 (mod p),
+�n
+i=0
+�m
+j=1 P′
+ij2xjxi − 1 = 0 (mod p).
+(28)
+The most naive way of solving such normalized system is to solve each equation and find a common solution. Each
+such equation is an instance of a Modular Diophantine Equation. The more obvious way to solve the system in Eq. (27)
+would be to use Gaussian elimination and transform the system to a single equation. Indeed, since the coefficients of
+the ciphertext are publicly known, and the noise variables are linear in the ciphertext, the adversary can express any
+noise variable using the remaining terms of the equation and reduce the system to a single equation of the form
+H(x, x1, . . . , xm−1) − 1 = 0
+(29)
+over Fp with m unknowns, where m > 1. We assume that the adversary favours the ciphertext form with less variables.
+Thus, from the perspective of the adversary the cipheretext has form as in Eq. (29). From here on forward, we consider
+the ciphertext in the form given in Eq. (29). We formally define said form below.
+Definition 4.2. Let m + 1 > 2. The ciphertext in its normalized reduced form is a single equation
+H(x, x1, . . . , xm−1) − 1 = 0
+(30)
+over Fp, where x corresponds to the plaintext variable and the remaining variables are noise variables.
+Note that even in its normalized reduced form the ciphertext is an instance of a Modular Diophantine Equation. Since
+m + 1 > 2 we can argue that the adversary does not benefit much by reducing the system in Eq. (28) to a single
+equation (30), and eliminating one variable. The number of expected solutions to the Eq. (30) remains pm−1, and
+the adversary is facing with the problem of deciding which solution is the correct one. That is, a brute-force search
+algorithm can find a list of solutions to the Eq. (30) by trying all the possible m − 1 variables values over Fp. The
+adversary is interested in a particular solution from the list.
+One might argue that the attacker is interested only in the plaintext variable x ∈ Fp. Thus, the adversary can simply
+guess the value x. The complexity of this guess is O(p). However, note that the guess has to be tested for correctness.
+This will require coming up with noise variables and testing whether the guess is correct. Moreover, NIST requires the
+size of the actual communicated secret to be 32 bytes. Thus, the secret that is transferred between two parties consists
+of K blocks, where each block is p bits. Each block corresponds to the HPPK secret x. The secret message is then
+K different values x concatenated together to form a 32 byte secret. Each such block x is encrypted separately using
+HPPK. The complexity of correctly guessing the transferred secret message is then O(p4).
+Theorem 4.1. The Modular Diophantine Equation Problem is NP-complete.
+Proof. The proof, using the Boolean Satisfiability Problem, is given by Moore and Meterns [54, Section 5.4.4].
+Theorem 4.1 states that a brute-force search algorithm can find a solution to the Modular Diophantine Equation by
+trying all the possible solutions. Thus, without loss of generality, we treat the ciphertext-only attack on a ciphertext in
+its normal reduced form as a Modular Diophantine Problem. Indeed, by Theorem 4.1 the algorithm to find a solution
+to a Modular Diophantine Equation does not simply terminate to give a solution, it is a brute-force search algorithm
+that considers every possible solution before producing a result. In other words, it goes through all the possibilities to
+choose the correct one.
+4.1.1
+IND-CPA Indistinguishability Property and Ciphertext only Attack
+We suppose that the adversary will choose to perpetrate the attack on the ciphertext in its normal reduced form as in
+Eq. (30), for its easier to attack. In the framework of HPPK, the public key elements are the coefficients of the ciphertext
+polynomials. Thus, if the ciphertext is presented in its reduced normalized form, as defined in Eq. (30), setting the
+coefficients of such polynomial to be the ciphertext does not disadvantage the adversary.
+Theorem 4.2 (HPPK has IND-CPA property). Let m > 1, where m is the total number of variables in the normalized
+reduced form of the ciphertext as in Eq. (30). If the Modular Diophantine Equation is NP-complete, the HPPK
+encryption system is provably secure in the IND-CPA security model with a reduction loss of pm−2.
+Proof. Assume that there exists an adversary A that (t, ϵ)-breaks the HPPK encryption system in the IND-CPA
+security model. We construct a simulator B that solves the Modular Diophantine Equation. Given as input, a Modular
+Diophantine Equation instance (p, H(x, x1, . . . , xm−1)), where H(x, x1, . . . , xm−1) is of the form (30) and m > 1,
+12
+
+Novel Homomorphic Functional Encryption over a Hidden Ring
+the simulator B runs A as follows. The simulator sets the normalized public key over Fp to the coefficients of the
+polynomial H(x, x1, . . . , xm−1). The challenge consists of the following game. The adversary A generates two distinct
+messages m0 and m1 ∈ Fp, and submits them to the simulator. The simulator B randomly chooses b in {0, 1} as well
+as random values r1, . . . , rm−1 for the noise variables, and sets the ciphertext to be the value
+¯H = H(mb, r1, . . . , rm−1).
+The challenge for the adversary then consists of the following equation to be solved for x:
+ˆH(x, x1, . . . , xm−1) − 1 = 0
+over Fp. Here,
+ˆH(x, x1, . . . , xm−1) = 1
+¯H H(x, x1, . . . , xm−1).
+The challenge remains to be the Modular Diophantine Equation H(x, x1, . . . , xm−1) − 1 = 0, since the value 1
+¯
+H can
+be pushed to the noise variables, which are random and do not influence the plaintext. Indeed, let hij be the coefficients
+of the polynomial H(x, x1, . . . , xm−1) for any i ∈ {0, . . . , n} and j ∈ {1, . . . , m − 1}, then
+l
+�
+i=0
+m−1
+�
+j=1
+hijxixj ×
+1
+H(mb, r1, . . . , rm−1) =
+l
+�
+i=0
+m−1
+�
+j=1
+hijxix′
+j,
+where x′
+j = xj
+¯
+H . The challenge in this case is correct, as it corresponds to the challenged plaintext and remains in the
+form of a Diophantine equation chosen by the simulator.
+The coefficients of the challenge equation come from the submitted Diophantine equation, and thus, from the point
+of view of the adversary are random. The values r1, . . . , rm−1 are selected at random. The adversary does not
+have knowledge of the values {x′
+1, . . . , x′
+m−1} and they can not be calculated from the other parameters given to the
+adversary. So the noise variables x′
+j for all j ∈ {1, . . . , m − 1} are random. Hence, the simulation holds randomness
+property. By construction, the simulation is indistinguishable from a real attack. That is, the adversary is challenged
+with solving the equation as in the Eq. (30), which is HPPK ciphertext in its normalized reduced form.
+There is no abort in the simulation. The adversary outputs a random guess b′ of b. When b′ is equal to b, the adversary
+wins. Otherwise, the adversary looses. The probability of simply guessing the value for x is Pr = 1
+2. We will calculate
+the probability of solving the IND-CPA challenge with the advantage of the adversary, that is Pr = 1
+2 + α. The
+advantage comes from the assumption that the adversary can break the HPPK cryptosystem.
+The challenge has a general form as in the Eq. (30), thus, the equation is expected to have pm−1 distinct solutions,
+considering all m variables. On the other hand, it is known that the variable x ∈ {m0, m1}. Assuming x = m0,
+there are now pm−2 possible solutions to choose the correct solution from. The same is true for x = m1. That is, the
+probability of finding correct solution of the equation H(x, x1, . . . , xm−1) − 1 = 0 is
+Pr(correct solution|x0 = m0) = Pr(correct solution|x = m1) =
+1
+pm−2 ,
+where Pr(correct solution) denotes probability of finding the correct solution to the equation H(x, x1, . . . , xm−1)−1 =
+0. Then by the law of total probability, the probability of solving the challenge equation is
+Pr(correct solution) = Pr(correct solution|x = m0)Pr(x = m0)+Pr(correct solution|x = m1)Pr(x = m1) =
+1
+pm−2 .
+Accounting for the advantage that the adversary has, the probability α is Pr(correct solution) =
+ϵ
+pm−2 . The total
+probability of solving the IND-CPA challenge is then 1
+2 +
+ϵ
+pm−2 .
+The simulation is indistinguishable from a real attack. So the adversary who can break the challenge ciphertext will
+uncover the solution to the given Modular Diophantine Equation problem. The probability of breaking the ciphertext is
+ϵ
+pm−2 .
+The advantage of solving the Diophantive Equation problem is then
+ϵ
+pm−2 . Let Ts denote the time cost of the simulation.
+We have Ts = O(1). The simulator B solves the Modular Diophantine Equation with time cost and advantage
+(t + Ts, ϵ/pm−2) = (t, ϵ/pm−2). Thus, contradicting the Theorem 4.1 so the initial assumption is wrong.
+The framework of the IND-CPA challenge entails known plaintext, in other words, the adversary knows that the secret
+x ∈ {m0, m1}. We now state the complexity of the unknown plaintext ciphertext-only attack.
+13
+
+Novel Homomorphic Functional Encryption over a Hidden Ring
+Lemma 4.3 (Ciphertext-only attack). Let m + 1 > 2. The classical complexity of finding the plaintext from the
+ciphertext is O(pm−1).
+Proof. Let the adversary favour the ciphertext in its reduced normal form (30). Without any knowledge about the
+plaintext, the adversary will need to solve the Eq. (30) to obtain the plaintext along with the noise variables. A single
+equation over Fp with m variables is expected to have pm−1 possible solutions over Fp. The correct one is among them.
+That is, the plaintext encapsulated in a single variable x is not the sole variable in the ciphertext equation. However, it is
+the only unknown of interest. The adversary can try and simply guess x, the complexity of the guess is O(p). However,
+they have to test whether their guess is correct. Moreover, the secret transferred between the communicating parties
+consist of 32 bytes as required by NIST. Thus, the adversary will have to guess K many values for x, where K = 32×8
+p
+.
+In this case, the complexity is O(pK). We expect K > m − 1. Quantum complexity of the described attack due to
+Grover’s search algorithm is O(p
+m−1
+2 ).
+4.2
+Private key attack
+Lemma 4.4. Let λ ≤ 2. There exists a polynomial time algorithm to find coefficients of univariate polynomials f1(x0)
+and f2(x0) given the plain central maps P1 and P2.
+Proof. Note that all the plain coefficients of the polynomials p1(x, x1, . . . , xm) and p2(x, x1, . . . , xm) as defined in
+Eq. (22) are defined over the prime field Fp. Thus, for any fixed j, it is possible to use Gaussian elimination to reduce
+the system of equations formed by the plain coefficients of p1(x, x1, . . . , xm) and p2(x, x1, . . . , xm) of the form
+�
+�
+�
+�
+�
+�
+�
+�
+�
+fz0b0j = p0jz,
+fz1b0j + fz0b1j = p1jz,
+...
+fzλbnbj = pnjz,
+(31)
+where z ∈ {1, 2} corresponding to either plain polynomial p1(x, x1, . . . , xm) or p2(x, x1, . . . , xm) for any given noise
+variable xj. Gaussian elimination would produce a single polynomial in λ variables, namely fz2
+fz0 and fz1
+fz0 for λ = 2 or
+fz1
+fz0 if λ = 1. Such univariate or bivariate equation is solvable. Depending on the HPPK parameters, the adversary can
+simply use radical solutions, Evdokimov’s algorithm [55], or resultants together with Evdokimov’s algorithm to solve
+such equation [55, 56]. Gaussian elimination can be performed in polynomial time, and finding solutions by radicals,
+Evdokimov’s algorithm and computing resultants all have polynomial time complexity [55, 56].
+Lemma 4.5. Let λ ≤ 2. Finding private key from the cipher public key in the framework of HPPK reduces to finding
+the homomorphic encryption key S, R1, and R2.
+Proof. The private key consists of the coefficients of the univariate polynomials f1(x), f2(x) as well as values S, R1, R2
+used to encrypt the plain public key to the cipher public key. By Lemma 4.4 once the values R1, R2 and S are known,
+the coefficients of f1(x), f2(x) can be found in polynomial time.
+Definition 4.3 (Diophantine set). The Diophantine set is a set S ⊂ N associated with a Diophantine equation
+P(b, a1, . . . , ak) ∈ Z[b, a1, . . . , ak], where k > 0 such that
+b ∈ S if and only if (∃a1, . . . , ak)(P(b, a1, . . . , am) = 0)
+Theorem 4.6 (MRDP Theorem). The Matiyasevich–Robinson–Davis–Putnam (MRDP) theorem states that every
+computably enumerable set is Diophantine, and every Diophantine set is computably enumerable.
+Proof. The result has been proven in various works, for instance [57].
+Theorem 4.7 (Hilbert’s tenth problem). Hilbert’s tenth problem asks whether the general Diophantine Problem is
+solvable. Due to MRDP, Hilbert’s tenth problem is undecidable.
+Proof. For proof see [57].
+Theorem 4.8. Private key attack is non-deterministic and has complexity of at least O(T 3), where T is the largest
+number with 2|p|2 + |L|2 bit-length.
+14
+
+Novel Homomorphic Functional Encryption over a Hidden Ring
+Proof. By Lemma 4.4 and 4.5 the attack on public key reduces to finding the values S, R1, and R2. From the perspective
+of the attacker, the values S, R1, and R2 could be treated as a one-time pad keys as they have been chosen at random,
+and can not be calculated from other parameters given to the attacker. An obvious attack would be a brute force search
+for all the three values, S, R1, and R2. The direct brute force search classical complexity would be greater than O(T 3)
+for the three values together, where T is the largest (2|p|2 + |L|2)-bit number. Due to Grover’s algorithm, the quantum
+complexity is greater than O(T
+3
+2 ). Note however, because of the condition gcd(S, R1) = gcd(S, R2) = 1, once S is
+found the search span for R1 and R2 reduces. Brute force search entails a non-deterministic result, however, we provide
+a more formal argument below.
+For each fixed chose of j, each public key coefficient can be written in the integer domain as follows
+�
+�
+�
+�
+�
+�
+�
+�
+�
+fz0b′
+z0j = r0jzS + P0jz,
+fz1b′
+z0j + fz0b′
+z1j = r1jzS + P1jz,
+...
+fzλb′
+znbj = rnjzS + Pnjz,
+(32)
+with j = 1, . . . , m, z = 1, 2, and b′
+zij = Rzbij. Here, Rz, and S are unknowns from the hidden ring ZS. Values rijz
+are merely some unknown integers. The only known values are those of the form Pijz. Using Gaussian eliminations,
+all unknowns of the form b′
+zij can be eliminated, and the equation system in Eq (32) can be reduced to a single equation
+over Z
+P(fz0, . . . , fzλ, r0jz, . . . , rnjz, S) − ¯P = 0.
+(33)
+Solving such equation by Theorem 4.6 and 4.7 is an NP-complete task. For each j, we can generate one such equation.
+Considering them all together, the adversary will arrive at an underdetermined system as the variables in the system
+depend on j. Each equation in such a system is a multivariate Diophantine equation. One way to solve this system
+is to solve each equation separately and search for common solutions. However, by Theorem 4.6 and 4.7 this is an
+NP-complete problem. Reducing the system to a single polynomial still produces a multivariate Diophantine equation,
+solving which is an NP-complete problem by Theorem 4.6 and 4.7.
+4.3
+Security Conclusion
+At large, the security of the HPPK cryptosystem relies on the problem of solving undetermined system of equations
+over Fp. Such system is expected to have pn−m possible solutions, where n is the number of variables and m is the
+number of equations in the system. The attacker can solve this system to find all possible solutions, however, it is the
+problem of determining the correct solution from all the possible solutions that makes HPPK secure.
+The ciphertext attack requires the adversary to solve an underdetermined system of equations over Fp, which can be
+reduced to a single Modular Diophantine equation. Solving this equation is an NP-complete problem.
+The public key attack aimed to unveil the plaintext reduces to a brute force search for three unknown values S, R1, R2.
+To find these values, the attacker can either use brute-force search or solve an underdetermined system of equations
+over the integers. The former yields non-deterministic results and the latter is an NP-complete problem.
+We conclude that from the point of view of the adversary, the following is true.
+Proposition 4.8.1. The best classical complexity to attack HPPK is O(pm−1).
+Proof. We assume that the malicious party will take the most advantageous path for them. Thus, by Lemma 4.3,
+Lemma 4.4, Lemma 4.5, and Theorem 4.8 we can conclude that the best attack is to obtain the plaintext from the
+ciphertext. Such attack is non-deterministic with classical complexity of O(pm−1).
+5
+Brief Benchmarking Performance
+To account for the best complexity of O(pm−1), we recommend the following configuration to achieve NIST security
+levels I, III, and V, as illustrated in Table 1.
+To measure the performance of the HPPK algorithm we used benchmarking toolkit, called the SUPERCOP. NIST PQC
+finalists used the SUPERCOP for their benchmarking and have contributed the results to the platform, thus, we take
+advantage of the available resources, and report on the performance of HPPK alongside with the NIST PQC schemes,
+namely, McEliece, Kyber, NTRU, and Saber algorithms. From now on we refer to them as the NIST finalists. For our
+work we used a 16-core Intel®Core™i7-10700 CPU at 2.90 GHz system. We have not, however, configured the AVX
+15
+
+Novel Homomorphic Functional Encryption over a Hidden Ring
+Table 1: Configuration of HPPK for different NIST Security Levels.
+Security Level
+Configuration
+Level I
+Level III
+Level V
+(log p, nb, λ, m)
+(64, 1, 1, 3)
+(64, 1, 1, 4),
+(64, 1, 1, 5)
+(log p, nb, λ, m)
+(64, 2, 1, 3)
+(64, 2, 1, 4),
+(64, 2, 1, 5)
+solution for optimized HPPK performance. Therefore, comparisons in this benchmarking performance are set to the
+reference mode for all finalists but in the same computing system.
+We start by illustrating the parameter set of all the measured primitives for all three security levels in the Table 2. As
+required by NIST, the secret is set at 32 bytes. The data illustrates that for each security level, HPPK offers considerably
+small public key sizes and ciphertext sizes for all three security levels compared to all NIST finalists, except for the
+ciphertext size of McEliece at level I and level III. We point out that, HPPK offers the same secret key size of 83 bytes
+and ciphertext size of 208 bytes for all three levels. In comparison with the NIST standardized KEM algorithm Kyber,
+HPPK’s public key sizes are less than half of respective public key sizes for Kyber for all three security levels. Secret
+key sizes for Kyber are about 20 times bigger at level I and close to 40 times bigger at level V than those of HPPK. As
+for the ciphertext sizes, Kyber demonstrates 3.7-7.5 times bigger ciphertext sizes than those of HPPK.
+Table 2: Parameter set of the measured primitives for NIST security levels I, III, and V, given the secret size of 32 bytes
+Crypto
+Size (Bytes)
+system
+Level I
+Level III
+Level V
+PK1
+SK1
+CT1
+PK
+SK
+CT
+PK
+SK
+CT
+McEliece2 [29]
+261120
+6492
+128
+524,160
+13,608
+188
+1,044,992
+13,932
+240
+NTRU4 [31]
+699
+935
+699
+930
+1,234
+930
+1,230
+1,590
+1,230
+Saber5 [33]
+672
+1568
+736
+1,312
+3,040
+1,472
+1,312
+3,040
+1,472
+Kyber3 [30]
+800
+1632
+768
+1,184
+2,400
+1,088
+1,568
+3,168
+1,568
+HPPK(nb = 1)6
+306
+83
+208
+408
+83
+208
+510
+83
+208
+HPPK(nb = 2)6
+408
+83
+208
+544
+83
+208
+680
+83
+208
+1 We denote the secret key as SK, the public key as PK, and the ciphertext as CT.
+2 mceliece348864 primitive was measured for Level I, mceliece460896 primitive was measured for Level III, and mceliece6688128
+for Level V
+3 Kyber512 primitive was measured for Level I, Kyber768 primitive was measured for Level III, and Kyber1024 for Level V
+4 NTRUhps2048509 primitive was measured for Level I, ntruhps2048677 primitive was measured for Level III, and ntruhps4096821
+for Level V
+5 Light Saber primitive was measured for Level I, Saber primitive was measured for Level III, and FireSaber for Level V
+6 For each security level, HPPK primitive is configured as shown in Table 1
+Table 3 provides the reader with median values in clock cycles of the measurement results for the key generation
+procedure of all the primitives configured to provide security levels I, III, and V. To provide a bigger picture we include
+results for RSA-2048. The results correspond only to security level I, as RSA-2048 provides 112 bits of entropy. The
+reader can see that HPPK key generation performance is rather fast, with median clock cycles of over 18, 000 for nb = 1
+and 22, 000 for nb = 2 for level I, over 21, 000 for nb = 1 and 28, 000 for nb = 2 for level III, and over 26, 000 for
+nb = 1 and 34, 000 for nb = 2 for level V. The fastest key generation performance among the NIST finalists is offered
+by Saber and Kyber, with median values of over 39, 000 and 72, 000 clock cycles respectively for level I, median values
+over 115000 for level III, and median values of 128, 000 clock cycles for level V. Compared to the standard algorithm
+Kyber, HPPK demonstrates a 3.5-6 times faster key generation performance. The remaining primitives measured,
+including RSA, display median values of over 6 million clock cycles for level I, over 10 million for level III, and over
+16 million clock cycles for level V.
+We provide Table 4 to illustrate encryption procedure performance of HPPK, NIST finalists, and RSA-2048. The table
+illustrates median values given in clock cycles. HPPK offers fast encryption with clock cycles from 17,000 for security
+level I to 25,000 for security level V, outperforming all mentioned NIST finalists. More specifically comparing the
+NIST standard algorithm, Kyber, to HPPK the table illustartes that Kyber offers 4-8 times slower encryption than HPPK
+for all levels. However, RSA-2048 offers the fastest encryption performance among all the other primitives measured
+for level I, due to its small public encryption key, usually chosen to be 65535.
+In Table 5 we illustrate median values given in clock cycles for the decryption procedure corresponding to the HPPK
+algortihm, the NIST finalists algorithms, and RSA-2048. Table 5 shows that HPPK offers fast decryption performance
+16
+
+Novel Homomorphic Functional Encryption over a Hidden Ring
+Table 3: Median values of the key generation performance for NIST security levels I, III, and V.
+Crypto
+Performance (Clock cycles)
+system
+Level I
+Level III
+Level V
+McEliece1 [29]
+152,424,455
+509,364,485
+1,127,581,201
+NTRU3 [31]
+6,554,031
+10,860,295
+16,046,953
+Saber4 [33]
+39,654
+128,935
+128,412
+Kyber2 [30]
+72,403
+115,654
+177,818
+HPPK (nb = 1)5
+18,034
+21,946
+26,603
+HPPK(nb = 2)5
+22,625
+28,360
+34,719
+RSA-2048
+91,985,129
+-
+-
+1 mceliece348864 primitive was measured for Level I, mceliece460896 primitive was measured for Level III, and mceliece6688128
+for Level V
+2 Kyber512 primitive was measured for Level I, Kyber768 primitive was measured for Level III, and Kyber1024 for Level V
+3 NTRUhps2048509 primitive was measured for Level I, ntruhps2048677 primitive was measured for Level III, and ntruhps4096821
+for Level V
+4 Light Saber primitive was measured for Level I, Saber primitive was measured for Level III, and FireSaber for Level V
+5 For each security level, HPPK primitive is configured as shown in Table 1
+Table 4: Median values of the key encapsulation performance estimated in clock cycles for NIST security levels I, III,
+and V.
+Crypto
+Performance (Clock cycles)
+system
+Level I
+Level III
+Level V
+McEliece1 [29]
+108,741
+172,538
+263,169
+NTRU3 [31]
+418,622
+703,046
+1,063,124
+Saber4 [33]
+62,154
+157,704
+157,521
+Kyber2 [30]
+95,466
+140,376
+205,505
+HPPK(nb = 1)5
+17,354
+20,951
+25,087
+HPPK(nb = 2)5
+23,073
+28,642
+35,173
+RSA-2048
+13,429
+-
+-
+1 mceliece348864 primitive was measured for Level I, mceliece460896 primitive was measured for Level III, and mceliece6688128
+for Level V
+2 Kyber512 primitive was measured for Level I, Kyber768 primitive was measured for Level III, and Kyber1024 for Level V
+3 NTRUhps2048509 primitive was measured for Level I, ntruhps2048677 primitive was measured for Level III, and ntruhps4096821
+for Level V
+4 Light Saber primitive was measured for Level I, Saber primitive was measured for Level III, and FireSaber for Level V
+5 For each security level, HPPK primitive is configured as shown in Table 1
+with median values for level I being at 28, 000 clock cycles. Meanwhile, median values for the faster NIST finalists
+are over 63, 000 and 117, 000 clcok cycles for Saber and Kyber respectively configured to provide level I security.
+Both RSA-2048 and NTRU have median values over 1 million clock cycles for level I. McEliece median value is over
+45 million clock cycles. For levels III and V the account is similar. HPPK offers median values for both of these
+levels which fall in the interval of [28000, 30000] clock cycles, while median values for Saber are over 170, 000 for
+levels III and V. Median values for Kyber fall into the interval of [166000, 238000] for levels III and V. NTRU displays
+median values of over 2 million clock cycles for level III and V. McEliece offers the slowest decryption procedure with
+median values being over 93 million for level III, and 179 million for level V. HPPK decryption demonstrates a stable
+performance at 30, 000 clock cycles for all security levels, due to its special decryption mechanism with a modular
+division.
+6
+Conclusion
+In this paper, we introduced a new Functional Homomorphic Encryption, which in contrast with conventional homomor-
+phic encryption, is intended to secure public keys of multivariate asymmetric cryptosystems. Functional homomorphic
+encryption is applied to polynomials, to leverage homomorphic properties and allow for user input through variables.
+The functional homomorphic encryption and decryption operators are multiplication operators modulo a hidden value S,
+with values R1 and R2 respectively. Such values R1 and R2 are chosen uniformly at random from the hidden ring ZS
+with certain conditions. We propose to use said homomorphic encryption in conjunction with Multivariate Polynomial
+17
+
+Novel Homomorphic Functional Encryption over a Hidden Ring
+Table 5: Median values of key decapsulation performance for NIST security levels I, III, and V.
+Crypto
+Performance (Clock cycles)
+system
+Level I
+Level III
+Level V
+McEliece1 [29]
+45,119,775
+93,121,708
+179,917,369
+NTRU3 [31]
+1,245,062
+2,099,254
+3,129,150
+Saber4 [33]
+63,048
+173,712
+177,109
+Kyber2 [30]
+117,245
+166,062
+237,484
+HPPK(nb = 1)5
+28,301
+28,759
+29,671
+HPPK (nb = 2)5
+29,791
+29,266
+29,364
+RSA-2048
+1,670,173
+-
+-
+1 mceliece348864 primitive was measured for Level I, mceliece460896 primitive was measured for Level III, and mceliece6688128
+for Level V
+2 Kyber512 primitive was measured for Level I, Kyber768 primitive was measured for Level III, and Kyber1024 for Level V
+3 NTRUhps2048509 primitive was measured for Level I, ntruhps2048677 primitive was measured for Level III, and ntruhps4096821
+for Level V
+4 Light Saber primitive was measured for Level I, Saber primitive was measured for Level III, and FireSaber for Level V
+5 For each security level, HPPK primitive is configured as shown in Table 1
+Public-key Cryptography, to secure the polynomial public keys, however, we do not study the encrypted MPKC in
+detail. Instead, we suggest a new variant of multivariate public-key cryptosystem with public keys encrypted using
+homomorphic encryption, called Homomorphic Polynomial Public Key or HPPK. We described the HPPK algorithm
+in detail, with the framework drawn from MPKC. HPPK public keys are product polynomials of a multivariate and
+univariate polynomials, encrypted with a homomorphic encryption operator. The ciphertext is created by the encrypting
+party through the input of plaintext and random noise as public polynomial variables. The decryption procedure
+involves first decrypting the public key, to nullify the homomorphic encryption and produce the original ciphertext.
+Said ciphertext is used to divide two product polynomials. By construction, such division cancels the base multiplicand
+polynomial with noise variable. and retains a single equation in one variable. Said variable is the plaintext, which
+can be found by radicals. We give a thorough security analysis of the HPPK cryptosystem, proving that the hardness
+of breaking the HPPK algorithm comes from the computational hardness of the Modular Diophantine Equation, and
+Hilbert’s tenth problem. We also show that HPPK holds IND-CPA property. We report briefly on benchmarking the
+performance of the HPPK cryptosystem, using the NIST-recognized SUPERCOP benchmarking tool, with nb = 1 and
+λ = 1. The benchmarking data illustrates that the HPPK offers rather small public keys and comparable ciphertext sizes.
+The key generation, key encapsulation, and key decapsulation procedure performance are efficient, being noticeably
+faster when considered together with the NIST PQC finalists. If the degree of univariate polynomial f1(x) and f2(x)
+are higher than 1, such as quadratic polynomials, the decryption would produce multiple roots, then an extra verification
+procedure is required. Moreover, the decryption speed would be dramatically slower than linear polynomials f1(x) and
+f2(x). In the future work, we will perform more detail benchmarking with variety of configurations as well as a more
+extensive security analysis, considering attacks that have not been described in the work.
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+21
+

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+International Journal on AdHoc Networking Systems (IJANS) Vol. 12, No. 4, October 2022 
+DOI: 10.5121/ijans.2022.12403                                                                                                                     35 
+ 
+SENSOR SIGNAL PROCESSING USING HIGH-LEVEL 
+SYNTHESIS AND INTERNET OF THINGS WITH A 
+LAYERED ARCHITECTURE 
+ 
+CS Reddy1 and Krishna Anand2 
+ 
+1Department of Mathematics, CIT - NC, VTU University, Bangalore, India  
+2Department of Computer Engineering, Anurag University, Hyderabad, India 
+ 
+ABSTRACT 
+ 
+Sensor routers play a crucial role in the sector of Internet of Things applications, in which the capacity for 
+transmission of the network signal is limited from cloud systems to sensors and its reversal process. It 
+describes a robust recognized framework with various architected layers to process data at high level 
+synthesis. It is designed to sense the nodes instinctually with the help of Internet of Things where the 
+applications arise in cloud systems. In this paper embedded PEs with four layer new design framework 
+architecture is proposed to sense the devises of IOT applications with the support of high-level synthesis 
+DBMF (database management function) tool. 
+ 
+KEYWORDS 
+ 
+Network Protocols, Wireless Network, Mobile Network, Internet of Things, Reconfigurable dynamic 
+processor, Sensor signal processing. 
+ 
+1. INTRODUCTION 
+ 
+Sensor routers play a crucial role in the sector of Internet of Things (IOT) applications, in which 
+the capacity for transmission of the network signal is limited from cloud systems to sensors and 
+its reversal process. Consequence to this volume of the data reduction is obligatory to combat 
+device computing functions between sensor nodes and transmitter to exchange the sufficient data 
+with the available networks [1-3]. Hence, low power consumption and small footprints are 
+commanded among sensor nodes to process information. One of the replacements for 
+microcontroller units is field programmable gate arrays to optimize the footprints size, so that it is 
+to be observed keenly the routing with configurable logic blocks and switches of look-up tables 
+which causes placement inefficiency [4, 7]. To fabricate Field Programmable Gate Arrays 
+(FPGA) it is wise to use High-level synthesis which will enable global optimization and 
+recompense the limitation of Routing and Placement [5, 6]. 
+ 
+In this paper embedded PEs with four layer new design framework architecture is proposed to 
+sense the devises of IOT applications with the support of high-level synthesis DBMF (database 
+management function) tool [8, 17, 19]. It exploits the repetitive high level synthesis process. 
+Macro blocks synthesized through high-level behavioural synthesis are registered in a database 
+before the system level synthesis, and the information in the database is used for the optimization 
+of resource consumption through the system level synthesis [10]. In this work authors tried to 
+investigate the dependencies of resource consumption on the granularity of coarse grained 
+function definitions using the extended database management function of Cyber Work Bench. 
+The evaluation results show that small footprint was achieved especially with dynamically 
+reconfigurable technique [9, 15]. 
+
+International Journal on AdHoc Networking Systems (IJANS) Vol. 12, No. 4, October 2022 
+36 
+Dynamically reconfigurable processors using high-level synthesis were proposed to improve the 
+efficiency, whereas the inefficiency of fixed mesh pointed out in still remains [11-14]. Fixed bit 
+widths of data paths, elementary blocks, and switch matrices aiming at mass production of the 
+devices were one example of the inefficiency. Sensors used in IOT applications have various data 
+interfaces, such as 8, 12, 14, or 16 bits [ 17, 19]. Predefined data path between arrays of 
+arithmetic logic units prevents behavioural synthesis tools from the optimization of layout size 
+and the reduction in power consumption. Therefore, optimized Arithmetic Logic Units (ALU) 
+and flexible data paths are required to embed processors in sensor units [13, 15-17]. 
+ 
+2. HIGH-LEVEL SYNTHESIS TOOLS  
+ 
+High-level synthesis is increasingly popular for the design of high-performance and energy-
+efficient heterogeneous systems, shortening time-to-market and addressing today’s system 
+complexity [18, 20, 25]. Early academic work extracted scheduling, allocation, and binding as the 
+basic steps for high-level-synthesis [22, 24, 32]. Scheduling partitions in the algorithm to control 
+steps that are used in the model are defined the states in the finite-state machine [21, 33].  
+ 
+First generation behavioural synthesis was introduced by Synopsys in 1994 as behavioural 
+Compiler and used Verilog or VHDL as input languages.  10 years later, in 2004, there emerged a 
+number of next generation commercial high-level synthesis products which provided synthesis of 
+circuits specified at C level to a register transfer level specification [23, 25, 26]. It was primarily 
+adopted in Japan and Europe in the early years. As of late 2008, there was an emerging adoption 
+in the United States. High-level synthesis (HLS) allows designers to work at a higher level of 
+abstraction by using a software program to specify the hardware functionality [28, 29]. 
+Additionally, HLS is particularly interesting for designing field-programmable gate array circuits, 
+where hardware implementations can be easily refined and replaced in the target device [27, 30-
+32]. Recent years have seen much activity in the HLS research community, with a plethora of 
+HLS tool offerings, from both industry and academia. 
+ 
+ 
+ 
+Fig.1. Classification of High-Level Synthesis Input Language. 
+
+Applicationdomains:
+HighLevelSynthesis
+Tool status:
+All domains
+Imaging
+Tools
+InUse
+Streaming
+Stream/lmage
+Abandoned
+Loop/Pipeline
+NIA
+DSP
+DataFlow
+.NET
+DomainSpecific
+Generic
+ODSE
+Languages
+Languages
+NEW
+C-extended
+Procedural
+ObjectOriented
+Languages
+Languages
+Languages
+Languages
+O CyberWorkBench (BDL)
+CoDeveloper (ImpulseC)
+O Vivado HLS
+CtoVerilog
+Maxeler (MaxJ)
+OBluespec (BSV)
+DK Design Suite (HandelC)
+oCatapult
+O C2H
+X
+KIWI (C#)
+PipeRench(DIL)
+ SA-C (SA-C)
+oCtos
+OSynphHLS
+SeaCucumber (Java)
+O HercuLeS (NAC)
+Garp (Cpragmas)
+SPARK
+MATCH
+Cynthesizer (SystemC)
+Napa-C (Cpragmas)
+O CHC
+AccelDSP
+eXCite (CSP pragmas)
+O LegUp
+O CHiMPS
+ROCCC (C extended)
+OBambu
+?DEFACTO
+GAUT
+ogcc2verilog
+TridentInternational Journal on AdHoc Networking Systems (IJANS) Vol. 12, No. 4, October 2022 
+37 
+HLS tools start from a software programmable high-level language to automatically produce a 
+circuit specification in HDL that performs the same function. The most common source inputs for 
+high-level synthesis are based on standard languages such as ANSI C/C++, System C and 
+MATLAB. HLS has also been recently applied to a variety of applications (e.g., medical 
+imaging, convolutional neural networks, and machine learning), with significant benefits in terms 
+of performance and energy consumption. 
+ 
+These HLS tools can leverage dedicated optimizations or micro architectural solutions for the 
+specific domain. However, the algorithm designer, who is usually a software engineer, has to 
+understand how to properly update the code. This approach is usually time-consuming and error 
+prone. For this reason, some HLS tools offer complete support for a standard HLL, such as C, 
+giving complete freedom to the algorithm designer. High level languages mainly classified into 
+two types that are shown in Fig. 1. One is Domain specific languages and other one is Generic 
+languages. In Domain specific languages it consist new languages and C- extended languages, 
+whereas, in Generic languages it consist Procedural and Object Oriented languages. HLS tool 
+select specific language for specific applications. 
+ 
+2.1. Five Key Challenges 
+ 
+In this section the important crucial challenges that arise in the process of signal synthesis in the 
+layered architecture.  
+ 
+ Hardware in inherently concurrent, whereas software representations are sequential. HLS 
+must map the sequential algorithm onto concurrent hardware. 
+ Timing is implicit in software in the sequence of instructions used. Synchronous hardware 
+must deal with timing constraints, with controlling and synchronizing operations at the clock 
+cycle level. 
+ In software, the word length is fixed (8, 16, 32 or 64bits), but in hardware, it is usually 
+optimized for the task being performed. 
+ The software model of memory is as a single block with a monolithic address space, with 
+almost all data items stored in memory. On an FPGA, local variables are stored in registers, 
+with multiple distributed memory blocks, each with their own independent address space. In 
+such an environment, pointers have little meaning, and dynamic memory allocation is very 
+difficult. Communication in software is usually through shared memory, whereas on an 
+FPGA it relies on constructing appropriate hardware, from implicit (within stream 
+processing), to simple token passing, to using dedicated FIFOs to manage flow control. 
+ 
+3. DATA LAYER INTERACTION ARCHITECTURE 
+ 
+ 
+Traditionally, the word architecture is concerned with the art or science of building, where it 
+relates to structural concepts and to styles of design. Here the authors will use it in the sense of 
+structured deceptions of a system from a number of compatible and complementary viewpoints 
+which, taken together, cover functional, design, fabrication and performance issues. The term 
+layer is used to refer to a particular descriptive viewpoint. Within a layer, descriptions will 
+generally be hierarchic to allow the containment of complexity or the exposure of detail by 
+suitable aggregation or decomposition techniques. Architecture(s) may be singular or plural 
+depending on whether it will be addressed as an individual layer or the complete set of 
+descriptions across all layers. It is clear from this definition that the architecture of a system is not 
+just confined to some sort of functional partitioning at the front end of a development, but 
+essentially embraces descriptions generated during all phases and stages of the development 
+process. 
+
+International Journal on AdHoc Networking Systems (IJANS) Vol. 12, No. 4, October 2022 
+38 
+The Data Interaction Architecture is a prime example of a layered architecture. It contains four 
+layers, each of which can be regarded as a model of the system from a particular viewpoint. 
+Fig.2. shows some small representational fragments for each of the four layers. The layers consist 
+of I/O circuitry, fine grained layer, coarse grained function definition layer, and bypass 
+connection layer, where they are listed from the bottom layer to the top layer, respectively. Well 
+defined notations and technical conventions exist for each of these layers and are closely 
+modelled on those of MASCOT, Modular Approach to Software Construction Operation and 
+Test. Where relevant the notations include composite structures to support hierarchical 
+representations over multiple levels within a layer. The motivation behind the layered 
+architecture method is to provide the means of moving from a functional model of a system to an 
+execution model. 
+ 
+ 
+ 
+Fig.2. Data Interaction Architecture 
+ 
+The emphasis on the identification of well-defined interactions in the functional model, and the 
+preservation of these speck tied interaction properties through the subsequent layers, provides the 
+structural conformance by which traceability is achieved. It follows that the functional model is 
+binding on subsequent development. Developers are not free to reverse engineer the functional 
+model to allow an alternative development path in which traceability would be lost. Of course the 
+functional model may need to be changed, either because system requirements have changed, or 
+because detail design has shown that an alternative functional model would be better. Whatever 
+the reason for change, consistency across the model set must be preserved. 
+ 
+The main distinguishing feature of Data Interaction Architecture (DIA) is the explicit 
+representation of shared information and shared data. Here i will concentrate on the sharing 
+
+Model
+Representation example
+Focus
+Functional
+What
+Generator
+User
+Design
+How
+Writer
+Reader
+Distribution
+Where
+Writer
+Reader
+Execution
+When
+Writer
+Reader
+V
+Kemel
+KermelInternational Journal on AdHoc Networking Systems (IJANS) Vol. 12, No. 4, October 2022 
+39 
+implicit in the bilateral interactions between processes, leaving discussion of the more general 
+information retention element. A summary description of the model in each layer shows how 
+bilateral interaction properties are tracked. 
+ 
+4. ARCHITECTURE OF PROCESSING ELEMENTS 
+ 
+In this paper authors proposed to establish a new design framework to solve issues described in 
+the preceding section by exploiting the binding process of high-level/behavioural synthesis. 
+ 
+4.1. Layered Architecture 
+ 
+The purpose of the conventional high-level synthesis tools is to generate finite state machines 
+with data paths. Here File System Meta-Data (FSMD) referred as fine grained layer. It is 
+expanded with the scope of high-level synthesis to coarse grained layer to exploit the functional 
+representations of circuitries. ALUs in coarse grained layer are defined by the functional 
+representations of high-level programming languages. Switches are added as bypass connection 
+layer for the scope of high-level synthesis intentionally. Implicit as-built meshes of switches put 
+constraints on high-level synthesis. In the processes are allowed as unprejudiced bypass switches 
+to remove the meshes. Communications between the PEs are limited between adjacent PEs, 
+whereas it is found no limitations for sensor applications with the limited communication paths. 
+ 
+The following explains these layers I/O circuitry is implemented with random logic gates and 
+mixed signal I/O circuitries. They are connected to the fine grained blocks in the above layer. The 
+fine grained layer mainly consists of finite state machines and data paths. They are replaceable to 
+follow the context described by high-level languages. The coarse grained function definition 
+layer is located on the fine grained layer. Optimized ALUs for a certain application are defined in 
+this layer. An operation primitive defined in the layer corresponds to an operation code (op code) 
+of a conventional multipoint control unit (MCU), whereas it does not have to be standardized for 
+over many applications. As for the topmost bypass connection layer, simple bypass switch 
+images can be specified over coarse grained blocks. The connections between inputs and outputs 
+of PEs are configurable with the bypass switches. 
+ 
+4.2. Design Flow 
+ 
+The following six steps compose the design flow of the design framework to design a PE through 
+layered scheme. 
+ 
+Step 1: Describe a system in high-level language. 
+Step 2: Define coarse grained operations. Typical dedicated functions for sensor signals are 
+signal compensation, feature point identification for image recognition, and image 
+compression in addition to basic arithmetic operations. The defined coarse grained 
+operations are exploited in high-level synthesis and behavioural synthesis process. 
+Step 3: Generate source codes written in hardware description language through behavioural 
+synthesis. The hard macros defined and implemented in the step 2 are exploited for 
+generating circuitries by the functions of CWB. 
+Step 4: This step is identical to conventional logic synthesis. 
+Step 5: This step is identical to conventional layout design. 
+Step 6: The verification and validation step include back annotation based on the result of delay 
+analysis. 
+ 
+
+International Journal on AdHoc Networking Systems (IJANS) Vol. 12, No. 4, October 2022 
+40 
+In the work authors used a data base management function of CWB to exploit the definitions of 
+operations in the coarse grained layer during the behavioural synthesis to treat a function as an 
+operator. The operations can be implemented as hard macros by using custom ASIC design tools 
+and/or LUTs of FPGAs. The operator definitions were exploited on the step 3 as macro blocks. 
+Once macro blocks are registered in a database, CWB can use the macro blocks during the 
+binding process of high-level synthesis automatically to reduce layout area size. The binding 
+process of high-level synthesis is regarded as nondeterministic polynomial (NP) hard and 
+heuristic approach was often employed. The design flow enables the automation of binding 
+process instead of the heuristic approach. 
+ 
+4.3. Typical Design of Processing Element 
+ 
+Two types of context were identified with reference to semantics or lambda calculus of functional 
+representations to define ALUs in coarse grained function definition layer. One is a configurable 
+static context often mentioned as stored information in a file, and the other is a dynamic context 
+as stored information in heap registers as shown in Fig. 3. The static contexts of an application 
+are expressed with finite state machines and data paths implemented by n sets of hard- ware 
+circuitry of PEs. The dynamic contexts of an application are specified as m sets of registers and 
+instructions. The instruction sets are optimized for an application, and each optimized instruction 
+set can be shared among some dynamic contexts. 
+ 
+ 
+ 
+Fig.3. Modified High-Level Synthesis Process. 
+ 
+The program code is implemented with a specific C-language comment description /* Cyber func 
+= operator */ for defining dynamic and static contexts, which are the sets of pairs of registers and 
+optimized instruction sets. The pair of a register set and an instruction set can be shared among 
+dynamic contexts using another specific C-language comment description as /* Cyber share name 
+= NAME */. NAME is an arbitrary designation. The binding process of high-level behavioural 
+
+Higher
+Hard
+Hard
+module
+Macro
+Macro
+Highlevel/Behavioral
+Highlevel /Behavioral
+Highlevel/Behavioral
+synthesis
+synthesis
+synthesis
+CDFGgeneration
+Allocation
+不
+LMSPEC
+RTL
+LMSPEC
+RTL
+Scheduling
+不
+Binding
+FSMDgeneration
+RTL
+Logic synthesis
+Place and Route
+RTL:RegisterTransferLevel
+不
+LMSPEC:LowerModuleSpecificationFile
+CDFG:ControlandDataFlowGraph
+Configuration
+FSMD:FiniteStateMachinewithDatapath
+DataInternational Journal on AdHoc Networking Systems (IJANS) Vol. 12, No. 4, October 2022 
+41 
+synthesis can be controlled by these descriptions to exploit the functional representations defined 
+in a high-level programming language. 
+ 
+5. RESULTS   
+ 
+Experiments are carried out to evaluate the design framework using convolution operations; those 
+are often used for sensor applications, with following three conditions. The matrix functions of 
+the filters are similar and the difference is the parameters of 3×3 matrices. It is observed one can 
+evaluate the layout area size reduction results by using an FPGA, Xilinx XC7A200T FPGA, 
+although the design framework was established aiming at improving ASIC design at first. 
+ 
+ Defining a convolution function as one operator. 
+ Implementing ALUs with basic operations, i.e., plus, minus, multiply, divide, and 
+comparison operations, using dynamically reconfiguration technique designated as 
+flexible reliability reconfigurable array (FRRA). 
+ Elaborating whole design with FPGA libraries without operator definitions and a 
+function definition database. 
+ 
+ 
+ 
+Fig.4. Comparison of LUT utilizations According to Operator Definitions 
+ 
+Derived LUT counts for each evaluation case are shown in Fig.4. The LUT counts of basic 
+operations were excluded for comparison. The larger the granularity of an operation was, the less 
+counts of LUTs were consumed. Remarkable difference of LUT counts was shown for cascaded 
+operations. The LUT counts of cascaded operations using FPGA library was more than double 
+compared to single operation. The results show that exploiting granularity in behavioural 
+synthesis was carried out by CWB automatically without specifying the reuse of macro blocks 
+explicitly. The difference between one operation and cascaded operations using FRRAs with 
+dynamically reconfiguration technique was 22% less than the difference using the operator 
+definition on convolution operations. 
+ 
+6. CONCLUSIONS 
+ 
+A novel model is developed to design a framework to reduce the footprints of programmable 
+functions of sensing devices for IOT applications. The design framework consists of four layered 
+structure of PE architecture and the extended database management function of a high-level 
+synthesis tool to exploit functional representations in high-level programming languages. 
+
+Laplacian Filter
+GaussianFilter+LaplacianFilter
+(a)
+(b)
+(c)
+0
+200
+400
+600
+800
+LUT countsInternational Journal on AdHoc Networking Systems (IJANS) Vol. 12, No. 4, October 2022 
+42 
+The layout size reduction is useful to embed a PE into a sensing device and to provide device 
+computing capabilities with a sensing device. The experimental results report the power 
+consumption reduction by adopting the design framework on a Nano Bridge FPGA. 
+ 
+REFERENCES 
+ 
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+Hihara, H., Iwasaki, A., Hashimoto, M., Ochi, H., Mitsuyama, Y., Onodera, H., ... & Sakamoto, T. 
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+R. Zhao, M. Tan, S. Dai, and Z. Zhang, “Area-efficient pipelining for FPGA-targeted high-level 
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+Saritha, S., Mamatha, E., Reddy, C. S., & Rajadurai, P. (2022). “A model for overflow queuing 
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+ 
+ 
+

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+page_content='12403                                                                                                                     35  SENSOR SIGNAL PROCESSING USING HIGH-LEVEL SYNTHESIS AND INTERNET OF THINGS WITH A LAYERED ARCHITECTURE  CS Reddy1 and Krishna Anand2  1Department of Mathematics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfrgXE/content/2301.03356v1.pdf'}
+page_content=' CIT - NC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfrgXE/content/2301.03356v1.pdf'}
+page_content=' VTU University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfrgXE/content/2301.03356v1.pdf'}
+page_content=' Bangalore,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfrgXE/content/2301.03356v1.pdf'}
+page_content=' India 2Department of Computer Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfrgXE/content/2301.03356v1.pdf'}
+page_content=' Anurag University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfrgXE/content/2301.03356v1.pdf'}
+page_content=' Hyderabad,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfrgXE/content/2301.03356v1.pdf'}
+page_content=' India  ABSTRACT  Sensor routers play a crucial role in the sector of Internet of Things applications,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfrgXE/content/2301.03356v1.pdf'}
+page_content=' in which the capacity for transmission of the network signal is limited from cloud systems to sensors and its reversal process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfrgXE/content/2301.03356v1.pdf'}
+page_content=' It describes a robust recognized framework with various architected layers to process data at high level synthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfrgXE/content/2301.03356v1.pdf'}
+page_content=' It is designed to sense the nodes instinctually with the help of Internet of Things where the applications arise in cloud systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfrgXE/content/2301.03356v1.pdf'}
+page_content=' In this paper embedded PEs with four layer new design framework architecture is proposed to sense the devises of IOT applications with the support of high-level synthesis DBMF (database management function) tool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfrgXE/content/2301.03356v1.pdf'}
+page_content='  KEYWORDS  Network Protocols, Wireless Network, Mobile Network, Internet of Things, Reconfigurable dynamic processor, Sensor signal processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfrgXE/content/2301.03356v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfrgXE/content/2301.03356v1.pdf'}
+page_content=' INTRODUCTION  Sensor routers play a crucial role in the sector of Internet of Things (IOT) applications, in which the capacity for transmission of the network signal is limited from cloud systems to sensors and its reversal process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfrgXE/content/2301.03356v1.pdf'}
+page_content=' Consequence to this volume of the data reduction is obligatory to combat device computing functions between sensor nodes and transmitter to exchange the sufficient data with the available networks [1-3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfrgXE/content/2301.03356v1.pdf'}
+page_content=' Hence, low power consumption and small footprints are commanded among sensor nodes to process information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfrgXE/content/2301.03356v1.pdf'}
+page_content=' One of the replacements for microcontroller units is field programmable gate arrays to optimize the footprints size, so that it is to be observed keenly the routing with configurable logic blocks and switches of look-up tables which causes placement inefficiency [4, 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfrgXE/content/2301.03356v1.pdf'}
+page_content=' To fabricate Field Programmable Gate Arrays (FPGA) it is wise to use High-level synthesis which will enable global optimization and recompense the limitation of Routing and Placement [5, 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfrgXE/content/2301.03356v1.pdf'}
+page_content='  In this paper embedded PEs with four layer new design framework architecture is proposed to sense the devises of IOT applications with the support of high-level synthesis DBMF (database management function) tool [8, 17, 19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfrgXE/content/2301.03356v1.pdf'}
+page_content=' It exploits the repetitive high level synthesis process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfrgXE/content/2301.03356v1.pdf'}
+page_content=' Macro blocks synthesized through high-level behavioural synthesis are registered in a database before the system level synthesis, and the information in the database is used for the optimization of resource consumption through the system level synthesis [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfrgXE/content/2301.03356v1.pdf'}
+page_content=' In this work authors tried to investigate the dependencies of resource consumption on the granularity of coarse grained function definitions using the extended database management function of Cyber Work Bench.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfrgXE/content/2301.03356v1.pdf'}
+page_content=' The evaluation results show that small footprint was achieved especially with dynamically reconfigurable technique [9, 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfrgXE/content/2301.03356v1.pdf'}
+page_content='  International Journal on AdHoc Networking Systems (IJANS) Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfrgXE/content/2301.03356v1.pdf'}
+page_content=' 12, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfrgXE/content/2301.03356v1.pdf'}
+page_content=' 4, October 2022 36 Dynamically reconfigurable processors using high-level synthesis were proposed to improve the efficiency, whereas the inefficiency of fixed mesh pointed out in still remains [11-14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfrgXE/content/2301.03356v1.pdf'}
+page_content=' Fixed bit widths of data paths, elementary blocks, and switch matrices aiming at mass production of the devices were one example of the inefficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfrgXE/content/2301.03356v1.pdf'}
+page_content=' Sensors used in IOT applications have various data interfaces, such as 8, 12, 14, or 16 bits [ 17, 19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfrgXE/content/2301.03356v1.pdf'}
+page_content=' Predefined data path between arrays of arithmetic logic units prevents behavioural synthesis tools from the optimization of layout size and the reduction in power consumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfrgXE/content/2301.03356v1.pdf'}
+page_content=' Therefore, optimized Arithmetic Logic Units (ALU) and flexible data paths are required to embed processors in sensor units [13, 15-17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfrgXE/content/2301.03356v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfrgXE/content/2301.03356v1.pdf'}
+page_content=' HIGH LEVEL SYNTHESIS TOOLS  High-level synthesis is increasingly popular for the design of high-performance and energy- efficient heterogeneous systems, shortening time-to-market and addressing today’s system complexity [18, 20, 25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfrgXE/content/2301.03356v1.pdf'}
+page_content=' Early academic work extracted scheduling, allocation, and binding as the basic steps for high-level-synthesis [22, 24, 32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfrgXE/content/2301.03356v1.pdf'}
+page_content=' Scheduling partitions in the algorithm to control steps that are used in the model are defined the states in the finite-state machine [21, 33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfrgXE/content/2301.03356v1.pdf'}
+page_content='  First generation behavioural synthesis was introduced by Synopsys in 1994 as behavioural Compiler and used Verilog or VHDL as input languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfrgXE/content/2301.03356v1.pdf'}
+page_content=' 10 years later, in 2004, there emerged a number of next generation commercial high-level synthesis products which provided synthesis of circuits specified at C level to a register transfer level specification [23, 25, 26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfrgXE/content/2301.03356v1.pdf'}
+page_content=' It was primarily adopted in Japan and Europe in the early years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfrgXE/content/2301.03356v1.pdf'}
+page_content=' As of late 2008, there was an emerging adoption in the United States.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfrgXE/content/2301.03356v1.pdf'}
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+page_content=' 4, October 2022 37 HLS tools start from a software programmable high-level language to automatically produce a circuit specification in HDL that performs the same function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfrgXE/content/2301.03356v1.pdf'}
+page_content=' The most common source inputs for high-level synthesis are based on standard languages such as ANSI C/C++, System C and MATLAB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfrgXE/content/2301.03356v1.pdf'}
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+page_content=', medical imaging, convolutional neural networks, and machine learning), with significant benefits in terms of performance and energy consumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfrgXE/content/2301.03356v1.pdf'}
+page_content='  These HLS tools can leverage dedicated optimizations or micro architectural solutions for the specific domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfrgXE/content/2301.03356v1.pdf'}
+page_content=' However, the algorithm designer, who is usually a software engineer, has to understand how to properly update the code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfrgXE/content/2301.03356v1.pdf'}
+page_content=' This approach is usually time-consuming and error prone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfrgXE/content/2301.03356v1.pdf'}
+page_content=' For this reason, some HLS tools offer complete support for a standard HLL, such as C, giving complete freedom to the algorithm designer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfrgXE/content/2301.03356v1.pdf'}
+page_content=' High level languages mainly classified into two types that are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfrgXE/content/2301.03356v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfrgXE/content/2301.03356v1.pdf'}
+page_content=' One is Domain specific languages and other one is Generic languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfrgXE/content/2301.03356v1.pdf'}
+page_content=' In Domain specific languages it consist new languages and C- extended languages, whereas, in Generic languages it consist Procedural and Object Oriented languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfrgXE/content/2301.03356v1.pdf'}
+page_content=' HLS tool select specific language for specific applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfrgXE/content/2301.03356v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfrgXE/content/2301.03356v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfrgXE/content/2301.03356v1.pdf'}
+page_content=' Five Key Challenges  In this section the important crucial challenges that arise in the process of signal synthesis in the layered architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfrgXE/content/2301.03356v1.pdf'}
+page_content='  \uf0d8 Hardware in inherently concurrent, whereas software representations are sequential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfrgXE/content/2301.03356v1.pdf'}
+page_content=' HLS must map the sequential algorithm onto concurrent hardware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfrgXE/content/2301.03356v1.pdf'}
+page_content=' \uf0d8 Timing is implicit in software in the sequence of instructions used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfrgXE/content/2301.03356v1.pdf'}
+page_content=' Synchronous hardware must deal with timing constraints, with controlling and synchronizing operations at the clock cycle level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfrgXE/content/2301.03356v1.pdf'}
+page_content=' \uf0d8 In software, the word length is fixed (8, 16, 32 or 64bits), but in hardware, it is usually optimized for the task being performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfrgXE/content/2301.03356v1.pdf'}
+page_content=' \uf0d8 The software model of memory is as a single block with a monolithic address space, with almost all data items stored in memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfrgXE/content/2301.03356v1.pdf'}
+page_content=' On an FPGA, local variables are stored in registers, with multiple distributed memory blocks, each with their own independent address space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfrgXE/content/2301.03356v1.pdf'}
+page_content=' In such an environment, pointers have little meaning, and dynamic memory allocation is very difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfrgXE/content/2301.03356v1.pdf'}
+page_content=' Communication in software is usually through shared memory, whereas on an FPGA it relies on constructing appropriate hardware, from implicit (within stream processing), to simple token passing, to using dedicated FIFOs to manage flow control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfrgXE/content/2301.03356v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfrgXE/content/2301.03356v1.pdf'}
+page_content=' DATA LAYER INTERACTION ARCHITECTURE  Traditionally, the word architecture is concerned with the art or science of building, where it relates to structural concepts and to styles of design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfrgXE/content/2301.03356v1.pdf'}
+page_content=' Here the authors will use it in the sense of structured deceptions of a system from a number of compatible and complementary viewpoints which, taken together, cover functional, design, fabrication and performance issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfrgXE/content/2301.03356v1.pdf'}
+page_content=' The term layer is used to refer to a particular descriptive viewpoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfrgXE/content/2301.03356v1.pdf'}
+page_content=' Within a layer, descriptions will generally be hierarchic to allow the containment of complexity or the exposure of detail by suitable aggregation or decomposition techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfrgXE/content/2301.03356v1.pdf'}
+page_content=' Architecture(s) may be singular or plural depending on whether it will be addressed as an individual layer or the complete set of descriptions across all layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfrgXE/content/2301.03356v1.pdf'}
+page_content=' It is clear from this definition that the architecture of a system is not just confined to some sort of functional partitioning at the front end of a development, but essentially embraces descriptions generated during all phases and stages of the development process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfrgXE/content/2301.03356v1.pdf'}
+page_content='  International Journal on AdHoc Networking Systems (IJANS) Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfrgXE/content/2301.03356v1.pdf'}
+page_content=' 12, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfrgXE/content/2301.03356v1.pdf'}
+page_content=' 4, October 2022 38 The Data Interaction Architecture is a prime example of a layered architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfrgXE/content/2301.03356v1.pdf'}
+page_content=' It contains four layers, each of which can be regarded as a model of the system from a particular viewpoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfrgXE/content/2301.03356v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfrgXE/content/2301.03356v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfrgXE/content/2301.03356v1.pdf'}
+page_content=' shows some small representational fragments for each of the four layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfrgXE/content/2301.03356v1.pdf'}
+page_content=' The layers consist of I/O circuitry, fine grained layer, coarse grained function definition layer, and bypass connection layer, where they are listed from the bottom layer to the top layer, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfrgXE/content/2301.03356v1.pdf'}
+page_content=' Well defined notations and technical conventions exist for each of these layers and are closely modelled on those of MASCOT, Modular Approach to Software Construction Operation and Test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfrgXE/content/2301.03356v1.pdf'}
+page_content=' Where relevant the notations include composite structures to support hierarchical representations over multiple levels within a layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfrgXE/content/2301.03356v1.pdf'}
+page_content=' The motivation behind the layered architecture method is to provide the means of moving from a functional model of a system to an execution model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfrgXE/content/2301.03356v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfrgXE/content/2301.03356v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfrgXE/content/2301.03356v1.pdf'}
+page_content=' Data Interaction Architecture  The emphasis on the identification of well-defined interactions in the functional model, and the preservation of these speck tied interaction properties through the subsequent layers, provides the structural conformance by which traceability is achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfrgXE/content/2301.03356v1.pdf'}
+page_content=' It follows that the functional model is binding on subsequent development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfrgXE/content/2301.03356v1.pdf'}
+page_content=' Developers are not free to reverse engineer the functional model to allow an alternative development path in which traceability would be lost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfrgXE/content/2301.03356v1.pdf'}
+page_content=' Of course the functional model may need to be changed, either because system requirements have changed, or because detail design has shown that an alternative functional model would be better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfrgXE/content/2301.03356v1.pdf'}
+page_content=' Whatever the reason for change, consistency across the model set must be preserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfrgXE/content/2301.03356v1.pdf'}
+page_content='  The main distinguishing feature of Data Interaction Architecture (DIA) is the explicit representation of shared information and shared data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfrgXE/content/2301.03356v1.pdf'}
+page_content=' Here i will concentrate on the sharing  Model Representation example Focus Functional What Generator User Design How Writer Reader Distribution Where Writer Reader Execution When Writer Reader V Kemel KermelInternational Journal on AdHoc Networking Systems (IJANS) Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfrgXE/content/2301.03356v1.pdf'}
+page_content=' 12, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfrgXE/content/2301.03356v1.pdf'}
+page_content=' 4, October 2022 39 implicit in the bilateral interactions between processes, leaving discussion of the more general information retention element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfrgXE/content/2301.03356v1.pdf'}
+page_content=' A summary description of the model in each layer shows how bilateral interaction properties are tracked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfrgXE/content/2301.03356v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfrgXE/content/2301.03356v1.pdf'}
+page_content=' ARCHITECTURE OF PROCESSING ELEMENTS  In this paper authors proposed to establish a new design framework to solve issues described in the preceding section by exploiting the binding process of high-level/behavioural synthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfrgXE/content/2301.03356v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfrgXE/content/2301.03356v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfrgXE/content/2301.03356v1.pdf'}
+page_content=' Layered Architecture  The purpose of the conventional high-level synthesis tools is to generate finite state machines with data paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfrgXE/content/2301.03356v1.pdf'}
+page_content=' Here File System Meta-Data (FSMD) referred as fine grained layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfrgXE/content/2301.03356v1.pdf'}
+page_content=' It is expanded with the scope of high-level synthesis to coarse grained layer to exploit the functional representations of circuitries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfrgXE/content/2301.03356v1.pdf'}
+page_content=' ALUs in coarse grained layer are defined by the functional representations of high-level programming languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfrgXE/content/2301.03356v1.pdf'}
+page_content=' Switches are added as bypass connection layer for the scope of high-level synthesis intentionally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfrgXE/content/2301.03356v1.pdf'}
+page_content=' Implicit as-built meshes of switches put constraints on high-level synthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfrgXE/content/2301.03356v1.pdf'}
+page_content=' In the processes are allowed as unprejudiced bypass switches to remove the meshes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfrgXE/content/2301.03356v1.pdf'}
+page_content=' Communications between the PEs are limited between adjacent PEs, whereas it is found no limitations for sensor applications with the limited communication paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfrgXE/content/2301.03356v1.pdf'}
+page_content='  The following explains these layers I/O circuitry is implemented with random logic gates and mixed signal I/O circuitries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfrgXE/content/2301.03356v1.pdf'}
+page_content=' They are connected to the fine grained blocks in the above layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfrgXE/content/2301.03356v1.pdf'}
+page_content=' The fine grained layer mainly consists of finite state machines and data paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfrgXE/content/2301.03356v1.pdf'}
+page_content=' They are replaceable to follow the context described by high-level languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfrgXE/content/2301.03356v1.pdf'}
+page_content=' The coarse grained function definition layer is located on the fine grained layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfrgXE/content/2301.03356v1.pdf'}
+page_content=' Optimized ALUs for a certain application are defined in this layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfrgXE/content/2301.03356v1.pdf'}
+page_content=' An operation primitive defined in the layer corresponds to an operation code (op code) of a conventional multipoint control unit (MCU), whereas it does not have to be standardized for over many applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfrgXE/content/2301.03356v1.pdf'}
+page_content=' As for the topmost bypass connection layer, simple bypass switch images can be specified over coarse grained blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfrgXE/content/2301.03356v1.pdf'}
+page_content=' The connections between inputs and outputs of PEs are configurable with the bypass switches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfrgXE/content/2301.03356v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfrgXE/content/2301.03356v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfrgXE/content/2301.03356v1.pdf'}
+page_content=' Design Flow  The following six steps compose the design flow of the design framework to design a PE through layered scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfrgXE/content/2301.03356v1.pdf'}
+page_content='  Step 1: Describe a system in high-level language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfrgXE/content/2301.03356v1.pdf'}
+page_content=' Step 2: Define coarse grained operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfrgXE/content/2301.03356v1.pdf'}
+page_content=' Typical dedicated functions for sensor signals are signal compensation, feature point identification for image recognition, and image compression in addition to basic arithmetic operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfrgXE/content/2301.03356v1.pdf'}
+page_content=' The defined coarse grained operations are exploited in high-level synthesis and behavioural synthesis process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfrgXE/content/2301.03356v1.pdf'}
+page_content=' Step 3: Generate source codes written in hardware description language through behavioural synthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfrgXE/content/2301.03356v1.pdf'}
+page_content=' The hard macros defined and implemented in the step 2 are exploited for generating circuitries by the functions of CWB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfrgXE/content/2301.03356v1.pdf'}
+page_content=' Step 4: This step is identical to conventional logic synthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfrgXE/content/2301.03356v1.pdf'}
+page_content=' Step 5: This step is identical to conventional layout design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfrgXE/content/2301.03356v1.pdf'}
+page_content=' Step 6: The verification and validation step include back annotation based on the result of delay analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfrgXE/content/2301.03356v1.pdf'}
+page_content='  International Journal on AdHoc Networking Systems (IJANS) Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfrgXE/content/2301.03356v1.pdf'}
+page_content=' 12, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfrgXE/content/2301.03356v1.pdf'}
+page_content=' 4, October 2022 40 In the work authors used a data base management function of CWB to exploit the definitions of operations in the coarse grained layer during the behavioural synthesis to treat a function as an operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfrgXE/content/2301.03356v1.pdf'}
+page_content=' The operations can be implemented as hard macros by using custom ASIC design tools and/or LUTs of FPGAs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfrgXE/content/2301.03356v1.pdf'}
+page_content=' The operator definitions were exploited on the step 3 as macro blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfrgXE/content/2301.03356v1.pdf'}
+page_content=' Once macro blocks are registered in a database, CWB can use the macro blocks during the binding process of high-level synthesis automatically to reduce layout area size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfrgXE/content/2301.03356v1.pdf'}
+page_content=' The binding process of high-level synthesis is regarded as nondeterministic polynomial (NP) hard and heuristic approach was often employed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfrgXE/content/2301.03356v1.pdf'}
+page_content=' The design flow enables the automation of binding process instead of the heuristic approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfrgXE/content/2301.03356v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfrgXE/content/2301.03356v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfrgXE/content/2301.03356v1.pdf'}
+page_content=' Typical Design of Processing Element  Two types of context were identified with reference to semantics or lambda calculus of functional representations to define ALUs in coarse grained function definition layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfrgXE/content/2301.03356v1.pdf'}
+page_content=' One is a configurable static context often mentioned as stored information in a file, and the other is a dynamic context as stored information in heap registers as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfrgXE/content/2301.03356v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfrgXE/content/2301.03356v1.pdf'}
+page_content=' The static contexts of an application are expressed with finite state machines and data paths implemented by n sets of hard- ware circuitry of PEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfrgXE/content/2301.03356v1.pdf'}
+page_content=' The dynamic contexts of an application are specified as m sets of registers and instructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfrgXE/content/2301.03356v1.pdf'}
+page_content=' The instruction sets are optimized for an application, and each optimized instruction set can be shared among some dynamic contexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfrgXE/content/2301.03356v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfrgXE/content/2301.03356v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfrgXE/content/2301.03356v1.pdf'}
+page_content=' Modified High Level Synthesis Process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfrgXE/content/2301.03356v1.pdf'}
+page_content='  The program code is implemented with a specific C-language comment description /* Cyber func = operator */ for defining dynamic and static contexts, which are the sets of pairs of registers and optimized instruction sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfrgXE/content/2301.03356v1.pdf'}
+page_content=' The pair of a register set and an instruction set can be shared among dynamic contexts using another specific C-language comment description as /* Cyber share name = NAME */.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfrgXE/content/2301.03356v1.pdf'}
+page_content=' NAME is an arbitrary designation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfrgXE/content/2301.03356v1.pdf'}
+page_content=' The binding process of high-level behavioural  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfrgXE/content/2301.03356v1.pdf'}
+page_content='Higher Hard Hard module Macro Macro Highlevel/Behavioral Highlevel /Behavioral Highlevel/Behavioral synthesis synthesis synthesis CDFGgeneration Allocation 不 LMSPEC RTL LMSPEC RTL Scheduling 不 Binding FSMDgeneration RTL Logic synthesis Place and Route RTL:RegisterTransferLevel 不 LMSPEC:LowerModuleSpecificationFile CDFG:ControlandDataFlowGraph Configuration FSMD:FiniteStateMachinewithDatapath DataInternational Journal on AdHoc Networking Systems (IJANS) Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfrgXE/content/2301.03356v1.pdf'}
+page_content=' 12, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfrgXE/content/2301.03356v1.pdf'}
+page_content=' 4, October 2022 41 synthesis can be controlled by these descriptions to exploit the functional representations defined in a high-level programming language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfrgXE/content/2301.03356v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfrgXE/content/2301.03356v1.pdf'}
+page_content=' RESULTS  Experiments are carried out to evaluate the design framework using convolution operations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfrgXE/content/2301.03356v1.pdf'}
+page_content=' those are often used for sensor applications, with following three conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfrgXE/content/2301.03356v1.pdf'}
+page_content=' The matrix functions of the filters are similar and the difference is the parameters of 3×3 matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfrgXE/content/2301.03356v1.pdf'}
+page_content=' It is observed one can evaluate the layout area size reduction results by using an FPGA, Xilinx XC7A200T FPGA, although the design framework was established aiming at improving ASIC design at first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfrgXE/content/2301.03356v1.pdf'}
+page_content='  \uf0d8 Defining a convolution function as one operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfrgXE/content/2301.03356v1.pdf'}
+page_content=' \uf0d8 Implementing ALUs with basic operations, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfrgXE/content/2301.03356v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfrgXE/content/2301.03356v1.pdf'}
+page_content=', plus, minus, multiply, divide, and comparison operations, using dynamically reconfiguration technique designated as flexible reliability reconfigurable array (FRRA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfrgXE/content/2301.03356v1.pdf'}
+page_content=' \uf0d8 Elaborating whole design with FPGA libraries without operator definitions and a function definition database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfrgXE/content/2301.03356v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfrgXE/content/2301.03356v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfrgXE/content/2301.03356v1.pdf'}
+page_content=' Comparison of LUT utilizations According to Operator Definitions  Derived LUT counts for each evaluation case are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfrgXE/content/2301.03356v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfrgXE/content/2301.03356v1.pdf'}
+page_content=' The LUT counts of basic operations were excluded for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfrgXE/content/2301.03356v1.pdf'}
+page_content=' The larger the granularity of an operation was, the less counts of LUTs were consumed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfrgXE/content/2301.03356v1.pdf'}
+page_content=' Remarkable difference of LUT counts was shown for cascaded operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfrgXE/content/2301.03356v1.pdf'}
+page_content=' The LUT counts of cascaded operations using FPGA library was more than double compared to single operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfrgXE/content/2301.03356v1.pdf'}
+page_content=' The results show that exploiting granularity in behavioural synthesis was carried out by CWB automatically without specifying the reuse of macro blocks explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfrgXE/content/2301.03356v1.pdf'}
+page_content=' The difference between one operation and cascaded operations using FRRAs with dynamically reconfiguration technique was 22% less than the difference using the operator definition on convolution operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfrgXE/content/2301.03356v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfrgXE/content/2301.03356v1.pdf'}
+page_content=' CONCLUSIONS  A novel model is developed to design a framework to reduce the footprints of programmable functions of sensing devices for IOT applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfrgXE/content/2301.03356v1.pdf'}
+page_content=' The design framework consists of four layered structure of PE architecture and the extended database management function of a high-level synthesis tool to exploit functional representations in high-level programming languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfrgXE/content/2301.03356v1.pdf'}
+page_content='  Laplacian Filter GaussianFilter+LaplacianFilter (a) (b) (c) 0 200 400 600 800 LUT countsInternational Journal on AdHoc Networking Systems (IJANS) Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfrgXE/content/2301.03356v1.pdf'}
+page_content=' 12, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfrgXE/content/2301.03356v1.pdf'}
+page_content=' 4, October 2022 42 The layout size reduction is useful to embed a PE into a sensing device and to provide device computing capabilities with a sensing device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfrgXE/content/2301.03356v1.pdf'}
+page_content=' The experimental results report the power consumption reduction by adopting the design framework on a Nano Bridge FPGA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfrgXE/content/2301.03356v1.pdf'}
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+page_content=' A Cellular bonding and adaptive load balancing based multi-sim gateway for mobile ad hoc and sensor networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfrgXE/content/2301.03356v1.pdf'}
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+Lattice models for ballistic aggregation: cluster-shape dependent exponents
+P. Fahad,1, 2, ∗ Apurba Biswas,3, 4, † V. V. Prasad,5, ‡ and R. Rajesh3, 4, §
+1Institut f¨ur Materialphysik im Weltraum, Deutsches Zentrum f¨ur Luft- und Raumfahrt (DLR), 51170 K¨oln, Germany
+2Institut f¨ur Theoretische Physik, Universit¨at zu K¨oln, Z¨ulpicher Strasse 77, 50937 K¨oln, Germany
+3The Institute of Mathematical Sciences, C.I.T. Campus, Taramani, Chennai 600113, India
+4Homi Bhabha National Institute, Training School Complex, Anushakti Nagar, Mumbai 400094, India
+5Department of Physics, Cochin University of Science and Technology, Cochin - 682022 India
+(Dated: January 27, 2023)
+We study ballistic aggregation on a two dimensional square lattice, where particles move ballis-
+tically in between momentum and mass conserving coalescing collisions. Three models are studied
+based on the shapes of the aggregates: in the first the aggregates remain point particles, in the sec-
+ond they retain the fractal shape at the time of collision, and in the third they assume a spherical
+shape. The exponents describing the power law temporal decay of number of particles and energy
+as well as dependence of velocity correlations on mass are determined using large scale Monte Carlo
+simulations. It is shown that the exponents are universal only for the point particle model. In the
+other two cases, the exponents are dependent on the initial number density and correlations vanish
+at high number densities. The fractal dimension for the second model is close to 1.49.
+I.
+INTRODUCTION
+There is a wide variety of physical phenomena at
+different length scales in which aggregation of parti-
+cles/clusters to form larger particles is the predominant
+dynamical process [1]. Examples include aerosols [2, 3],
+agglomeration of soot
+[4, 5], gelation [6], cloud forma-
+tion [7], astrophysical problems [8], aggregation of dust
+particles in planetary discs [9–11], dynamics of Saturn’s
+rings [10, 12], polyelectrolytes [13, 14], networks [15],
+etc. A minimal model that focuses only on the effects
+of aggregation is the cluster-cluster aggregation (CCA)
+model in which particles that come into contact undergo
+mass conserving coalescence (reviews may be found in
+Refs. [16–18]).
+In addition to its relevance for differ-
+ent physical phenomena, CCA has also been studied as
+a nonequilibrium system undergoing scale invariant dy-
+namics that is described by exponents that depend only
+on very generic features of the transport process. This
+universal feature allows applications of results for CCA in
+seemingly unrelated systems like Burgers turbulence [19–
+23], Kolmogorov self-similar scaling [24–26], granular sys-
+tems [27–29], hydrodynamics of run and tumble par-
+ticles [30], evolution of planetesimals [31], geophysical
+flows [32], etc.
+Among the different transport processes,
+ballistic
+transport is of particular importance and the resultant
+CCA is known as the ballistic aggregation (BA) model,
+the focus of this paper. In the BA model, momentum
+is additionally conserved in collisions.
+The BA model
+with spherical particles has been studied using mean field
+theory, large scale simulations in two and three dimen-
+sions, and is exactly solvable in one dimension.
+It is
+∗ fahad.puthalath@dlr.de
+† apurbab@imsc.res.in
+‡ prasad.vv@cusat.ac.in
+§ rrajesh@imsc.res.in
+found that the number of particles, n(t), and energy,
+e(t), decrease with time, t, as power-laws: n(t) ∝ t−θn,
+e(t) ∝ t−θe. These exponents have been determined in
+d-dimensions within a mean field approximation which
+assumes that the particle density is small, that the parti-
+cles are compact spherical clusters of equal density, and
+that the velocities of the particles constituting a cluster
+are uncorrelated. Within these assumptions, scaling ar-
+guments predict the existence of a growing length scale
+Lt ∼ t1/zmf with zmf = (d + 2)/2d and mean field ex-
+ponents, θmf
+n
+= 2d/(d + 2) and θmf
+e
+= θmf
+n
+[33].
+The
+correlations in the initial velocities of the constituents of
+a cluster is characterized by an exponent η: ⟨v2
+m⟩ ∼ m−η,
+where ⟨v2
+m⟩ is the mean square velocity of a particle of
+mass m. In the mean field approximation, by assump-
+tion, ηmf = 1. The mean field results for the exponents
+are of particular significance to the study of the unrelated
+problem of freely cooling granular gas in which ballis-
+tic particles undergo energy-dissipating, momentum con-
+serving binary collisions. It has been shown that expo-
+nent characterizing the energy decay in the granular gas
+is equal to θmf
+e
+in dimensions upto three [27, 29, 34].
+In one dimension, BA is exactly solvable and the ex-
+ponents match with the mean-field exponents [20, 35–
+37]. However, in two and three dimensions, it has been
+shown that the exponents for BA with spherical parti-
+cles depend on the initial number density n0.
+In two
+dimensions and for dilute systems (n0 → 0), it has been
+shown that the numerically obtained θn is 17% larger
+than θmf
+n
+because of strong velocity correlations between
+colliding aggregates, with η decreasing from η ≈ 1.33 for
+low densities to η ≈ 1 = ηmf for high densities [38–41].
+In three dimensions, it is found that as n0 increases from
+0.005 to 0.208, θe decreases from θe = 1.283 to 1.206
+and appears to converge to the θmf
+e
+= 1.2 with increasing
+n0, and η decreases from η ≈ 1.23 for low densities to
+η ≈ 1 = ηmf for high densities [29, 41]. It is remark-
+able that the mean field results describe well only the
+systems with large n0, while its derivation assumes the
+arXiv:2301.11250v1  [cond-mat.stat-mech]  26 Jan 2023
+
+2
+limit n0 → 0. This counterintuitive result has been ar-
+gued to be due to the randomization of the velocities at
+higher densities due to avalanche of coagulation events
+that occur due to the overlap of a newly created spheri-
+cal particle with already existing particles, as the number
+density is increased.
+While the kinetics of BA with spherical particles are
+reasonably understood, much less is known for the ex-
+ponents when clusters have non-spherical shapes. The
+scaling analysis can be extended to the case when the
+mass scales with radius with a fractal dimension df [39]
+(also see Sec. III where we review scaling theory). The
+scaling theory leads to hyperscaling relations between
+the different exponents independent of the mean field as-
+sumptions. Fractal shapes are of particular importance in
+the case of the experiments on aggregates of soot [5, 42],
+mammary epithelial cells [43, 44], spray flames [45], etc.,
+where the aggregates have a fractal dimension different
+from those of compact structures (df = 2, 3).
+While
+the fractal dimensions seen in experiments [5, 43] are
+sometimes close to that for diffusion limited aggregation
+(df ≈ 1.7), there are many examples for which it is very
+different (for instance 1.54 for sprays [45], 1.5 for cells [44]
+or 2.4 for soot [42]). The fractal dimension of aggregates
+formed by ballistic motion is not known to the best of
+our knowledge.
+In addition, it is also not known how
+the exponents for BA change when the shape of the clus-
+ters deviates from spherical. Neither is it known whether
+the mean field limit is reached for any particular limit
+of number density when the clusters are fractal. Finally,
+in the characterization of mass distribution, a relevant
+exponent is the scaling of mass distribution with small
+mass, namely N(m) ∼ mζ [also see definition in Eq. (6)].
+The exponent ζ is an independent exponent and cannot
+be obtained from scaling theory, and is not known even
+for BA with spherical particles.
+To answer these questions, we study three differently
+shaped clusters (named as models A, B, C) undergoing
+BA on the square lattice. We choose a lattice approach
+as it allows us to maintain fractal shapes in a computa-
+tionally efficient manner. Lattice models are known to
+reproduce the same results as the continuum for BA in
+one dimension [23, 46], and we expect the equivalence to
+hold true for two and higher dimensions. In model A, the
+clusters occupy a single site irrespective of its mass. This
+limiting model allows us to separate the dependence of
+the velocity correlations on the initial density from the
+dependence on mass-dependent shape. In model B, we
+study clusters where the clusters maintain the shape at
+the time of contact. Such clusters turn out to be frac-
+tal. In model C, we study “spherical” clusters in which
+the lattice approximation to the disc is maintained. This
+model allows us to study lattice effects by comparing the
+results on the lattice with the continuum results. In ad-
+dition, we obtain the value of the exponent ζ for all the
+three models. The results for the three models are sum-
+marized in Table II (model A), Table III and Fig. 17
+(model B), Table IV and Fig. 23 (model C). For model
+A, we show that the exponents are universal, in the sense
+that it is independent of the initial number density, n0
+and it is different from the mean field results. For models
+B and C, we find that the exponents are dependent on
+n0 and approach the mean field assumptions of uncorre-
+lated velocities only in the limit of large n0. The fractal
+dimension for model B, on the other hand, is universal,
+with df ≈ 1.49.
+The remainder of the paper is organized as follows.
+Section II contains a definition of the different models
+as well as a description of the simulation methods. We
+briefly review the scaling theory for BA with differently
+shaped particles in Sec. III. In Sec. IV, for the three mod-
+els, we describe the results for the different exponents
+obtained from large scale Monte Carlo simulations. Sec-
+tion V contains a summary and discussion of the results.
+II.
+MODEL
+In this section, we define the three models that we
+study in this paper.
+Consider a square lattice of size
+L × L with periodic boundary conditions.
+Initially N
+particles, each of mass 1, are randomly distributed with a
+site having utmost one particle. Each particle is assigned
+a velocity whose magnitude is drawn from a uniform dis-
+tribution in [0, 1) and whose direction is chosen uniformly
+in [0, 2π). The velocity of the center of mass is set to be
+zero by choosing an appropriate frame of reference. The
+system evolves stochastically in time as follows. A par-
+ticle with velocity (vx, vy) hops in the x-direction with
+rate |vx| in the positive (negative) direction depending on
+whether vx is positive (negative). Likewise, it hops along
+the y-axis with rate |vy| in the direction determined by
+the sign of vy. When two particles collide, they aggregate
+to form a new particle. The mass of the new particle is
+the sum of the constituent particles while the new veloc-
+ity is determined by conservation of linear momentum.
+The shape of the new particle is determined based on
+three different rules, leading to three different models.
+Model A: Point particles
+In model A, when a particle hops onto a site which
+is already occupied, then the two particles coalesce, con-
+serving mass and momentum. The new particle occupies
+the same lattice site. We call this model the point parti-
+cle model, since the sizes of all the particles are the same
+(one lattice site) irrespective of their mass.
+Model B: Fractal clusters
+In model B, the particles, also referred to as clusters,
+are extended objects consisting of a collection of sites
+that are linked to each other by nearest neighbor bonds.
+
+3
+(a)
+(b)
+(c)
+(d)
+FIG. 1.
+Snapshots of the configurations at different times
+t for model B (fractal clusters), where different clusters are
+shown by different colors. The different panels correspond to
+(a) t = 50, (b) t = 500, (c) t = 5000 and (d) t = 25790.
+The data are for system size L = 200 and initial number of
+N = 2000 particles (n0 = 0.05).
+When a cluster hops, if any of the lattice sites belong-
+ing to it becomes adjacent to a site belonging to another
+cluster, then the two clusters coalesce.
+The new clus-
+ter maintains the shape at the time of coalescing, till it
+collides with another cluster at a future time. The new
+velocity of the cluster is determined through momentum
+conservation. Snapshots of the configuration at different
+times are shown in Fig. 1. The clusters are extended and
+will be shown to be fractals.
+Model C: Spherical clusters
+In model C, like in model B, particles are extended
+clusters. However, the shape of these particles are con-
+strained to be spherical. When two particles come into
+contact, they are replaced by a new spherical particle.
+The center of mass of the new particle is chosen to be
+lattice site closest to the center of mass of the constituent
+particles. To construct a spherical cluster on the square
+lattice, we fill all lattice sites within circles of increas-
+ing radius. The sites in the outermost shell, if not fully
+occupied, are chosen at random. This rearrangement of
+sites to form a spherical shape will, at times, lead to the
+new cluster overlapping with other nearby clusters, trig-
+(a)
+(b)
+FIG. 2. Snapshots of the configurations at different times t
+for model C (spherical clusters), where different clusters are
+shown by different colors. The different panels correspond to
+(a) t = 100 and (b) t = 500. The data are for system size
+L = 200 and initial number of N = 4000 particles (n0 = 0.1).
+TABLE I. Simulation details.
+Model
+L’s simulated
+Number densities (n0)
+A
+upto 1000
+0.01 - 1.00
+B
+upto 10000
+0.001 - 0.01
+C
+upto 10000
+0.0001 - 0.16
+gering an avalanche of coalescence events. Snapshots of
+a typical time evolution are shown in Fig. 2.
+Details of simulation
+In model B and model C, where extended clusters
+hop as a single unit, we identify the different clusters
+and their merging using the Hoshen-Kopelman algo-
+rithm [47]. Simulations were carried out for different sys-
+tem sizes varying from L = 100 upto L = 10000 for all
+three models and a wide range of number densities n0.
+The simulation is continued till all the clusters aggregate
+together to form the final single cluster. The details of
+the densities and the lattice sizes used for simulations of
+the three models are given in Table I.
+III.
+REVIEW OF SCALING THEORY
+In this section, we review the scaling theory for BA,
+described initially in Ref. [33]. Here, we give a scaling
+argument based on the Smoluchowski equation for aggre-
+gation (see Refs. [16, 17] for reviews). Different scaling
+arguments, leading to the same results, may be found in
+Refs. [39, 40]. Let N(m, t) denote the average density of
+
+4
+clusters of mass m at time t. N(m, t) evolves in time as
+dN(m, t)
+dt
+= −N(m, t)
+� ∞
+0
+dm1K(m, m1)N(m1, t)
++ 1
+2
+� m
+0
+dmK(m1, m − m1)N(m, t)N(m − m1, t), (1)
+where the kernel K(m1, m2) is the rate at which particles
+of masses m1 and m2 collide. The first term in the right
+hand side of Eq. (1) describes a loss term where a particle
+of mass m collides with another particle, while the second
+term describes a gain term where two particles collide to
+form a particle of mass m.
+We restrict ourselves to homogeneous kernels, which
+are known to describe many physical systems, examples
+of which may be found in Refs. [16, 17]. Homogeneous
+kernels have the property
+K(hm1, hm2) = hλK(m1, m2),
+h > 0,
+(2)
+where λ is called the homogeneity exponent. For λ < 1,
+and for large masses and times, it can be shown that
+Eq. (1) is solved by a N(m, t) which has the scaling form
+N(m, t) ≃
+1
+t2θn Φ
+� m
+tθn
+�
+.
+(3)
+For x ≫ 1, Φ(x) vanishes exponentially. For x ≪ 1, Φ(x)
+is a power law
+Φ(x) ∼ xζ,
+x ≪ 1.
+(4)
+Thus, there are two exponents θn and ζ characterizing
+the mass distribution N(m, t).
+The exponent θn describes how the mean density of
+particles n(t) =
+�
+m N(m, t)dm decreases with time. In-
+tegrating Eq. (3), we obtain
+n(t) ∼ t−θn.
+(5)
+The exponent ζ describes the power law dependence of
+N(m, t) on mass for small masses:
+N(m, t) ∼
+mζ
+tθn(2+ζ) ,
+m ≪ tθn.
+(6)
+The dependence of θn on the homogeneity exponent λ
+can be obtained by substituting Eq. (3) into Eq. (1), and
+is known to be (for example, see Refs. [16, 17])
+θn =
+1
+1 − λ.
+(7)
+We now focus on the collision kernel that corresponds
+to BA. Assuming a homogeneous mixture of clusters of
+all masses, the rate of collision between two masses m1
+and m2 is proportional to (r1 + r2)d−1|⃗v1 − ⃗v2| where
+r1 and r2 are the radii of the particles, ⃗v1 and ⃗v2 the
+velocities, and d is the dimension. The relative velocity
+may be approximated as |⃗v1 − ⃗v2| ≈
+�
+v2
+1 + v2
+2. Thus, the
+collision kernel for BA may be written as
+K(m1, m2) ∝ (r1 + r2)d−1
+�
+v2
+1 + v2
+2.
+(8)
+To express the radii and velocities in terms of the masses,
+we assume that the typical speed, vm, of particles of mass
+m, scales with mass as
+v2
+m ∼ m−η.
+(9)
+The radii are related to mass though the fractal dimen-
+sion, df, of a cluster:
+r ∝ m1/df .
+(10)
+Thus, the kernel in Eq. (8) reduces to
+K(m1, m2) ∝
+�
+m1/df
+1
++ m1/df
+2
+�d−1 �
+m−η
+1
++ m−η
+2 . (11)
+This kernel is homogeneous in its arguments with ho-
+mogeneity exponent given by
+λ = d − 1
+df
+− η
+2.
+(12)
+From Eq. (7), we then obtain
+θn =
+2df
+2df − 2(d − 1) + ηdf
+.
+(13)
+Another quantity of interest is the mean kinetic energy
+e(t), defined as
+e(t) ≃
+�
+dm1
+2mv2
+mN(m, t).
+(14)
+The energy density decreases in time as a power law
+e(t) ∼ t−θe.
+Substituting v2
+m ∼ m−η, we obtain the
+scaling relation
+θe = ηθn.
+(15)
+We now reproduce the results obtained for BA in
+Ref. [33] which we refer to as the mean field BA ex-
+ponents. Here, it is assumed that the clusters that are
+formed are spherical (df = d) and that the velocities of
+the constituent particles of a given cluster are uncorre-
+lated implying that η = 1. Substituting these values into
+Eqs. (13) and (15), we reproduce the results
+θmf
+n
+= θmf
+e
+=
+2d
+d + 2,
+(16)
+where the superscript mf denotes mean field. Note that
+the main simplifying assumption is that η = 1. In one
+dimension η continues to be 1 as the order of particles is
+maintained and a cluster made up of m initial neighbor-
+ing particles will have uncorrelated velocities. However,
+η need not be 1 in higher dimensions.
+
+5
+We now summarize the scaling theory predictions for
+the models studied in this paper.
+For model A, since
+particles are point-like objects we have r ∼ m0 or df =
+∞.
+Similarly, in model C since clusters are spherical
+df = d, which is spatial dimension itself. We thus obtain
+θn =
+�
+�
+�
+�
+�
+2
+2+η,
+model A,
+2df
+2df −2+ηdf ,
+model B,
+2
+1+η,
+model C,
+(17)
+with θe = ηθn.
+It is useful to have a relation between θn and θe that
+does not involve η. This will enable us to verify scaling
+theory without having to numerically measure the differ-
+ent exponents. Eliminating η, we obtain
+2θn + θe =2,
+model A,
+2θn
+2θn + θe − 2 =df,
+model B,
+(18)
+θn + θe =2,
+model C.
+IV.
+RESULTS
+In this section, we describe the results, obtained from
+extensive Monte Carlo simulations, for models A, B, and
+C. For all the three models, we will independently de-
+termine the exponents θn, θe, η and ζ. For model B the
+fractal dimension df is also measured. Their dependence
+on number density, the scaling relations between them,
+as well as deviation from the mean field results, are de-
+termined.
+A.
+Model A: Point particles
+We first determine θn from the power law decay of
+the mean density of particles, n, with time t. The data
+for different initial number density n0 and initial mean
+speed v0 collapse onto one curve when scaled, based on
+dimensional analysis, according to
+n(t, n0) ≃ n0f(tn0v0),
+(19)
+as shown in Fig. 3.
+After an initial crossover time
+tc ∼ n−1
+0 , n(t) decreases as a power law. From the excel-
+lent collapse of the data for different n0 onto one curve,
+we conclude that the power law exponent is independent
+of the initial number density. From fitting a power law
+to the data, we obtain θn = 0.633(7), which describes the
+data well over 5 decades. In the inset of Fig. 3, the com-
+pensated curve tθnn(t) is shown for n0 = 1. The mean
+slope of the curve changes from negative to positive as θn
+varies from 0.626 to 0.640, consistent with our estimate
+of θn from direct measurement.
+We now numerically determine θn using different anal-
+yses, both for the sake of consistency as well as for bench-
+marking different methods that will be more useful in
+determining exponents for models B and C.
+10-6
+10-5
+10-4
+10-3
+10-2
+10-1
+100
+10-2
+100
+102
+104
+106
+108
+1010
+1012
+1014
+n/n0
+tn0v0
+ n0 = 0.01
+ n0 = 0.10
+n0 = 0.20
+n0 = 0.40
+n0 = 0.70
+n0 = 1.00
+ 0.9
+ 1
+ 1.1
+ 1.2
+102
+103
+104
+105
+106
+107
+tθn n(t)
+t
+θn = 0.640
+θn = 0.633
+θn = 0.626
+FIG. 3.
+The data (model A) for mean number density of par-
+ticles, n(t), for different initial number densities n0 collapse
+onto a single curve when n(t) and t are scaled as in Eq. (19).
+The solid line is a power law t−0.633. Inset: The compensated
+data n(t)tθn is shown for three different choices of θn differing
+by 0.007 for n0 = 1.0. The curve is flat for θn = 0.633. The
+data are obtained for L = 1000. All data have been averaged
+over 300 different initial conditions.
+10-4
+10-3
+10-2
+10-1
+100
+101
+102
+103
+104
+10-2
+10-1
+100
+101
+102
+103
+104
+t 2θn N(m,t)
+m/tθn
+t = 250
+t = 1000
+t = 4000
+t = 16000
+t = 64000
+t = 256000
+t = 1024000
+10-3
+10-1
+101
+103
+105
+100
+101
+102
+103
+104
+105
+N(m,t)
+m
+t = 250
+t = 2000
+t = 16000
+t = 128000
+t = 1024000
+FIG. 4.
+The mass distribution N(m, t) for different times
+collapse onto a single curve when scaled as in Eq. (3), with
+θn = 0.633. The data are for model A, with initial number
+density n0 = 1.0, and system size L = 1000 lattice. Inset:
+The unscaled data for N(m, t) for different times t.
+First we check that the measured value of θn is consis-
+tent with the mass distribution N(m, t) and then we use
+the finite size scaling for large times. The dependence
+of N(m, t) on time and mass are shown in the inset of
+Fig. 4. When scaled as in Eq. (3) with θn = 0.633, the
+data for different times, that span three decades, collapse
+onto a single curve (see Fig. 4).
+Finally, we examine finite size effects. For very large
+times, when the number of clusters is order one, we ex-
+pect that n(t) ∼ L−2, where L is the system size. As-
+suming finite size scaling, we can write
+n(t) ≃ 1
+L2 fn
+�
+t
+L2/θn
+�
+,
+(20)
+where the scaling function fn(x) ∼ x−θn for x ≪ 1, and
+
+6
+10-1
+100
+101
+102
+103
+104
+105
+106
+10-8
+10-6
+10-4
+10-2
+100
+n L2
+t/L2/θn
+L = 1000
+L = 750
+L = 500
+L = 250
+t -θn
+FIG. 5.
+The number density n(t) for different system sizes L
+collapse onto a single curve when scaled as in Eq. (20), with
+θn = 0.633. The data are for model A, and initial number
+density n0 = 1.0.
+fn(x) ∼ constant for x ≫ 1. The data for n(t) for differ-
+ent L, when scaled as in Eq. (20) with θn = 0.633, col-
+lapse onto a single curve, as shown in Fig. 5. For model
+B and C, we will find the analysis of the data based on
+N(m, t) and finite size scaling very useful for determining
+the exponents.
+We now determine θe from the power law decay of
+the mean energy density e with time t.
+The data for
+energy for different initial number density n0, initial
+speed v0 and initial mean energy e0 collapse onto one
+curve when scaled, based on dimensional analysis, as
+e(t) ≃ e0fe(tn0v0), as can be seen in Fig. 6. After an
+initial crossover time tc ∼ n−1
+0 , e(t) decreases as a power
+law. From the excellent data collapse, we conclude that
+the power law exponent is independent of the initial num-
+ber density. From fitting a power law to the data, we
+obtain θe = 0.728(5), which describes the data well over
+5 decades. In the inset of Fig. 6, the compensated curve
+tθee(t) is shown for n0 = 1.0.
+The mean slope of the
+curve changes from negative to positive as θn varies from
+0.723 to 0.733, consistent with our direct measurement
+of θe.
+The exponent θe can also be determined from finite
+size scaling. As for number density, e(t) is expected to
+obey finite size scaling of the form
+e(t) ≃
+1
+L2θe/θn fe
+�
+t
+L2/θn
+�
+,
+(21)
+where the scaling function fe(x) ∼ x−θe for x ≪ 1, and
+fe(x) ∼ constant for x ≫ 1. The simulation data for
+different L collapse onto a single curve (see Fig. 7) when
+e(t) and t are scaled as in Eq. (21) with θn = 0.633 and
+θe = 0.728. The power law extends over 4 decades.
+We now determine the exponent η relating the scaling
+of velocity with mass as v2
+m ∼ m−η [see Eq. (9)]. As seen
+from Fig. 8, ⟨v2⟩ for a fixed mass scales as a power law
+with m. We obtain η = 1.1505(3).
+Note that η is not an independent exponent, but re-
+lated to θn and θe through scaling theory, to be η = θe/θn
+10-14
+10-12
+10-10
+10-8
+10-6
+10-4
+10-2
+100
+10-2
+100
+102
+104
+106
+108
+1010
+ e/e0
+tn0 v0
+ n0 = 0.01
+ n0 = 0.10
+n0 = 0.20
+n0 = 0.40
+n0 = 0.70
+n0 = 1.00
+ 0.14
+ 0.15
+ 0.16
+ 0.17
+102
+103
+104
+105
+106
+tθee(t)
+t
+θe = 0.733
+θe = 0.728
+θe = 0.723
+FIG. 6.
+The data for mean energy density, e(t), at time t for
+different initial number densities n0 in model A collapse onto
+a single curve when e(t) and t are scaled as shown in figure.
+The solid line is a power law t−0.728. Inset: The compensated
+data n(t)tθe is shown for three different choices of θe differing
+by 0.005 for n0 = 1.0. The curve is flat for θe = 0.728. The
+data are obtained for L = 1000. All data have been averaged
+over 300 different initial conditions.
+10-6
+10-4
+10-2
+100
+102
+104
+106
+10-8
+10-6
+10-4
+10-2
+100
+L2θe /θn e(t)
+t/L2/θn
+L = 1000
+L = 750
+L = 500
+L = 250
+t -θe
+FIG. 7.
+The mean energy e(t) for different system sizes L
+collapse onto a single curve when scaled as in Eq. (21), with
+θn = 0.633 and θe = 0.728. The data are for model A, and
+initial number density n0 = 1.0.
+[see Eq. (17)]. From the measured values of θe = 0.728
+and θn = 0.633, we obtain η = 1.15, consistent with
+the value from direct measurement η = 1.1505(3), thus
+providing support for the correctness of scaling theory.
+We now provide a more direct evidence of scaling the-
+ory being correct. From Eqs. (17) and (15), we obtain,
+by eliminating η, a relation between θe and θn as given in
+Eq. (18). If this relation is true, it implies that t2n2(t)e(t)
+should not depend on time t. In Fig. 9, we show the vari-
+ation of tan2(t)e(t) with a = 1.98, 2.00, 2.02. It is clear
+that only for a = 2.0, the curve is horizontal. This gives
+us a way of validating the scaling relations without the
+need to measure any exponent directly.
+Finally, we determine the exponent ζ defined in Eq. (6)
+for small masses: N(m, t) ∼ mζt−θn(2+ζ). Note that ζ
+is not related to θn or θe and is an independent expo-
+
+7
+10-8
+10-7
+10-6
+10-5
+10-4
+10-3
+10-2
+10-1
+100
+100
+101
+102
+103
+104
+105
+〈v2〉
+m
+m-1.1505
+FIG. 8.
+The variation of the mean square velocity ⟨v2⟩ with
+mass m. The solid line is power law t−η with η = −1.1505.
+The data are for model A, with n0 = 1.0 and system size
+L = 1000.
+ 0.04
+ 0.05
+ 0.06
+ 0.07
+ 0.08
+ 0.09
+ 0.1
+102
+103
+104
+105
+106
+107
+108
+t ae(t)n(t) 2
+t
+a = 2.02
+a = 2.00
+a = 1.98
+FIG. 9.
+The variation of tan2(t)e(t) with time t for three
+different values of a close to 2.
+The compensated curve
+is horizontal for a = 2.0, validating the scaling relation in
+Eq. (18). The data are for model A, with initial number den-
+sity n0 = 1.0 and system size L = 1000.
+nent. To determine ζ, we study the temporal behavior
+of N(m, t) for fixed mass m = 2, 4, 8, 12, 16. As shown
+in Fig. 10, the data for the different masses for large
+times collapse onto one curve when N(m, t) is scaled as
+N(m, t)/mζ, with ζ = 0.270(5). We additionally check
+that the scaled data are consistent with the power law
+t−θn(2+ζ) for large times.
+The numerically obtained values of the exponents for
+model A are summarized in Table II.
+B.
+Model B: Fractal Clusters
+In this subsection, we determine the exponents θn, θe,
+η and ζ for model B. We first show that the clusters in
+model B are fractal with a fractal dimension, df, that
+lies between 1 and 2. To determine df, we consider the
+final cluster in each of the simulations for a given initial
+number density n0. df of this cluster is measured using
+10-8
+10-7
+10-6
+10-5
+10-4
+10-3
+10-2
+10-1
+100
+100
+101
+102
+103
+104
+105
+N(m,t)/mζ
+t
+m = 2
+m = 4
+m = 8
+m = 12
+m = 16
+t -θn(2+ζ)
+FIG. 10. The data for N(m, t) for different masses for large
+times collapse onto one curve when the number density is
+scaled as N(m, t)/mζ with ζ = 0.270.
+The solid line is a
+power law t−θn(2+ζ) with θn = 0.633. The data are for model
+A, with initial number density n0 = 1.0.
+TABLE II. Summary of the numerically obtained values of
+the exponents for model A. The values are independent of
+initial density n0.
+exponent
+value
+θn
+0.633(7)
+θe
+0.728(5)
+η
+1.1505(3)
+ζ
+0.270(5)
+the box counting method [48]. In this method, the lattice
+is tiled with square boxes of length ℓ. Let M be the num-
+ber of non-empty boxes. Then M ∼ ℓ−df . The results
+for three different n0 are shown in Fig. 11. The data for
+different n0 fall on top of each other for intermediate box
+sizes. The same is true for other n0 and we conclude that
+df is independent of n0. We estimate df to be 1.49(3).
+Consider now the decay of the density of particles n(t)
+with time t. We find that for model B, it is difficult to
+accurately determine θn directly from the data for n(t)
+because of strong crossover effects. This can be seen from
+Fig. 12 where the variation of n(t) with t is shown for two
+different initial densities n0 = 0.00125 and n0 = 0.01.
+The data for the two densities overlap for short times but
+deviate for larger times. The solid lines, which are the
+estimates for θn from finite size scaling (to be discussed
+below) match with the data only for late times. The con-
+vergence to the asymptotic answer can also be seen from
+measuring the instantaneous slope θn = −d ln n(t)/d ln t
+for each time (see inset of Fig. 12). We find that the expo-
+nent θn saturates only at late times for the larger initial
+densities. We find that the same issue is present for the
+temporal decay of energy e(t), making it also difficult to
+measure θe directly.
+We determine θn from finite size scaling.
+For finite
+systems, n(t) has the finite size scaling form given in
+
+8
+10-6
+10-5
+10-4
+10-3
+10-2
+10-1
+100
+101
+100
+101
+102
+103
+M /(n0 L2)  
+l 
+n0=0.0001
+n0=0.0025
+n0=0.01
+l -1.49
+FIG. 11.
+Determination of the fractal dimension of the
+largest cluster in model B using the box counting method.
+The number of non-empty boxes, M, varies with the size ℓ
+of the boxes used to tile the lattice as M ∼ ℓ−df . We find
+df ≈ 1.49(3) (power law shown by solid line) irrespective of
+the initial density. The data are for L = 5000.
+10-5
+10-4
+10-3
+10-2
+10-1
+100
+10-3
+10-2
+10-1
+100
+101
+102
+103
+104
+t -1.10
+t -1.01
+n/n0
+tn0v0
+n0 = 0.00125
+n0 = 0.01
+ 0.2
+ 0.6
+ 1
+ 1.4
+102
+104
+106
+θn
+t
+n0=0.00125
+n0=0.01
+FIG. 12. The variation of the mean density of clusters n(t) in
+model B with time t is shown for two different initial densi-
+ties. The exponents for the power laws, shown by solid lines,
+have been obtained from finite size scaling. Inset: The time
+dependent exponent θn obtained from θn = −d ln n(t)/d ln t is
+shown. θn saturates for the larger initial densities only at late
+times. Data are for L = 2000 and averaged over 300 different
+initial conditions.
+Eq. (20), namely n(t) ≃ L−2fn(t/L2/θn). In Fig. 13, we
+show the results for two representative initial densities
+n0 = 0.00125 and n0 = 0.01. The data for different L,
+when scaled as in Eq. (20), collapse onto a single curve
+with θn = 1.01(1) for n0 = 0.00125 and θn = 1.10(1) for
+n0 = 0.01. The results for other n0 are listed in Table III,
+based on which we conclude that θn depends on n0 and
+converges to θn = 1 as n0 → 0. We also check that the
+same value of θn leads to the collapse of the data for
+N(m, t) for different times when scaled as in Eq. (3).
+The limiting value of θn = 1 for n0 → 0 coincides
+with θmf
+n
+= 1. However, it is not clear whether the mean
+field result is obtained because correlations vanish. We
+check for correlations by measuring the exponent η. In
+Fig. 14, we show the dependence of the mean squared
+100
+101
+102
+103
+104
+105
+106
+107
+108
+10-6
+10-4
+10-2
+100
+102
+n0=0.01
+n0=0.00125
+n(t)L2
+t/L2/θn
+L = 10000
+L = 5000
+L = 2000
+L = 1000
+L = 500
+L = 250
+FIG. 13. Finite size scaling of n(t) for model B: The number
+density n(t) for different system sizes L collapse onto a single
+curve when scaled as in Eq. (20), with θn = 1.01 and θn =
+1.10 for the initial densities n0 = 0.00125 and n0 = 0.01
+respectively.
+The data for n0 = 0.01 has been shifted for
+clarity.
+10-10
+10-8
+10-6
+10-4
+10-2
+100
+100
+101
+102
+103
+104
+105
+m-1.293
+m-1.204
+〈v2〉
+m
+n0=0.00125
+n0=0.01
+FIG. 14.
+The variation of the mean square velocity ⟨v2⟩
+with mass m for different initial densities.
+The solid lines
+are power-laws m−η with η = 1.293(4) for n0 = 0.00125 and
+η = 1.204(3) for n0 = 0.01. The data are for model B with
+system sizes L = 10000 for n0 = 0.00125 and L = 5000 for
+n0 = 0.01. The data for n0 = 0.01 has been shifted for clarity.
+velocity on the mass m for two initial densities.
+The
+power law dependence extends over three decades and we
+obtain exponents that depend on the initial density n0
+with η = 1.293(4) for n0 = 0.00125 and η = 1.204(3) for
+n0 = 0.01. The results for other n0 are listed in Table III,
+based on which we conclude that η also depends on n0
+and differs significantly from one for small n0. However,
+as n0 increases, we find that η → 1.
+Since it is difficult to measure θe directly from e(t),
+we estimate θe using the scaling relation θe = ηθn [see
+Eq. (15)]. To check for consistency, we confirm that for
+this choice of θe, the data for different system sizes col-
+lapse onto one curve when e(t) and t are scaled using
+finite size scaling as described in Eq. (21). The data col-
+lapse for two different n0, shown in Fig. 15, is satisfactory.
+The results of θe for different n0 are listed in Table III.
+
+9
+10-2
+100
+102
+104
+106
+108
+1010
+10-6
+10-4
+10-2
+100
+102
+n0=0.01
+n0=0.00125
+e(t)L2θe /θn 
+t/L2/θn 
+L = 10000
+L = 5000
+L = 2000
+L = 1000
+L = 500
+L = 250
+FIG. 15. Finite size scaling of e(t) for model B: The mean
+energy e(t) for different system sizes L collapse onto a single
+curve when scaled as in Eq. (21). Results for two different
+densities n0 = 0.00125 and n0 = 0.01(vertically shifted for
+visualization) is shown with θn obtained using finite size scal-
+ing [Eq. (20)] whereas θe is obtained using the hyperscaling
+relation [Eq. (15)].
+10-8
+10-6
+10-4
+10-2
+100
+102
+100
+101
+102
+103
+104
+105
+106
+107
+108
+n0=0.01
+n0=0.00125
+ N(m,t)/mζ
+ t
+m = 2
+m = 4
+m = 8
+m = 12
+m = 16
+t-θn(2+ζ)
+FIG. 16. The data for N(m, t) in model B for fixed masses
+collapse onto one curve when the number density is scaled as
+N(m, t)/mζ with ζ = −0.422(6) for n0 = 0.00125 whereas
+ζ = −0.538(24) for n0 = 0.01. The solid line is a power law
+t−θn(2+ζ) with θn taking values 1.01 and 1.10 for the initial
+densities 0.00125 and 0.01 (vertically shifted for visualization)
+respectively.
+Finally, we determine the exponent ζ defined in Eq. (6)
+for small masses. Similar to model A, in order to deter-
+mine ζ, we study the temporal behavior of N(m, t) for
+fixed mass m = 2, 4, 8, 12, 16. Here, we illustrate the be-
+havior of ζ for two different initial densities. As shown
+in Fig. 16, the data for the different masses collapse
+onto one curve for the respective initial densities when
+N(m, t) is scaled as N(m, t)/mζ, with ζ = −0.4229(6)
+for n0 = 0.00125 and ζ = −0.538(24) for n0 = 0.01. As
+an additional check, the scaled data are consistent with
+the power law with an exponent t−θn(2+ζ).
+Thus, the
+exponent ζ is dependent on the initial density n0. Also,
+they are negative, as compared to model A where the
+exponent is positive.
+TABLE III. Summary of the numerically obtained values of
+the exponents for model B.
+n0
+θn
+η
+θe
+df
+df
+ζ
+(= ηθn)
+[Eq. (18)]
+0.00100 1.01(5) 1.291(4) 1.30(7) 1.49(3) 1.54(17) -0.41(5)
+0.00125 1.01(8) 1.293(4) 1.30(10) 1.49(3) 1.54(23) -0.42(1)
+0.00250 1.03(4) 1.261(7) 1.30(6) 1.49(3) 1.52(14) -0.46(2)
+0.00500 1.08(4) 1.231(2) 1.33(5) 1.49(3) 1.46(12) -0.49(2)
+0.01
+1.10(2) 1.204(3) 1.32(3) 1.49(3)
+1.45(7)
+-0.54(2)
+0.04
+—
+1.10
+—
+—
+—
+—
+0.08
+—
+1.08
+—
+—
+—
+—
+0.16
+—
+1.05
+—
+—
+—
+—
+ 1
+ 1.1
+ 1.2
+ 1.3
+ 1.4
+ 1.5
+10-3
+10-2
+10-1
+θn, θe , η
+n0
+θn
+θe
+η
+FIG. 17.
+The variation of the exponents θn, θe and η are
+shown as function of the initial density n0. The data are for
+model B. θn and θe approach an asymptotic limit 1.0 and 1.3
+respectively for lowest densities. The exponent η ≈ 1.3 in the
+low density limit and approaches the mean field result (η = 1)
+for higher density.
+The results for the exponents θn, θe, η, df and ζ are
+summarized in Table III and their dependence on num-
+ber density n0 is shown in Fig. 17. For higher densities,
+it is difficult to get the exponents θn and hence θe due
+to increasing finite-size effects. However, the exponent η
+can be calculated for the densities larger than 0.01. From
+Table III, we observe that, when n0 → 0, the exponents
+tend to the limiting values θn → 1, η → 1.3 and θe → 1.3.
+When the density increases, we find that η → 1, thus ap-
+proaching its mean field value ηmf = 1. We conclude that
+velocity correlations vanish as density increases. We note
+that in model B, there are no avalanche of coalescence
+events caused due to two clusters colliding. We also ver-
+ify that the exponents satisfy the hyperscaling relation
+given by Eq. (18). In Table III, the fractal dimension
+determined numerically is compared with that obtained
+by Eq. (18) [see columns 5 and 6]. For all densities, the
+values are equal within error bars, thus consistent with
+the scaling theory.
+
+10
+10-7
+10-6
+10-5
+10-4
+10-3
+10-2
+10-1
+100
+10-4
+10-2
+100
+102
+104
+106
+t -0.83
+t -0.93
+n(t)/n0
+tn0v0
+n0 = 0.0001
+n0 = 0.16
+ 0.2
+ 0.6
+ 1
+102
+104
+106
+θn
+t
+FIG. 18. The variation of the mean density of clusters n(t) in
+model C with time t is shown for two different initial densi-
+ties. The exponents for the power laws, shown by solid lines,
+have been obtained from finite size scaling. Inset: The time
+dependent exponent θn obtained from θn = −d ln n(t)/d ln t is
+shown. θn saturates for the larger initial densities only at late
+times. The dashed lines are the reference for the exponents
+0.83 and 0.93. Data are for L = 2000 and averaged over 300
+different initial conditions.
+C.
+Model C: Spherical clusters
+We now determine the exponents θn, θe, η and ζ for
+model C. We first show that the exponent θn depends on
+initial densities n0. Figure 18 shows the variation of n(t)
+with time t for two different initial densities, one small
+and one large. The time dependent θn = −d ln n(t)/d ln t,
+shown in the inset, saturates at different values for the
+different initial densities. Like for model B, it is difficult
+to measure θn directly as n(t) shows strong crossover ef-
+fects. For this reason, we determine θn from finite size
+scaling (see below) following which we obtain θn = 0.83
+for n0 = 0.0001 and θn = 0.93 for n0 = 0.16. The ex-
+ponents obtained from finite size scaling are shown in
+Fig. 18 for comparison and they describe the data for
+large times well.
+We determine the exponent θn using the finite size scal-
+ing n(t) ≃ L−2fn(t/L2/θn) [see Eq. (20)]. Two represen-
+tative cases are shown in Fig. 19. The data of n(t) for
+different L, when scaled as in Eq. (20) collapse onto a
+single curve for θn = 0.83 for n0 = 0.001 and θn = 0.93
+for n0 = 0.16. The results for other n0 are listed in Ta-
+ble IV, based on which we conclude that θn depends on
+n0 and increases to the mean field result θmf
+n
+= 1 with
+increasing n0. We also check that the same value of θn
+leads to the collapse of the data for N(m, t) for different
+times when scaled as in Eq. (3).
+It is possible that the mean field result is obtained at
+higher n0 because the correlations vanish. Two repre-
+sentative cases are shown in Fig. 20.
+We find that η
+depends on the initial density n0 with η = 1.283(13) for
+n0 = 0.0001 and η = 1.114(2) for n0 = 0.16. The results
+for other n0 are listed in Table IV and it shows that η
+decreases to its mean field prediction ηmf = 1 as density
+100
+101
+102
+103
+104
+105
+106
+107
+108
+109
+10-8
+10-6
+10-4
+10-2
+100
+n0=0.0001
+n0=0.16
+L2n(t)
+t/L2/θn 
+L10000
+L5000
+L2000
+L1000
+L500
+FIG. 19. Finite size scaling of n(t) for model C: The number
+density n(t) for different system sizes L collapse onto a single
+curve when scaled as in Eq. (20), with θn = 0.83(4) and θn =
+0.93(5) for the initial densities n0 = 0.0001 and n0 = 0.16
+respectively.
+The data for n0 = 0.16 has been shifted for
+clarity.
+10-12
+10-10
+10-8
+10-6
+10-4
+10-2
+100
+100
+101
+102
+103
+104
+105
+m-1.283
+m-1.114
+〈v2〉
+m
+n0=0.0001
+n0=0.16
+FIG. 20.
+The variation of the mean square velocity ⟨v2⟩
+plotted as function of mass m for different initial densities.
+The solid lines are power-laws m−η with η = 1.283(13) and
+η = 1.114(2) for n0 = 0.0001 and n0 = 0.16 respectively.
+The data are for model C with system sizes L = 10000 and
+L = 2000 for the densities n0 = 0.0001 and n0 = 0.16 respec-
+tively. The data for n0 = 0.16 has been shifted for clarity.
+increases.
+We find that it is difficult to measure θe directly from
+the power-law decay of e(t). Hence, we measure θe using
+the scaling relation, θe = ηθn [see Eq. (15)]. To check
+for the consistency of the result for θe obtained using
+the scaling relation [Eq. (15)], we confirm that for this
+choice of θe, the data for different system sizes can be
+collapsed onto one curve using finite size scaling e(t) ≃
+L−2θe/θnfe(t/L2/θn) [see Eq. (21)]. The data collapse is
+satisfactory as shown in Fig. 21 for the two different n0.
+The results of θe for different n0 are listed in Table IV
+which shows that θe is close to the mean field limit, θmf
+e
+=
+1 for all n0.
+Finally, we determine the exponent ζ [defined in
+Eq. (6)] for small masses. In order to determine ζ, we
+
+11
+10-8
+10-6
+10-4
+10-2
+100
+102
+104
+106
+108
+10-8
+10-7
+10-6
+10-5
+10-4
+10-3
+10-2
+10-1
+100
+n0=0.0001
+n0=0.16
+L2θe / θn n(t)
+t/L 2/θn 
+L10000
+L5000
+L2000
+L1000
+L500
+FIG. 21.
+Finite size scaling of e(t) for model C: The mean
+energy density e(t) for different system sizes L collapse onto
+a single curve when scaled as in Eq. (21). Results for two
+different densities n0 = 0.0001 and n0 = 0.16 is shown with
+θn obtained using finite size scaling [Eq. (20)] whereas θe ob-
+tained using the hyperscaling relation [Eq. (15)]. The data
+for n0 = 0.0001 has been shifted for clarity.
+10-8
+10-7
+10-6
+10-5
+10-4
+10-3
+10-2
+10-1
+100
+101
+100
+101
+102
+103
+104
+105
+106
+107
+108
+109
+n0=0.0001
+n0=0.16
+ N(m,t)/mζ
+ t
+m = 2
+m = 4
+m = 8
+m = 16
+t-θn(2+ζ)
+FIG. 22. The data for N(m, t) in model C for fixed masses
+collapse onto one curve when the number density is scaled as
+N(m, t)/mζ with ζ = −0.248(26) for n0 = 0.0001 whereas
+ζ = −0.563(10) for n0 = 0.16. The solid line is a power law
+t−θn(2+ζ) with θn as 0.83 and 0.93 for the initial densities
+0.0001 and 0.16 respectively. The data for n0 = 0.0001 has
+been shifted for clarity.
+study the temporal behavior of N(m, t) for fixed mass
+m = 2, 4, 8, 16. Here, we illustrate the behavior of ζ for
+two different initial densities. As shown in Fig. 22, the
+data for the different masses collapse onto one curve for
+the respective initial densities when N(m, t) is scaled as
+N(m, t)/mζ, with ζ = −0.248(26) for n0 = 0.0001 and
+ζ = −0.563(10) for n0 = 0.16. As an additional check,
+the scaled data are consistent with the power law with
+an exponent t−θn(2+ζ). The results of ζ for other densi-
+ties are listed in Table IV. We conclude that ζ is strongly
+dependent on n0.
+We find that the exponents θn, θe, η and ζ are den-
+sity dependent [see Table IV and Fig. 23(a)].
+θn in-
+creases with the increase in density and approaches the
+TABLE IV. Summary of the numerically obtained values of
+the exponents for model C.
+n0
+θn
+η
+θe(= ηθn)
+ζ
+0.0001
+0.83(4)
+1.283(13)
+1.06(6)
+-0.248(26)
+0.00125
+0.84(5)
+1.275(10)
+1.07(7)
+-0.350(27)
+0.01
+0.85(5)
+1.241(2)
+1.05(6)
+-0.364(6)
+0.04
+0.87(6)
+1.174(3)
+1.02(7)
+-0.403(4)
+0.16
+0.93(5)
+1.114(2)
+1.04(6)
+-0.563(10)
+(a)
+(b)
+FIG. 23.
+(a) The variation of the exponents θn, θe and η with
+initial density, n0, for model C. The horizontal dotted line is
+the mean field prediction, θmf
+n
+= θmf
+e
+= ηmf = 1. (b) Com-
+parison of the exponent θn with results of earlier simulations
+of BA in the continuum [39, 41].
+mean field predictions θmf
+n
+= 1. An opposite trend is ob-
+served in the variation of exponent η with density where
+it decreases with the increase in initial density but, ap-
+proaches the mean field prediction ηmf = 1 with the in-
+crease in density. On the other hand, θe has a rather weak
+dependence on the initial density and is always close to
+the mean field result θmf
+e
+= 1 irrespective of the initial
+density. We compare our results with those for BA in the
+continuum [39, 41] in Fig. 23(b). We find that the data
+are in good agreement, suggesting that the stochasticity
+introduced in the temporal evolution of the lattice model
+is not relevant.
+
+12
+V.
+CONCLUSION
+In this paper, we studied the problem of ballistic ag-
+gregation in two dimensions using three different lat-
+tice models. In all the three models, particles move, on
+an average, in a straight line and undergo momentum-
+conserving aggregation on contact. The three models dif-
+fer in the shape of the particles. In Model A, the particles
+are point-sized and occupy a single lattice site. In model
+B, the shape of the aggregate is the combined shape of
+the two aggregating particles at the time of collision, and
+is a fractal. In model C, the shape of the particles are
+spherical, to the closest lattice approximation. For the
+three models, from large scale Monte Carlo simulations,
+we determine the exponents characterizing the power-law
+decay of the number density of particles, the mean en-
+ergy, the fractal dimension, the correlation between the
+velocities of the particles constituting an aggregate and
+the scaling function for the mass distribution. The re-
+sults for the three models are summarized in Table II
+(model A), Table III and Fig. 17 (model B), Table IV
+and Fig. 23 (model C).
+We find that the values of the exponents are indepen-
+dent of the initial number density only for model A. For
+models B and C, the exponents are weakly dependent on
+the initial number density, making them non-universal.
+The fractal dimension in model B is, however, indepen-
+dent of the initial number density, within the numeri-
+cal accuracy that we could achieve.
+In model C, the
+trends in the dependence of the exponents on n0 are con-
+sistent with the corresponding simulations for spherical
+particles in the continuum [29, 38–41]. While the expo-
+nent θn matches closely with the continuum results [see
+Fig. 23(b)], we find that the numerical values of the ex-
+ponent θe is less than the continuum result [41] and ap-
+proaches the mean field result faster. This discrepancy
+could be due to difficulties in measuring θe accurately
+due to strong crossovers seen in the data. We have shown
+that the results for the exponents in all the models, irre-
+spective of its dependence on n0, satisfy the hyperscaling
+relations derived from scaling theory.
+The fractal dimension of clusters formed by aggrega-
+tion is of interest in many experiments (for example,
+see [5, 42–45]). While it is to be expected that the expo-
+nents θn and θe will depend on the nature of transport
+and the shapes of the clusters, it is not clear whether the
+fractal dimension depends on transport. Fractal dimen-
+sion of the cluster in two-dimensional diffusion-limited
+aggregation (DLA) models, where clusters grow from a
+nucleating center, show df ≃ 1.70 [49, 50].
+However,
+fractal dimension of clusters, when there is no nucleating
+center but all the aggregates undergo diffusive motion, is
+different from that of DLA. In the case when the diffu-
+sion constant of larger masses decreases with mass or is
+mass-independent, df has been been shown to be in the
+range df ≃ 1.38−1.52 [51–53]. This result is close to our
+result for ballistic aggregation (model B) for which we
+found df ≃ 1.49. While close, it is not clear whether the
+fractal dimension is different for the diffusive and ballis-
+tic models. The value 1.49 is very close to that observed
+in sprays (1.54) [45], and cells (1.5) [44].
+It would be
+interesting to explore this connection further as well as
+understand the dependence of the fractal dimension on
+different mass dependent velocities, especially the limit
+where larger masses move faster.
+The mean field approximation assumes that the ve-
+locities of the particles forming a cluster are uncorre-
+lated. The correlations are characterized by the power-
+law dependence of the speed on the mass of the aggre-
+gate: ⟨v2(m)⟩ ∼ m−η, with ηmf = 1. Earlier simulations
+of spherical particles in the continuum show that η de-
+creases to η = ηmf as the initial number density of parti-
+cles, n0, is increased [29, 38–41]. This lack of correlation
+was attributed to the increased avalanche of coagulation
+events that occur due to the overlap of a newly created
+spherical particle with already existing particles, as the
+number density is increased.
+In this paper, we deter-
+mined η for the three models.
+For model A, we find
+that η ≈ 1.15 is independent of n0 and hence there is no
+limit in which velocities become uncorrelated. For mod-
+els B and C, we find that η → ηmf with increasing n0
+(see Tables III and IV). However, in model B there are
+no avalanche of collisions while model C has avalanche
+of collisions.
+Thus, contrary to earlier conjecture, the
+avalanche of collisions cannot be a necessary condition
+for velocities to become uncorrelated.
+In contrast to BA in the continuum where the dy-
+namics is deterministic, the temporal evolution in the
+lattice models is stochastic.
+Each particle moves in a
+straight line only on an average. In the continuum models
+stochasticity enters only through the initial conditions.
+However, for BA in one dimension, it has been shown
+that the stochasticity in the dynamics not only does not
+affect scaling laws, the lattice models reproduce many
+details of the trajectory like shock positions for the same
+initial conditions [23, 46]. For model C, we find that the
+results for θn match with the earlier continuum results in
+two dimensions for all n0. We thus conclude that stochas-
+ticity in the initial conditions dominate the fluctuations
+induced by the dynamics. This is in sharp contrast to
+diffusive systems where diffusive fluctuations dominate
+randomness in initial conditions.
+For all the three models, we measure the exponent ζ
+[see definition in Eq. (6)] which characterizes the behavior
+of smaller mass aggregates. The exponent ζ is not easily
+obtained from scaling arguments and for the correspond-
+ing diffusive problem requires renormalisation group cal-
+culations [54–56]. For model A, we find that ζ is positive,
+implying that there is a typical time dependent mass.
+This is in contrast to point particles in one dimension
+where the mass distribution is a power law. For models
+B and C, we find that ζ is dependent on n0. However,
+it is negative for all values of n0, implying that the mass
+distribution is a power law in mass, for a given time.
+
+13
+ACKNOWLEDGMENTS
+The simulations were carried out on the supercomputer
+Nandadevi at The Institute of Mathematical Sciences
+(IMSc). P.F would like to thank IMSc for the visiting stu-
+dentship. VVP acknowledges SERB SRG 2022/001077
+for support.
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+

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+ 
+*Corresponding author: erimyanik@gmail.com 
+One-shot domain adaptation in video-based assessment of surgical skills 
+Erim Yanik1*, Steven Schwaitzberg2,  Gene Yang2, Xavier Intes1, and Suvranu De1,3 
+1 Center for Modeling, Simulation & Imaging in Medicine, Rensselaer Polytechnic Institute, NY, 
+USA 
+2 School of Medicine and Biomedical Sciences, University at Buffalo, NY, USA 
+3 College of Engineering, Florida A&M University and The Florida State University, FL, USA 
+ 
+Deep Learning (DL) has achieved automatic and objective assessment of surgical skills. 
+However, DL models are data-hungry and restricted to their training domain. This prevents 
+them from transitioning to new tasks where data is limited. Hence, domain adaptation is 
+crucial to implement DL in real life. Here, we propose a meta-learning model, A-VBANet, 
+that can deliver domain-agnostic surgical skill classification via one-shot learning. We 
+develop the A-VBANet on five laparoscopic and robotic surgical simulators. Additionally, 
+we test it on operating room (OR) videos of laparoscopic cholecystectomy. Our model 
+successfully adapts with accuracies up to 99.5% in one-shot and 99.9% in few-shot settings 
+for simulated tasks and 89.7% for laparoscopic cholecystectomy. For the first time, we 
+provide a domain-agnostic procedure for video-based assessment of surgical skills. A 
+significant implication of this approach is that it allows the use of data from surgical 
+simulators to assess performance in the operating room.
+There is growing interest in using deep learning (DL) approaches in surgical skill assessment1,2. 
+DL models1–20 enable real-time objective assessment of surgical skills with sufficient procedure-
+specific data. However, surgical data is scarce20–23, expensive to collect2 in real environments, and 
+time-consuming to process/annotate24. Thus, current models are typically developed and tested for 
+one specific task, limiting their utility to the community at large. To generalize such models to 
+other surgical tasks – or domains – manually-intensive post-processing methodologies, such as 
+transfer learning25,26, are required. This is highly impractical and inefficient as the number of 
+surgical procedures performed and their variations are vast. Therefore, a major hurdle for the wide 
+dissemination of DL models and impacting clinical practice is for them to provide robust 
+performances while adapting to new surgical procedures for which limited data are available. 
+Herein, we propose a domain-agnostic DL model, Adaptive Video-Based Assessment Network 
+(A-VBANet), for surgical skill assessment using video streams. Fig. 1 details our approach. We 
+utilized few-(one-)shot26–30 meta-learning26,31–36 and investigated adaptability in five physical 
+simulators and laparoscopic cholecystectomy in the operating room (OR). Existing literature in 
+surgical skill assessment is not linked to meta-learning so far, and the closest study was adaptive 
+tool detection in robotic surgery30. This renders our pipeline the first in the field. A-VBANet has 
+the potential for broad implementation for surgical training, assessment, and credentialing. 
+ 
+Results 
+Metaset characteristics. Datasets. Our metaset comprised six surgical tasks from four cohorts 
+that include laparoscopic pattern cutting (cohort 1), laparoscopic suturing (cohort 2), robotic 
+suturing23, needle passing23, knot tying23 (cohort 3), and laparoscopic cholecystectomy (test 
+cohort). This yielded 29 students and 43 surgeons of 16 skill classes performing more than 2,300 
+trials (Fig. 1). The skill classes for laparoscopic pattern cutting and cholecystectomy were based 
+
+ 
+2 
+ 
+on trial-wise performance. For the remaining tasks, we labeled using the surgical expertise of the 
+subjects. 
+Preprocessing. We utilized SimCLR37 to preprocess surgical videos. For each cohort, the model 
+was trained separately. Then, the trained models were used in their respective cohorts to extract 
+self-supervised features (SSFs) per frame in temporal order. This generated spatiotemporal feature 
+set used as input to the meta-learner. Our study included increasing SSFs from 2 to 64 as multiples 
+of two. 
+ 
+Fig. 1 | Overview of the A-VBANet pipeline. a. Surgical tasks and cohorts of the metaset. Here, 
+laparoscopic cholecystectomy is an OR surgery, while the remaining are simulators. b. The self-
+supervision network and the corresponding spatiotemporal feature extraction. Here, T denotes 
+temporal length. c. The meta-learner pipeline. The model adapts to one task at a time in a round-
+robin fashion, using the residual backbone designed for sequential inputs. At each turn, the trained 
+models are tested in the validation task and laparoscopic cholecystectomy. d. The t-SNE plot 
+shows the distribution of spatiotemporal feature sets. As seen, different cohorts generate different 
+clusters. e. The metaset characteristics. 
+ 
+
+Pattern cutting
+Suturing (Lap.)
+Suturing (Rob.)
+Knot tying
+Needle passing
+Cholecystectomy
+ynet
+Irain
+Validate
+(+Test)
+Proto-MAML
+Datasets
+Subjects
+Classes / Number of trials
+Fail
+Pass
+Pattern Cutting Residents
+213
+1,842
+Novice
+Expert
+Suturing (Lap.)
+Both
+24
+39
+Novice Intermediate Expert
+Suturing (Rob.)
+38
+20
+20
+Needle Passing
+Surgeons
+22
+16
+18
+Knot Tying
+32
+20
+20
+Low perform.
+High perform.
+Cholecystectomy Surgeons
+12
+3 
+3 
+ 
+A-VBANet adapts to simulation tasks. We trained and validated our pipeline in a round-robin 
+fashion via one-shot learning. For instance, when pattern cutting was the target domain, the 
+remaining tasks were the source domain, i.e., the training domain of the network. At each round, 
+the validation task was also used for testing. Notably, laparoscopic cholecystectomy was excluded 
+from this scheme. Instead, we used it to further test the trained model’s adaptability in real-life 
+surgery at each round (Fig. 1). In this study, the results of a task are given for the best SSF set via 
+one-test-shot (k=1), an average of 100 repetitions for cohorts 1-3, and the best of 100 repetitions 
+for the test cohort. 
+ 
+A-VBANet adapts to binary-class tasks. In pattern cutting, the adaptation accuracy was 
+0.900±.023 (Table 1). We also report the area under curve (AUC) of the Receiver Operating 
+Characteristics (ROC) to be 0.955±.020. In laparoscopic suturing, these values were 0.995±.008 
+and 0.999±.005 for accuracy and AUC. Fig. 2a illustrates the ROC curves for pattern cutting and 
+suturing. In addition, accuracy increased with k, i.e., few-test-shot setting, in both tasks (Table 1 
+and Extended Table 1). 
+ 
+ 
+Besides performance, we evaluated the reliability of the true predictions in each skill class 
+using NetTrustScore (NTS)38. (Supplementary Information / NetTrustScore). In pattern cutting, 
+NTSs were 0.989±.068 for Fail and 0.991±.047 for Pass. In laparoscopic suturing, these values 
+were 0.991±.009 for Novice and 0.998±.005 for Expert. Moreover, NTS increased with k in both 
+tasks (Extended Table 2). Fig. 2b details the trust density distribution over Softmax. 
+ 
+AVBA-Net adapts to multi-class tasks. The adaptation accuracies were 0.651±.040, 0.626±.027, 
+and 0.688±.022 in robotic suturing, needle passing, and knot tying (Table 1). Here, the model’s 
+performance increased with k in all tasks (Extended Table 1). Notably, k = 16 was not observed 
+in needle passing due to insufficient data. In addition, for true predictions, the NTSs were 
+0.998±.003, 0.994±.008, and 0.994±.008 for Novice, Intermediate, and Expert in robotic suturing. 
+In needle passing, these values were  0.981±.020, 0.978±.022, and 0.965±.026. Finally, in knot 
+tying, we obtained 0.921±.039, 0.868±.052, and 0.817±.065. NTS increased with k in all tasks 
+(Extended Table 2). Fig. 3 illustrates the trust density distribution over SoftMax.
+ 
+ 
+ 
+ 
+Table 1 | Task adaptation accuracies. 
+Val. and Test Dataset 
+k = 1 
+k = 2 
+k = 4 
+k = 8 
+k = 16 
+Pattern Cutting 
+0.900±.023 0.910±.022 
+0.920±.018 0.925±.019 
+0.929±.017 
+Suturing (Lap.) 
+0.995±.008 0.995±.008 
+0.995±.006 0.997±.006 
+0.999±.005 
+Suturing (Robotic) 
+0.651±.040 0.664±.027 
+0.697±.039 0.716±.035 
+0.761±.044 
+Needle Passing 
+0.626±.027 0.645±.022 
+0.690±.033 0.727±.038 
+N/A 
+Knot Tying 
+0.688±.022 0.697±.031 
+0.714±.042 0.763±.057 
+0.835±.077 
+
+ 
+4 
+ 
+ 
+Fig 2. | a. ROCs and b. trust spectrums, resulting from 100 repetitions in pattern cutting (purple) 
+and laparoscopic suturing (turquoise).
+ 
+Fig 3. | Trust spectrums for k = 1 cumulative of 100 runs in a. robotic suturing, b. needle passing 
+and c. knot tying. 
+a 
+b 
+c 
+a 
+b 
+
+160
+100
+Novice
+100
+140
+Intermediate
+Expert
+80
+80
+60
+09
+80
+60
+40
+40
+40
+20
+20
+20
+0
+0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+SoftmaxDistribution
+Softmax Distribution
+Softmax Distribution
+80
+70
+Novice
+Intermediate
+60
+Expert
+60
+50
+·50
+40
+40
+30
+30
+20
+20
+10
+10
+0
+0
+0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Softmax Distribution
+Softmax Distribution
+SoftmaxDistribution
+70
+Novice
+25
+Intermediate
+17.5
+Expert
+60
+15.0
+20
+12.5
+40
+15
+10.0
+10
+7.5
+5.0
+10
+5
+2.5
+0
+0
+0.0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Softmax Distribution
+Softmax Distribution
+Softmax Distribution1.0
+1e3
+Fail
+Pass
+4
+4
+0.6 
+3
+2
+2
+1
+1
+0.0
+0
+0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0.0
+0.2
+0.4
+0.6
+8'0
+1.0
+0'0
+0.2
+0.4
+0.6
+0.8
+1.0
+False Positive Rate
+Softmax
+Distribution
+Softmax
+Distribution
+1.0
+160
+Novice
+200
+Expert
+140
+150
+0.6
+100
+80
+100
+0.4
+60
+40
+50
+20
+0.0
+0
+0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+False Positive Rate
+Softmax Distribution
+Softmax Distribution 
+5 
+ 
+A-VBANet adapts to an operating room procedure. After being validated on a different 
+simulator at each round, we tested how well the A-VBANet can perform on laparoscopic 
+cholecystectomy. The accuracies are reported in Table 2. We obtained an overall accuracy of 0.867 
+and an AUC of 0.840. We did not analyze the few-shot setting due to limited data. When we broke 
+down the performance in individual validation tasks, we observed consistency between the tasks 
+(Table 2 and Extended Table 3). In addition, the NTSs for true predictions were 1.0 for both the 
+Low Performance and High Performance classes (Extended Table 4). Fig. 4 shows the trust density 
+distributions over SoftMax. 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+Fig 4. | Trust spectrums for each validation task for k = 1 in laparoscopic cholecystectomy for 
+Low Performance (left) and High Performance (right) classes. 
+ 
+Discussion 
+For over two decades, global rating tools, i.e., OSATS24 and FLS39 scoring, have been the gold 
+standard in assessing surgical skills. Current surgical skill assessment models rely on these rating 
+tools. However, such models are data-intensive and domain-specific. Further, surgical data is 
+limited2, and more than a few domains exist in real life. Furthermore, extracting fundamental 
+features of what constitutes surgical skills is challenging to determine manually. Hence, for DL 
+models to be useful, they must be capable of extracting information from simulator data and 
+adapting that to operating room procedures. 
+The main contribution of this paper is to utilize meta-learning effectively to pave the way for 
+DL approaches for surgical skill assessment without the need for extensive data. The A-VBANet 
+is able to adapt to surgical simulations by seeing only one sample (Table 1). This is the first step 
+Table 2 | Adaptation accuracies on laparoscopic cholecystectomy for k = 1 
+Validation dataset 
+Accuracy 
+AUC 
+Pattern Cutting 
+0.872 
+0.818 
+Suturing (Lap.) 
+0.872 
+0.848 
+Suturing (Rob.) 
+0.821 
+0.833 
+Needle Passing 
+0.872 
+0.838 
+Knot Tying 
+0.897 
+0.864 
+Overall 
+0.867 
+0.840 
+
+35
+Pattern cutting
+15.0
+Pattern cutting
+30
+Suturing (Lap.)
+Suturing (Lap.)
+ity
+25
+Suturing (Rob.)
+Suturing (Rob.)
+Needle passing
+10.0
+Needle passing
+Knot tying
+7.5
+Knot tying
+Trust
+5.0
+5
+2.5
+0
+0.0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Softmax Distribution
+Softmax Distribution 
+6 
+ 
+in the real-life deployment of such technology in surgical training2,40–42 and credentialing39,43 
+outside the operating room. Both laparoscopic pattern cutting and suturing are prerequisites for 
+board certification in general and ob/GYN surgery39. Our model’s adaptation accuracy was 0.900 
+and 0.995 in these tasks. 
+Domain-agnostic assessment in the operating room is the second goal of this paper. Assessing 
+real-life expertise is essential for lifelong learning44 and continuous certification45–47. However, it 
+is inherently difficult and time-consuming to collect and annotate data from unregulated 
+environments such as the operating room2. Companies and medical societies engaged in collecting 
+limited datasets and expensive manual annotations typically restrict free public access to those 
+annotated data. For the widespread application of DL to surgical skill evaluation, it is crucial to 
+overcome these data-related challenges. The A-VBANet, trained and validated on simulators, was 
+tested on laparoscopic cholecystectomy. We investigated the best overall performance our model 
+can attain and obtained promising adaptation accuracy (overall 0.867 up to 0.897) and AUC of the 
+ROC (0.840 up to 0.964) (Table 2) from raw video data collected in the operating room without 
+annotation. This showed for the first time that adapting to the operating room is feasible with only 
+one sample via taking advantage of simulation task data. These results are preliminary under the 
+context of given surgical tasks. By adding more diverse representation, i.e., different real-life 
+procedures and cohorts, to the source set, the performance can be further improved. 
+A critical consideration for DL model development is that the network should be trustworthy, 
+i.e., ensure consistent performance for unseen trials. Hence, we also measured the network's 
+confidence in true predictions using NTS38. In Figs. 2, 3, and 4, we observed high density 
+distribution towards high Softmax values, indicating robust NTS (>0.8) for all tasks. In other 
+words, the prediction probabilities for the actual classes were well-separable from the probabilities 
+of other classes. Thus, the adapted models were reliable and highly likely to perform consistently. 
+In addition, we investigated the few-test-shot (k>1) setting for all the tasks other than 
+laparoscopic cholecystectomy. The performance of the A-VBANet increased with k (Extended 
+Tables 1). This is expected as increasing k implies more information for the network to adapt to a 
+new domain. However, the drawback of using larger k is that it decreases the number of testing 
+samples, increasing epistemic uncertainty48. This can be seen in increasing standard deviation in 
+datasets with limited sample size, i.e., JIGSAWS tasks. On the other hand, the NTS got better in 
+laparoscopic pattern cutting and suturing but did not follow a trend for the rest of the tasks as the 
+sample size was limited (Extended Tables 2). The models need to be tested with more data before 
+a conclusive statement can be made for NTS. 
+We did not observe a linear correlation between SSF and model performance (Extended Tables 
+1 and 3). However, increased SSF led to increased NTS (Extended Tables 2 and 4). This signifies 
+that more information leads to higher confidence in true predictions. However, it decreases the 
+possibility of cross-overs, i.e., false prediction being predicted correctly and vice versa. This makes 
+the model less flexible. Thus, more prone to overfitting. 
+Another strength of our study is using videos over sensor-based kinematics, as the latter is 
+more expensive to collect and often unavailable2. On the other hand, videos are increasingly more 
+available 49. In addition, video-based assessment (VBA) is currently the main focus of national 
+institutions44,50 to replace traditional intraoperative training1,24,49. Besides, videos enabled us to 
+derive additional information from unlabeled data. For instance, even though there were 15 labeled 
+laparoscopic cholecystectomy trials, the self-supervision model corresponding to this cohort was 
+trained on 198 (183 of which were unlabeled) videos. Although we had limited data, we did not 
+implement the snippeting technique15,18,51 to augment the data size. This is because it inflates the 
+
+ 
+7 
+ 
+score prediction12,24. Moreover, it causes inconsistent labeling as it is uncertain that performance 
+is isotropic within every trial12. 
+Some limitations of our study include cohort-specific preprocessing and limited testing data 
+and tasks. In future studies, we plan to incorporate meta-learning into self-supervision to provide 
+cohort-agnostic feature extraction from videos. This allows an end-to-end pipeline for domain 
+adaptation. Further, we aim to incorporate a broader range of surgical tasks. 
+This study demonstrated for the first time that one-shot domain adaptation is feasible in 
+surgical skill assessment, and a DL model can successfully adapt to multiple domains effectively 
+and automatically. This brings DL models one step closer to real-life implementation for surgical 
+training, assessment, and credentialing. 
+ 
+Methods 
+Metaset generation. The pattern cutting and suturing data were collected separately by our group in collaboration 
+with the University at Buffalo. They are subtasks of the Fundamentals of Laparoscopic Surgery (FLS) program, 
+which is a prerequisite for board certification39. For both, Institutional Review Board (IRB) approval was sought at 
+Rensselaer Polytechnic Institute and University at Buffalo. Further, informed consent was collected from each 
+subject. 
+Pattern cutting enrolled 21 residents (6 males / 15 females), ages between 21 and 30 (Mean: 23.95 / Std.: 1.69), 
+with no laparoscopy background. Here, one subject was left-handed. The subjects executed the task for 12 days, 
+generating 2,055 trials. We labeled each trial Pass or Fail (Fig. 1e) based on the FLS-based cut-off threshold52. This 
+produced 1,842 Pass and 213 Fail samples. Videos were collected at 640 x 480 resolution at 30 FPS via the FLS box 
+camera. 
+Laparoscopic suturing included 10 surgeons (5 males / 5 females) and 8 residents (5 males / 3 females), with 
+ages ranging from 23 to 56 (Mean: 31 / Std.: 7.9). All the surgeons were experienced in FLS with years of 
+experience varying between 1 to 20 years, while no resident had prior expertise in laparoscopy. This totaled 63 
+suturing trials (Fig. 1e). Notably, using the same methodology as pattern cutting, we ended up with only three Fail 
+samples. Thus, instead, we labeled the trials by residents as Novice and surgeons as Expert. This generated 24 
+Novice and 39 Expert samples. The videos were recorded at 720x480 resolution at 30 FPS via the FLS box camera. 
+We also employed robotic suturing, needle passing, and knot tying from the publicly available JIGSAWS 
+dataset23. All the tasks were conducted via the Da Vinci Surgical System. For each task, 8 surgeons performed 
+approximately five times. The class labels were assigned based on the surgical expertise in robotic surgery. 
+Surgeons with less than 10 hours of experience were labeled Novices, while more than 100 hours were Experts. 
+Surgeons in between were Intermediates. This led to 4 Novice, 2 Intermediate, and 2 Expert surgeons (Fig. 1e). In 
+addition, two separate video streams were collected per task from different angles at 640x480 resolution and 30 FPS. 
+We assumed each view as a separate trial to augment the data size. 
+Laparoscopic cholecystectomy videos were collected at Kaleida Health in Buffalo, New York, totaling 198 
+trials. In this study, 15 trials were annotated as Low Performance and High Performance, based on the OSATS 
+scores, yielding 12 and 3 samples (Fig. 1e) in each category. The criterion for a trial labeled as High Performance 
+was having an OSATS score greater than 23 (out of 25) (See Extended Table 5 for the OSATS breakdown). Next, 
+the surgical videos were collected via laparoscopes of varying resolutions at 30 FPS. 
+ 
+Model development. Developing feature extractor. SimCLR37 is a self-supervised contrastive network used to 
+extract comprehensive spatiotemporal features. We used SimCLR to reinforce our pipeline against corrupted frames, 
+e.g., blurry frame and background interference, such as changing light conditions and jitter. SimCLR uses a 
+backbone, 𝑓𝑏(. ) ∈ ℝ𝐷, to aggregate D-dimensional feature sets, i.e., representations37 from the video frames. In this 
+study, the backbone is ResNet34, with D = 512. Then, using 
+a linear classifier, it maps the representations into K-(128-) dimensional hidden space, 𝑓ℎ(. )∈ ℝ𝐾. The aim is to 
+maximize the likelihood of the classifier finding the augmented versions of the input frame in a large batch of 
+uncorrelated frames in 𝑓ℎ(.)37. Once trained, the classifier is removed. Extended Fig. 1 illustrates the SimCLR 
+architecture and deployment. 
+Generating spatiotemporal features. To generate spatiotemporal features (𝑿), we applied the trained backbone  
+𝑓𝑏(. ) to each frame in a surgical video53 in temporal order, i.e., 𝑿𝑖 = [𝑓𝑏(𝑥𝑖1),… ,𝑓𝑏(𝑥𝑖𝑗),…, 𝑓𝑏(𝑥𝑖𝑇)] ∈ ℝ𝑇𝑥𝐷, 
+where 𝑥𝑖 ∈ ℝ𝑇𝑥3 is the list of frames of the 𝑖𝑡ℎ trial. Here, T is the temporal length and 𝑥𝑖𝑗 is the 𝑖𝑡ℎ trial’s 𝑗𝑡ℎ frame. 
+
+ 
+8 
+ 
+Finally, 𝑿𝑖 ∈ ℝ𝑇𝑥𝐷 is the spatiotemporal feature set for the 𝑖𝑡ℎ trial. In addition, we used 1D Global Average 
+Pooling (GAP)54 to downsample 𝐷-dimensional representations: 𝐺𝐴𝑃(𝑿𝑖) ∈ ℝ𝑇𝑥𝐷 → 𝑿𝒊
+′ ∈ ℝ𝑇𝑥𝐷′ where 𝐷′ is of 
+2,4,8,16,32, and 64-dimensions. 
+The meta-learner methodology. ProtoMAML35 is the combination of Prototypical Network (ProtoNet)34 and 
+Model-agnostic meta-learning (MAML)36. ProtoNet is a metric-based55 meta-learning model, learning to learn 
+prototypical (class) centers, 𝑣𝑐, in nonlinear embedding space34. MAML, on the other hand, is a model-based55 meta-
+learner that offers fast and flexible adaptability to the target domains by learning the "global" optimal initialization 
+parameters (𝜃)36. One shortcoming of MAML is the lack of robust initialization to the output layer35. Proto-MAML 
+addresses this by combining MAML’s flexible adaptability with the prototypical center methodology from ProtoNet 
+and reports the best overall performance in multiple image datasets35. Specifically, ProtoMAML works by splitting 
+the training and validation sets into support and query sets. The support sets were used to optimize the parameter 
+space. On the other hand, the query sets were used to compute the train and validation losses. (Supplementary 
+Information / ProtoMAML implementation). 
+Developing the backbone of the meta-learner. The backbone of the ProtoMAML was developed based on our 
+previously published state-of-the-art model, the VBA-Net12, and the residual networks proposed by He et al.53. The 
+backbone consisted of two attention-infused56 residual blocks and 1x1 convolutional layer54 in between to adjust the 
+dimension. In addition, each block had two convolutional layers and an identity shortcut12. Further, the 
+convolutional layers were diluted to expand the receptive field without losing temporal resolution57. Notably, 
+dilation proved helpful in improving model performance when working with sequential data12. 
+ The residual layers were followed by a classifier adjusted to work with the meta-learner. Meta-learning models 
+are used for object classification30,34–36. In such models, the input is spatial, 𝑥 ∈ ℝ𝐵𝑥𝐻𝑥𝑊𝑥3 (B: batch size, H: height, 
+W: width). and reduced to 𝑥̂ ∈ ℝ𝐵𝑥𝐷𝑜 by a flattening layer where 𝐷𝑜 is the output dimension. However, our input is 
+spatiotemporal, 𝑥 ∈ ℝ𝐵𝑥𝑇𝑥𝐷′. Hence, the residual blocks were followed by a 1D GAP layer in our design to obtain 
+𝑥̂ ∈ ℝ𝐵𝑥𝐷𝑜. GAP also enabled us to use entire sequences. Following GAP, a fully-connected layer generated the 
+embedding space, 𝑓(𝑥̂) ∈ ℝ𝐷𝑜. Finally, a linear classifier, initialized via 𝑣𝑐, outputted predictions35. In this study, 
+𝐷𝑜 varied based on 𝐷′. (See Supplementary Information / Hyperparameter selection for more information and 
+Extended Fig. 2 for the backbone architecture). 
+ 
+Training. Feature extractor. When training SimCLR, we used a train/validate split of 143,287/17,373 frames in 
+pattern cutting. These values were 21,191/3,315 in laparoscopic suturing and 447,314/66,836 for the JIGSAWS 
+dataset. In laparoscopic cholecystectomy, the split was 353,168/46,310. To generate the augmented version of the 
+input frames, we used the contrastive transformations suggested by the SimCLR developers37. This included 
+horizontal flip, random resized crop, jittering, grayscaling, and Gaussian blur. All the images were normalized prior 
+to training. 
+During the training, we set the minimum number of epochs to 200. Further, we implemented early stopping 
+with the patience of 10, i.e., training is terminated when there is no improvement in accuracy for ten consequent 
+epochs. Notably, self-supervised learning benefits from high batch size37. It increases the negative samples in the 
+batch, making it harder for the network to find the augmented pairs. This encourages the network to extract salient 
+features. As a result, we set the mini-batch size to 256 for pattern cutting and 512 for the rest of the tasks. 
+Meta-learner. Before the training, we downsampled each video stream to 1 FPS. This lowers computational 
+cost20 while retaining the salient information, as shown by our recently published DL model12 that achieved state-of-
+the-art performance for the JIGSAWS tasks at 1 FPS. In addition, training and validation sets were normalized 
+separately using min-max normalization. During the training, we set the minimum epochs to 40. Also, we used early 
+stopping with a patience of 10. The mini-batch size was 8. 
+One restriction to Proto-MAML is the need for an equal number of samples per class34. The absence of this rule 
+causes an inflated representation of some classes over the others. This leads to biased, i.e., domain-specific, 
+estimations. However, in our study, each skill class had a different sample size (Fig. 1e). Therefore, we ran each 
+round 100 times with different seeds. We removed the outlier performances based on accuracies using the Tukey 
+Fences58 method. Also, for each repetition, we randomly sampled Ntrain trials from each class. Here, Ntrain is the 
+smallest sample size in a class among all the classes in the training set. For the validation set, this value was Nval. 
+Another limitation is that the model needs the same input size for each mini-batch. However, our data was 
+spatiotemporal with varying lengths, both inter- and intra-tasks. Hence, we incorporated mini-batch zero padding to 
+Proto-MAML. Here, we did not zero-pad the entire input based on the longest sequence, as some tasks were 
+considerably shorter than others. This difference would lead to an abundance of zeros in those datasets, increasing 
+computational cost. 
+
+ 
+9 
+ 
+ 
+Evaluation. We used the round-robin scheme to evaluate the A-VBANet. Specifically, one task was used at each 
+round to validate and test the model, while the remaining trained the network. This was repeated until every task in 
+the cohorts other than the test cohort was the target. At each round, we also tested the trained A-VBANet on 
+laparoscopic cholecystectomy. Then, after all the rounds, we averaged the results to obtain the overall adaptation 
+performance. 
+During testing, few-test-shots (k) were used to adapt to the new domain. The rest of the samples were utilized to 
+compute the performance. For multi-class tasks, the accuracies were micro-averaged. 
+In this study, the models were developed on Pytorch, and training was conducted via the IBM Artificial 
+Intelligence Multiprocessing Optimized System (AiMOS) at Rensselaer Polytechnic Institute on 8 NVIDIA Tesla 
+V100 GPUs, each with 32 GB capacity. 
+ 
+Data availability 
+The laparoscopic pattern cutting and suturing datasets were collected by our group under IRB regulations, and the 
+deidentified source frames and class labels will be released upon publication. The JIGSAWS dataset is available at 
+https://cirl.lcsr.jhu.edu/research/hmm/datasets/jigsaws_release/. 
+ 
+Code availability 
+The code for developing the models will be made public upon publication. 
+ 
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+ 
+ 
+ 
+
+ 
+12 
+ 
+ Acknowledgments 
+The authors graciously acknowledge Dr. Yuanyuan Gao for assisting with the pattern cutting video data collection 
+and Dr. Lora Cavuoto for spearheading the laparoscopic suturing experiments and data collection. 
+ 
+Author contributions 
+E.Y. and S.D. conceived the idea. E.Y. designed the analysis, developed the network architecture, trained the pipeline, 
+and drafted the manuscript. S.D. and X.I. were responsible for supervising and revising the manuscript. S.S. and G.Y. 
+collected the laparoscopic cholecystectomy videos and G.Y. provided corresponding OSATS scores. 
+ 
+Competing Interests 
+The authors declare no competing interests. 
+ 
+Additional Information 
+Supplementary information is available. 
+Supplementary figures and tables are available. 
+Correspondence and requests for materials should be addressed to S.D. 
+ 
+ 
+Supplementary Information 
+NetTrustScore (NTS). NTS is a trustworthiness estimation based on the Softmax of predictions38. 
+NTS builds around the following steps: 
+ 
+Question-answer trust, 𝑄𝑧(𝑥, 𝑦). It quantifies the reliability of the predicted label (𝑦) for a 
+given sample (𝑥) via model 𝑀. As seen in Eqn 1, for true predictions( 𝑅𝑦=𝑧, 𝑧 being the actual 
+label / class), the Softmax values, 𝐶(𝑦|𝑥), are aggregated via a reward coefficient (𝛼). For the 
+false predictions (𝑅𝑦≠𝑧), the Softmax values were subtracted from 1 with a penalty coefficient 
+(𝛽). In this study, both 𝛼 and 𝛽 are 1. 
+ 
+𝑄𝑧(𝑥,𝑦) = {
+𝐶(𝑦|𝑥) 𝛼
+(1 − 𝐶(𝑦|𝑥)) 𝛽
+𝑖𝑓 𝑥 𝜖 𝑅𝑦=𝑧|𝑀
+𝑖𝑓 𝑥 𝜖 𝑅𝑦≠𝑧|𝑀 
+(1) 
+ 
+ 
+For conditional trustworthiness59, i.e., reliability of a condition such as true predictions or 
+false predictions, we use Eqn. 2 differently than in the original paper (Eqn. 1), as the false 
+predictions are handled separately from the true ones without the need to penalize them. In Eqn. 
+2, 𝑅𝑐 is the condition space. 
+ 
+𝑄𝑐(𝑥,𝑦) = 𝐶(𝑦|𝑥)𝛼 𝑖𝑓 𝑥 𝜖 𝑅𝑐|𝑀 
+(2) 
+ 
+ 
+Trust Density, 𝐹(𝑄𝑐). It is the trust behavior of the model for all the samples (𝑥𝑠) in a 
+given condition. It is obtained using non-parametric density estimation through Gaussian kernel38. 
+Here, the bandwidth of the kernel is 
+𝛾
+√𝑁 with 𝛾 = 0.5 and 𝑁 = 𝑙𝑒𝑛𝑔𝑡ℎ(𝑥). 
+ 
+Trust Spectrum, 𝑇𝑀(𝑐). It is the trust behavior of the network for all the conditions in a 
+dataset, as given in Eqn. 3. In the equation, 𝑇𝑀(𝑐), outputs a list of overall trustworthiness for each 
+condition. 
+
+ 
+13 
+ 
+𝑇𝑀(𝑐) = 1
+𝑁 ∫𝑄𝑐(𝑥)𝑑𝑥 
+(3) 
+ 
+NetTrustScore, NTS. Based on the original proposal, NTS is the overall trustworthiness 
+score of the network via all the predictions and classes and scales from 0 to 1. However, in this 
+study, when we report NTS, it is not global but for a condition instead as governed by Eqns. 2 and 
+3. 
+ 
+Proto-MAML implementation. Proto-MAML35 is a meta-learner that combines Model-agnostic 
+meta-learning (MAML)36 and Prototypical Networks (ProtoNet)34. 
+ 
+MAML. In MAML, the model (𝑓) learns the best parameter space (𝜃) to provide fast and 
+flexible adaptability. In detail, first, 𝜃 is randomly initialized, and the input is passed forward (𝑓𝜃) 
+for task 𝑇𝑖. Then based on the computed loss (𝐿𝑇𝑖) in the embedded space, backpropagation is 
+applied to update weights (𝛻𝜃) as shown in Eqn. 4. 
+ 
+𝜃𝑖
+′ = 𝜃 −  𝜶𝛻𝜃𝐿𝑇𝑖 (𝑓𝜃) 
+(4) 
+ 
+Ideally, 𝑓(𝜃𝑖
+′) represents the 𝑇𝑖 robustly after several updates, i.e., 𝑁𝑤: the number of 
+updates, same as conventional training. However, in meta-learning, the objective is not to find the 
+optimal parameters for a task but to find the parameters that ensure adaptation. Therefore, we apply 
+Eqn. 4 to each task in the task distribution, 𝑃(𝑇),  and obtain respective parameter spaces (𝜃′), 
+which are then passed forward, 𝑓𝜃′, to compute the new loss as seen in Eqn. 5. This way, we obtain 
+the optimal parameter space that minimizes the joint cost function. This step is called the inner 
+loop. Thus, in Eqn. 4, 𝜶 is the inner learning rate. 
+ 
+𝜃 = 𝑎𝑟𝑔𝑚𝑖𝑛𝜃 ∑
+𝐿𝑇𝑖(
+𝑇𝑖~𝑃(𝑇)
+𝑓𝜃𝑖
+′) 
+(5) 
+ 
+ 
+Next, we update 𝜃 based on the optimal parameters from the inner loop, as illustrated in 
+Eqn. 6. This step is called the outer loop, and 𝜷 is the outer learning rate. 
+ 
+𝜃 = 𝜃 −  𝜷𝛻𝜃 ∑
+𝐿𝑇𝑖 (
+𝑇𝑖~𝑃(𝑇)
+𝑓𝜃𝑖
+′) 
+(6) 
+ 
+ 
+As seen in Eqn. 6, a gradient's gradient is computed, i.e., Hessian-vector product36, which 
+is computationally expensive. Thus, the authors of the MAML article 36 proposed first-order 
+MAML (fo-MAML), which only uses the first-order gradients. We also followed this paradigm, 
+hence updated Eqn. 6 as follows: 
+𝜃 = 𝜃 −  𝜷 ∑
+𝛻𝜃𝑖
+′𝐿𝑇𝑖 (
+𝑇𝑖~𝑃(𝑇)
+𝑓𝜃𝑖
+′) 
+(7) 
+ 
+ 
+
+ 
+14 
+ 
+ProtoNet. The way the ProtoNet works is detailed as follows. First, the training set is split 
+into 
+support 
+set, 
+𝑆 = [(𝑥1,𝑦1),… , (𝑥𝑠,𝑦𝑠),… ,(𝑥𝑁,𝑦𝑁)] 
+and 
+query 
+set, 
+𝑄 =
+[(𝑥1,𝑦1), …, (𝑥𝑞,𝑦𝑞),…, (𝑥𝑁,𝑦𝑁)] with 𝑁 samples. Here, 𝑥𝑠,𝑥𝑞  ∈  ℝ𝐷 are inputs whereas 𝑦𝑠 and 
+𝑦𝑞 are the corresponding labels in the support and query sets. Then, the model, 𝑓𝜃, embeds the 
+inputs into 𝑀-dimensional feature set, 𝑓𝜃(. ): ℝ𝐷 →  ℝ𝑀. Next, using the embedded support set 
+samples, the prototypical center (𝑣𝑐) is computed as given in Eqn. 8. In the equation, 𝑆𝑐 is all the 
+(𝑥𝑠,𝑦𝑠) pairs in the support set with 𝑦 = 𝑐. Here 𝑐 ∈ 𝑪 | 𝑪: all the classes represented in 𝑆. 
+ 
+𝑣𝑐 =  1
+|𝑆𝑐|
+∑
+𝑓𝜃(𝑥𝑠,𝑖)
+(𝑥𝑠,𝑖,𝑦𝑠,𝑖)∈𝑆𝑐
+ 
+(8) 
+ 
+The query set is then used to compute the loss function (𝐿) based on the distance between 
+the query samples, 𝑥𝑞, and 𝑣𝑐 via the distance function,  𝑑𝜑: the Euclidean Distance. 
+ 
+ProtoMAML. It follows the MAML, specifically fo-MAML, methodology to adapt35, while 
+for the final layer, i.e., the layer that outputs for a specific task, the weights (𝑊𝑐) and bias (𝑏𝑐) are 
+initialized based on the 𝑣𝑐 as computed in Eqn. 8, instead of random initialization as used by the 
+vanilla fo-MAML. Particularly, the initialization occurs as follows: 𝑊𝑐 = 2𝑣𝑐 and 𝑏𝑐 = −||𝑣𝑐||2. 
+For more information please refer to the original paper35. 
+ 
+Hyperparameter selection. The SimCLR network uses the pretrained ResNet3453 - on 
+ImageNet60 – as its backbone. Moreover, the pipeline aims to minimize the loss function, InfoNCE 
+(NT-Xent)37, via the Adam optimizer with a learning rate of 0.0005. Further, the non-linearity is 
+added via ReLU. 
+ 
+The ProtoMAML minimizes Cosine Similarity Loss61 based on Euclidean distance34. The 
+inner and outer loop optimizers are Stochastic Gradient Descent (SGD) and Adam, respectively, 
+while the learning rates are 0.1 and 0.01. Moreover, we used a learning rate scheduler in which the 
+rate was factored by 0.6 for every 10 epochs without improving the validation accuracy. Further, 
+𝑁𝑤, i.e., number of inner loop weight updates, is 1 in training and 20 in testing. Finally, for self-
+supervised feature (SSF) sets from 2 to 32, the 𝑣𝑐 is a 512-dimensional feature vector; for the SSF 
+set of 64, this value is 1,024. 
+In addition, the meta-learner utilizes an in-house Residual Neural Network (RNN) as the 
+backbone. The filter size is 5 for the convolutional layers of the first residual block. This value is 
+3 for the second. On the other hand, the dilation rates are 1 and 2, and the stride is 1. Finally, the 
+non-linearity is added using ReLU. 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+
+ 
+15 
+ 
+Supplementary Tables 
+Extended Table 1 | Accuracies for task adaptation. k: number of test shots. Bold values are 
+reported in the manuscript. For needle passing, k = 16 was not investigated as it leaves no 
+Intermediate class for the query set. 
+Validation 
+and 
+Testing 
+Dataset 
+SSF 
+set 
+No. of test-shots 
+k = 1 
+k = 2 
+k = 4 
+k = 8 
+k = 16 
+Pattern 
+Cutting 
+2 
+0.858±.045 
+0.871±.044 
+0.881±.039 
+0.888±.039 
+0.892±.037 
+4 
+0.889±.037 
+0.903±.033 
+0.910±.032 
+0.916±.030 
+0.918±.029 
+8 
+0.900±.023 
+0.910±.022 
+0.920±.018 
+0.925±.019 
+0.929±.017 
+16 
+0.899±.035 
+0.910±.034 
+0.914±.033 
+0.915±.033 
+0.914±.035 
+32 
+0.868±.042 
+0.888±.043 
+0.896±.039 
+0.898±.038 
+0.900±.038 
+64 
+0.821±.066 
+0.839±.062 
+0.847±.055 
+0.853±.048 
+0.853±.049 
+Suturing 
+(Lap.) 
+2 
+0.987±.012 
+0.989±.011 
+0.989±.011 
+0.990±.011 
+0.997±.009 
+4 
+0.991±.013 
+0.988±.019 
+0.993±.009 
+0.994±.008 
+0.999±.005 
+8 
+0.995±.008 
+0.995±.008 
+0.995±.006 
+0.997±.006 
+0.998±.008 
+16 
+0.990±.013 
+0.992±.011 
+0.993±.010 
+0.992±.011 
+0.999±.006 
+32 
+0.992±.011 
+0.992±.012 
+0.992±.011 
+0.992±.011 
+0.999±.005 
+64 
+0.966±.036 
+0.969±.033 
+0.971±.032 
+0.974±.029 
+0.964±.042 
+Suturing 
+(Robotic) 
+2 
+0.618±.014 
+0.649±.018 
+0.649±.029 
+0.631±.036 
+0.708±.058 
+4 
+0.625±.014 
+0.657±.021 
+0.669±.029 
+0.686±.041 
+0.692±.063 
+8 
+0.650±.015 
+0.659±.018 
+0.661±.018 
+0.675±.024 
+0.662±.035 
+16 
+0.650±.023 
+0.664±.027 
+0.697±.039 
+0.716±.035 
+0.761±.044 
+32 
+0.644±.022 
+0.653±.021 
+0.663±.026 
+0.686±.038 
+0.691±.050 
+64 
+0.651±.040 
+0.679±.050 
+0.692±.052 
+0.688±.068 
+0.740±.091 
+Needle 
+Passing 
+2 
+0.607±.021 
+0.626±.025 
+0.666±.033 
+0.694±.042 
+N/A 
+4 
+0.616±.018 
+0.645±.022 
+0.690±.033 
+0.727±.038 
+8 
+0.621±.026 
+0.637±.029 
+0.645±.042 
+0.697±.046 
+16 
+0.626±.027 
+0.640±.034 
+0.666±.040 
+0.692±.042 
+32 
+0.614±.027 
+0.632±.046 
+0.639±.045 
+0.683±.054 
+64 
+0.581±.019 
+0.587±.026 
+0.611±.038 
+0.618±.049 
+Knot 
+Tying 
+2 
+0.676±.023 
+0.691±.023 
+0.698±.032 
+0.707±.044 
+0.707±.042 
+4 
+0.688±.022 
+0.694±.020 
+0.698±.024 
+0.742±.036 
+0.766±.062 
+8 
+0.670±.015 
+0.686±.021 
+0.713±.025 
+0.730±.034 
+0.786±.057 
+16 
+0.688±.028 
+0.697±.031 
+0.714±.042 
+0.749±.060 
+0.831±.064 
+32 
+0.673±.026 
+0.694±.029 
+0.708±.042 
+0.763±.057 
+0.835±.077 
+64 
+0.653±.033 
+0.673±.043 
+0.692±.050 
+0.715±.060 
+0.733±.078 
+ 
+ 
+
+ 
+16 
+ 
+Extended Table 2 | NTS for true predictions in task adaptation. k: number of test shots. Bold 
+values are reported in the manuscript, selected based on best accuracies in Extended Table 1. For 
+needle passing, k = 16 was not investigated as it leaves no Intermediate class for the query set. 
+Val. and 
+Testing 
+Dataset 
+SSF 
+set 
+Classes 
+No. of test-shots 
+k = 1 
+k = 2 
+k = 4 
+k = 8 
+k = 16 
+Pattern 
+Cutting 
+2 
+Fail 
+0.984±.011 0.984±.012 0.985±.012 0.986±.012 0.986±.013 
+Pass 
+0.988±.009 0.989±.009 0.991±.008 0.991±.008 0.991±.008 
+4 
+Fail 
+0.986±.007 0.988±.007 0.988±.007 0.989±.007 0.989±.007 
+Pass 
+0.989±.005 0.991±.005 0.992±.005 0.992±.004 0.992±.004 
+8 
+Fail 
+0.989±.007 0.990±.007 0.990±.007 0.991±.006 0.991±.006 
+Pass 
+0.991±.005 0.993±.004 0.994±.003 0.994±.003 0.994±.003 
+16 
+Fail 
+0.998±.003 0.999±.002 0.999±.003 0.999±.002 0.999±.003 
+Pass 
+0.999±.002 0.999±.001 0.999±.001 0.999±.001 0.999±.002 
+32 
+Fail 
+0.997±.004 0.998±.004 0.998±.004 0.998±.005 0.998±.005 
+Pass 
+0.998±.004 0.998±.003 0.999±.003 0.999±.003 0.999±.003 
+64 
+Fail 
+1.0 
+1.0 
+1.0 
+1.0 
+1.0 
+ 
+Pass 
+1.0 
+1.0 
+1.0 
+1.0 
+1.0 
+Suturing 
+(Lap.) 
+2 
+Novice 
+0.968±.027 0.971±.028 0.971±.029 0.973±.027 0.979±.029 
+Expert 
+0.967±.043 0.968±.046 0.966±.049 0.966±.049 0.972±.044 
+4 
+Novice 
+0.981±.017 0.983±.015 0.984±.014 0.986±.014 0.997±.007 
+Expert 
+0.989±.020 0.988±.024 0.988±.024 0.990±.022 0.990±.025 
+8 
+Novice 
+0.991±.009 0.992±.010 0.993±.009 0.993±.008 0.987±.018 
+Expert 
+0.998±.005 0.997±.007 0.996±.010 0.996±.009 0.998±.008 
+16 
+Novice 
+0.997±.005 0.997±.005 0.997±.006 0.997±.006 
+1.0 
+Expert 
+0.999±.004 0.999±.003 0.999±.002 0.999±.002 0.999±.003 
+32 
+Novice 
+0.998±.003 0.999±.003 0.998±.003 0.998±.004 0.999±.006 
+Expert 
+1.0 
+1.0 
+1.0 
+1.0 
+1.0 
+64 
+Novice 
+1.0 
+1.0 
+1.0 
+1.0 
+1.0 
+ 
+Expert 
+1.0 
+1.0 
+1.0 
+1.0 
+1.0 
+Suturing 
+(Robotic) 
+2 
+Novice 
+0.884±.041 0.861±.046 0.888±.054 0.897±.058 0.910±.064 
+Interm. 
+0.818±.057 0.774±.066 0.724±.075 0.759±.096 0.562±.056 
+Expert 
+0.782±.063 0.693±.073 0.661±.071 0.721±.086 0.585±.070 
+4 
+Novice 
+0.878±.042 0.868±.049 0.886±.054 0.891±.059 0.894±.065 
+Interm. 
+0.820±.058 0.753±.072 0.691±.084 0.637±.098 0.623±.110 
+Expert 
+0.794±.059 0.759±.067 0.692±.073 0.666±.085 0.592±.098 
+8 
+Novice 
+0.897±.041 0.918±.043 0.929±.042 0.927±.043 0.938±.051 
+Interm. 
+0.757±.062 0.725±.076 0.681±.078 0.582±.060 0.598±.100 
+Expert 
+0.806±.058 0.768±.065 0.681±.073 0.607±.059 0.577±.058 
+16 
+Novice 
+0.984±.022 0.986±.023 0.989±.028 0.989±.028 0.990±.030 
+Interm. 
+0.959±.045 0.954±.057 0.901±.110 0.881±.140 0.816±.180 
+Expert 
+0.959±.039 0.947±.061 0.907±.088 0.895±.110 0.855±.140 
+
+ 
+17 
+ 
+32 
+Novice 
+0.977±.030 0.979±.029 0.984±.025 0.984±.027 0.985±.031 
+Interm. 
+0.950±.045 0.933±.067 0.919±.087 0.850±.130 0.862±.170 
+Expert 
+0.947±.048 0.925±.067 0.908±.092 0.817±.140 0.761±.210 
+64 
+Novice 
+0.998±.003 0.999±.003 0.999±.003 0.999±.002 0.999±.006 
+Interm. 
+0.994±.008 0.992±.017 0.989±.023 0.992±.020 0.963±.089 
+Expert 
+0.994±.008 0.992±.013 0.992±.017 0.989±.024 0.965±.083 
+Needle 
+Passing 
+2 
+Novice 
+0.907±.041 0.881±.061 0.869±.080 0.871±.110 
+N/A 
+Interm. 
+0.927±.038 0.889±.054 0.894±.068 0.871±.110 
+Expert 
+0.878±.050 0.804±.071 0.760±.095 0.698±.110 
+4 
+Novice 
+0.925±.037 0.910±.048 0.894±.078 0.903±.095 
+Interm. 
+0.935±.040 0.912±.054 0.893±.064 0.903±.095 
+Expert 
+0.885±.050 0.857±.066 0.777±.095 0.764±.120 
+8 
+Novice 
+0.939±.035 0.920±.044 0.905±.057 0.894±.083 
+Interm. 
+0.953±.028 0.909±.048 0.906±.065 0.946±.068 
+Expert 
+0.896±.049 0.847±.067 0.811±.089 0.796±.120 
+16 
+Novice 
+0.981±.020 0.981±.023 0.979±.026 0.984±.027 
+Interm. 
+0.978±.022 0.974±.031 0.969±.035 0.984±.031 
+Expert 
+0.965±.026 0.955±.042 0.947±.056 0.946±.070 
+32 
+Novice 
+0.981±.018 0.970±.034 0.972±.034 0.977±.042 
+Interm. 
+0.985±.017 0.984±.021 0.983±.028 0.969±.051 
+Expert 
+0.970±.028 0.954±.044 0.946±.056 0.936±.075 
+64 
+Novice 
+0.996±.067 0.998±.005 0.997±.009 0.997±.010 
+Interm. 
+0.997±.006 0.996±.006 0.993±.021 0.993±.025 
+Expert 
+0.994±.009 0.995±.008 0.993±.015 0.993±.015 
+Knot 
+Tying 
+2 
+Novice 
+0.875±.053 0.871±.055 0.859±.068 0.894±.070 0.883±.080 
+Interm. 
+0.817±.064 0.789±.075 0.728±.091 0.720±.100 0.691±.120 
+Expert 
+0.776±.072 0.743±.085 0.699±.096 0.639±.097 0.629±.100 
+4 
+Novice 
+0.921±.039 0.924±.040 0.907±.041 0.935±.047 0.934±.059 
+Interm. 
+0.868±.052 0.844±.061 0.814±.079 0.687±.120 0.666±.140 
+Expert 
+0.817±.065 0.806±.063 0.796±.071 0.682±.120 0.692±.120 
+8 
+Novice 
+0.926±.038 0.923±.045 0.923±.043 0.928±.058 0.949±.044 
+Interm. 
+0.875±.064 0.831±.080 0.791±.095 0.740±.130 0.736±.110 
+Expert 
+0.808±.073 0.787±.079 0.774±.082 0.749±.086 0.658±.130 
+16 
+Novice 
+0.970±.029 0.965±.034 0.966±.036 0.970±.041 0.977±.039 
+Interm. 
+0.960±.049 0.951±.057 0.943±.075 0.863±.140 0.984±.130 
+Expert 
+0.919±.068 0.928±.067 0.923±.075 0.892±.120 0.866±.160 
+32 
+Novice 
+0.980±.020 0.982±.022 0.975±.030 0.982±.030 0.985±.034 
+Interm. 
+0.962±.034 0.955±.044 0.956±.049 0.891±.120 0.933±.110 
+Expert 
+0.930±.056 0.932±.058 0.937±.060 0.901±.120 0.894±.140 
+64 
+Novice 
+0.995±.008 0.997±.006 0.997±.007 0.996±.013 0.997±.010 
+Interm. 
+0.992±.011 0.991±.016 0.991±.020 0.976±.062 0.981±.064 
+
+ 
+18 
+ 
+Expert 
+0.989±.014 0.990±.016 0.989±.020 0.974±.059 0.972±.080 
+ 
+Extended Table 3 | Accuracies and AUC in cholecystectomy. k: number of test shots. Bold values 
+are reported in the manuscript. 
+Validation Dataset 
+SSF 
+set 
+No. of test-shots 
+k = 1 
+ 
+ 
+Accuracy 
+AUC 
+Pattern Cutting 
+2 
+0.692 
+0.798 
+4 
+0.718 
+0.803 
+8 
+0.795 
+0.848 
+16 
+0.821 
+0.803 
+32 
+0.795 
+0.788 
+64 
+0.872 
+0.818 
+Suturing (Lap.) 
+2 
+0.667 
+0.747 
+4 
+0.718 
+0.841 
+8 
+0.821 
+0.864 
+16 
+0.872 
+0.848 
+32 
+0.846 
+0.848 
+64 
+0.821 
+0.818 
+Suturing (Robotic) 
+2 
+0.718 
+0.788 
+4 
+0.718 
+0.788 
+8 
+0.769 
+0.848 
+16 
+0.615 
+0.652 
+32 
+0.795 
+0.833 
+64 
+0.821 
+0.833 
+Needle Passing 
+2 
+0.872 
+0.838 
+4 
+0.846 
+0.859 
+8 
+0.846 
+0.876 
+16 
+0.692 
+0.677 
+32 
+0.692 
+0.758 
+64 
+0.872 
+0.818 
+Knot Tying 
+2 
+0.667 
+0.755 
+4 
+0.564 
+0.621 
+8 
+0.795 
+0.838 
+16 
+0.872 
+0.864 
+32 
+0.615 
+0.715 
+64 
+0.897 
+0.864 
+ 
+ 
+ 
+
+ 
+19 
+ 
+Extended Table 4 | NTS for true predictions in cholecystectomy. k: number of test shots. Bold 
+values are used to obtain average NTSs, as reported in the manuscript. They are selected based on 
+the best accuracies in Extended Table 3. 
+Validation 
+Dataset 
+SSF 
+set 
+Classes 
+No. of test-shots 
+k = 1 
+Pattern 
+Cutting 
+2 
+Low Performance 
+1.0 
+High Performance 
+1.0 
+4 
+Low Performance 
+1.0 
+High Performance 
+1.0 
+8 
+Low Performance 
+1.0 
+High Performance 
+1.0 
+16 
+Low Performance 
+1.0 
+High Performance 
+1.0 
+32 
+Low Performance 
+1.0 
+High Performance 
+1.0 
+64 
+Low Performance 
+1.0 
+High Performance 
+1.0 
+Suturing 
+(Lap.) 
+2 
+Low Performance 
+1.0 
+High Performance 
+1.0 
+4 
+Low Performance 
+1.0 
+High Performance 
+1.0 
+8 
+Low Performance 
+1.0 
+High Performance 
+1.0 
+16 
+Low Performance 
+1.0 
+High Performance 
+1.0 
+32 
+Low Performance 
+1.0 
+High Performance 
+1.0 
+64 
+Low Performance 
+1.0 
+High Performance 
+1.0 
+Suturing 
+(Robotic) 
+2 
+Low Performance 
+1.0 
+High Performance 
+1.0 
+4 
+Low Performance 
+1.0 
+High Performance 
+1.0 
+8 
+Low Performance 
+1.0 
+High Performance 
+1.0 
+16 
+Low Performance 
+1.0 
+High Performance 
+1.0 
+32 
+Low Performance 
+1.0 
+High Performance 
+1.0 
+64 
+Low Performance 
+1.0 
+High Performance 
+1.0 
+
+ 
+20 
+ 
+Needle 
+Passing 
+2 
+Low Performance 
+1.0 
+High Performance 
+1.0 
+4 
+Low Performance 
+1.0 
+High Performance 
+1.0 
+8 
+Low Performance 
+1.0 
+High Performance 
+1.0 
+16 
+Low Performance 
+1.0 
+High Performance 
+1.0 
+32 
+Low Performance 
+1.0 
+High Performance 
+1.0 
+64 
+Low Performance 
+1.0 
+High Performance 
+N/A 
+Knot 
+Tying 
+2 
+Low Performance 
+1.0 
+High Performance 
+1.0 
+4 
+Low Performance 
+1.0 
+High Performance 
+1.0 
+8 
+Low Performance 
+1.0 
+High Performance 
+1.0 
+16 
+Low Performance 
+1.0 
+High Performance 
+1.0 
+32 
+Low Performance 
+1.0 
+High Performance 
+1.0 
+64 
+Low Performance 
+1.0 
+High Performance 
+1.0 
+Mean 
+ 
+Low Performance 
+1.0 
+High Performance 
+1.0 
+ 
+Extended Table 5 | OSATS scores breakdown. 
+OSATS score 
+Number of trials 
+Assigned label 
+13 
+1 
+Low performance 
+15 
+1 
+16 
+2 
+18 
+1 
+19 
+1 
+20 
+1 
+21 
+2 
+22 
+1 
+23 
+2 
+24 
+2 
+High performance 
+25 
+1 
+ 
+
+ 
+21 
+ 
+Supplementary Figures 
+Extended Fig. 1 | SimCLR architecture and spatiotemporal feature set generation. 𝐷 represents 
+the output dimension of the SimCLR once trained while 𝐷′ is the dimension after the 1D GAP 
+layer. 𝑇 is the temporal length of a given sample. Pattern cutting frames were used to represent the 
+pipeline. 
+Extended Fig. 2 | The backbone of the pipeline. 𝐷 and 𝐾 represent the dimension of the 
+convolutional layers. In this study, 𝐷 is equal to the output dimension of the SimCLR–. 𝐾 is 16 
+for 𝐷 = 2,4,8 and 𝐾 is 64 for 𝐷 = 16,32, and 𝐾 is 256 for 𝐷 = 64. 
+
+ResNet Block1
+ResNet Block2
+D
+SimCLR
+→
+kernel size = 5,
+kernel size = 1,
+kernel size = 3,
+dilation = 1
+dilation=1
+dilation=2
+: 1D Conv. layer
+: 1D GAPlayer
+:Attention layer
+: Fully -connected layerTransform.2
+IdenticalResNet34s
+Removedaftertraining
+Training
+Maximize
+aggrement
+Transform.1
+D-dimensional
+representations
+1DGAP
+ResNet34
+Post-training
+TxD'-dimensional
+spatiotemporal
+featureset

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+arXiv:2301.00574v1  [quant-ph]  2 Jan 2023
+Environmental-induced work extraction
+Rasim Volga Ovali†,1, Shakir Ullah†,2, Mehmet Günay†,3, and Mehmet Emre Tasgin2,∗
+† Contributed equally
+∗ correspondence: metasgin@hacettepe.edu.tr and metasgin@gmail.com
+1Department of Physics, Recep Tayyip Erdogan University, 53100, Rize, Turkey
+2Institute of Nuclear Sciences, Hacettepe University, 06800 Ankara, Turkey and
+3Department of Nanoscience and Nanotechnology, Faculty of Arts and Science,
+Mehmet Akif Ersoy University, 15030 Burdur, Turkey
+A measurement can extract work from an entangled, e.g., two-mode system. Here, we inquire
+the extracted work when no intellectual creature, like an ancilla/daemon, is present. When the
+monitoring is carried out by the environmental modes, that is when no measurement-apparatus is
+present, the measurement-basis becomes the coherent states. This implies a Gaussian measurement
+with a fixed strength λ = 1. For two-mode Gaussian states, extracted work is already independent
+from the measurement outcome. After the strength is also fixed, this makes nature assign a particular
+amount of work to a given entanglement degree. Extracted work becomes the entanglement-degree
+times the entire thermal energy at low temperatures —e.g., room temperature for optical modes.
+Environment, nature itself, converts entanglement to an ordered, macroscopic, directional (kinetic)
+energy from a disordered, microscopic, randomized thermal energy. And the converted amount is
+solely determined by the entanglement.
+Quantum entanglement enables technologies which are
+not possible without them [1].
+Measurements below
+the standard quantum limit [2, 3], quantum enhanced
+imaging [4–6], quantum radars [7], quantum teleporta-
+tion (QT) [8], and quantum computation [9] are all en-
+abled by entangled —more generally nonclassical [10]—
+states. Entanglement can also be utilized as a resource
+for quantum heat engines [11–17] which makes them op-
+erate more efficiently compared to their classical coun-
+terparts [18]. As an example, an ancilla can utilize en-
+tanglement for extracting a larger amount of work by
+maximizing the efficiency [19]. It is quite intriguing that
+entanglement can even be directly transformed into work
+via measurements [20, 21]. The energy can be extracted
+from a single heat bath [20, 21] [22]. This phenomenon
+—we focus here— takes place as follows, e.g., in a two-
+mode entangled state.
+Work extraction as a measurement backaction.— Let
+us assume that one of the modes (mode a) belongs to
+an optical cavity which includes, for example, a free-to-
+move board or a piston inside the cavity [20, 21]. The
+second (b) mode relies somewhere outside of the cavity
+and it is entangled with the a-mode. Both modes are
+in thermal equilibrium with the environment at temper-
+ature T. When a measurement is carried out on the b-
+mode (outside), entropy of the cavity (a) mode decreases
+to S(meas)
+V
+because of the measurement backaction. After
+the measurement, the state (a-mode) thermalizes back
+to equilibrium and assigns a higher entropy S(ther)
+V
+[23].
+During the rethermalization with the heat bath the ex-
+pansion of the a-mode pushes the board located inside
+the cavity [20]. An amount of W = kBT (S(ther)
+V
+−S(meas)
+V
+)
+work can be extracted from the single heat bath. That
+is, W amount of thermal energy can be converted into
+“directional” kinetic energy (KE) of the board. In case
+of maximum entanglement, the state of the a-mode is
+completely determined from the outcome of the b-mode
+and entropy of the a-mode vanishes, i.e., S(meas)
+V
+= 0.
+Thus, the extractable work becomes kBT S(ther)
+V
+, i.e., the
+complete internal energy. The amount of extracted work
+can be employed for witnessing/quantifying the entan-
+glement [24–27].
+In general, the extracted work depends on the nature
+of the measurement and its outcome. Yet, interestingly,
+the state of the a-mode (cavity field) turns out to be in-
+dependent of the outcome of the b-mode as long as Gaus-
+sian states and measurements are concerned [28–31]. The
+state, into which the a-mode collapses, depends only on
+the strength (λ) of the Gaussian measurement/operation
+carried out on the b-mode [32].
+Thus, also the ex-
+tracted work depends only on λ.
+Here, λ ∈ [0, ∞] is
+the quadrature-noise belonging to the Gaussian opera-
+tion [33].
+That is, if λ (for a reason) assigns a fixed
+value, a given degree of entanglement extracts a particu-
+lar amount of work. This is the phenomenon we explore
+here.
+The observable that is measured in an experiment (this
+can be, for example, number of photons) is determined
+by the quantum apparatus employed in the measurement
+of the b-mode.
+More explicitly, monitoring of the en-
+vironment on the apparatus (i.e., decoherence) destroys
+the superpositions among the natural pointer states (the
+measurement-basis). This makes us observe one of the
+values in the measurement-basis. Refs. [34–39] provide
+mathematical and numerical demonstrations of the mon-
+itoring process.
+The
+question.—
+Here
+we
+examine
+the
+following
+already-answered-question in the context of work extrac-
+tion process. What happens if no measurement appa-
+ratus is present?
+In other words, what is the pointer
+(measurement) basis if no intellectual being, such as a
+human, a daemon, or an ancilla, is present? In this case,
+environmental monitoring sets the measurement-basis as
+
+2
+the the coherent states [40–47]. Measurement strength is
+λ = 1 for any one of the coherent states.
+Environmental monitoring.— At this stage, we better
+re-depict the picture of the system we have in mind in a
+more clear way. The cavity (a) mode is entangled with
+the b-mode. It is worth noting that, in general, the a
+and b modes do not need to be in interaction [48]. The b-
+mode is monitored by the environment. Environment (a
+collection of infinite number of modes) can monitor the
+b-mode only indirectly as two light beams do not interact
+directly. Monitoring can be performed over common in-
+teractions with masses of particles present around which
+induces an effective interaction between the environment
+and the b-mode [49]. It is worth noting that pointer states
+are still coherent states, for instance, in case a harmonic
+chain [42] is considered. Therefore, in total, environment
+monitors (measures) the b-mode in one of the coherent
+states. So, the measurement is a Gaussian one.
+It is straightforward to realize that the measurement
+strength is fixed λ = 1. Moreover, a rotated R(φ) form
+of the coherent state basis —R(φ) is present in the most
+general form of a Gaussian measurement [28–31]— is also
+a coherent state basis. Putting things together, nature
+itself makes a Gaussian measurement on one of the two
+entangled modes. The measurement basis is composed
+of coherent states. As the basis possesses a fixed λ: a
+particular amount of work (KE) becomes assigned to a
+given degree of entanglement as long as Gaussian states
+are concerned.
+The thermodynamical energy, proba-
+bilistic and disordered in nature, is converted into an
+ordered (mechanical) form of energy now belonging to
+the board [20, 21]. This sets an observer-independent,
+nature-assigned, association between entanglement and
+directional/ordered energy.
+We call this phenomenon
+environment-induced work extraction (EIWE).
+One can gain a better understanding by examining the
+phenomenon in low-T limit —such as room temperature
+for optical resonances [50]. At this limit, a simple-looking
+analytical result can be obtained, because the kBT term,
+present in the W formula, cancels with a 1/kBT term
+appearing within the entropy difference [51].
+Please,
+see Eqs. (S11) and (S13) in the Supplementary Mate-
+rial (SM) [52].
+The extracted work W = ξ(r) (¯nℏωa) depends only on
+the degree of the entanglement ξ(r) = [1−2/(1+cosh2r)]
+which runs from 0 to 1 as entanglement increases. ¯n is
+the occupation of the a (cavity) mode of resonance ωa.
+So, (¯nℏωa) is already the “entire” thermodynamical (it is
+probabilistic) energy present inside the cavity either be-
+fore the measurement or after the rethermalization pro-
+cess. Here we use the von Neumann entropy SV in dif-
+ference to Ref. [20] where Réyni entropy is employed. r
+is the two-mode squeezing rate which is proportional to
+the time the entanglement device is kept open [53].
+We present the derivations in the SM [52]. In the rest
+of the paper, we aim to put this result into words in order
+to develop a physical understanding.
+We observe that the extracted work (KE of the board
+or piston) is: the degree of entanglement times “all of
+the thermal energy” present inside the cavity [54].
+It
+gets closer to (¯nℏωa) in the case of maximum (max)
+entanglement [55].
+In other words, max entanglement
+converts the entire thermodynamical energy of the (a)
+photon mode into the kinetic (directional) energy of the
+board/piston [56]. As we will see below, this is true also
+for other max entangled states, e.g., max two-mode en-
+tangled state (|1, 0⟩ + |0, 1⟩)/
+√
+2 and symmetrization en-
+tanglement of identical particles [57]. That is, we cross-
+check our EIWE result with other incidences.
+What is peculiar to EIWE is that the work is extracted
+by itself. That is, an observer-independent entanglement-
+energy correspondence appears. The converted energy is
+proportional to the degree of the entanglement ξ [58] and
+depends only on the excitation spectrum.
+Before making the comparison with other systems, we
+would like to bring two important issues into attention.
+First, we note that W is calculated using a density ma-
+trix which involves classical (thermodynamical) probabil-
+ities —grand canonical ensemble. It is a result weighted
+over classical probabilities. For this reason, we prefer to
+use the words “all of the thermal energy” present inside
+the cavity is converted into the KE of the board/piston.
+The energy present inside the cavity after the rether-
+malization is also probabilistic. Conservation of energy,
+however, tells us the following. If the energy realized in-
+side the cavity after the rethermalization assigns one of
+the classically probable ones, the extracted work needs
+to be equal to that value.
+Second, we note that al-
+most all of the notion (e.g., entanglement-work conver-
+sion and entanglement-energy analogy [59]) and the cal-
+culations carried out here are already discussed in other
+studies [20, 21].
+Here, in difference, we introduce the
+notion of entanglement-work correspondence due to the
+presence of nature-originated measurement.
+Comparison with other work extraction phenomena.—
+We first compare the (i) EIWE result with the one
+for another (max) entangled state (ii) |e⟩ = (|1, 0⟩ +
+|0, 1⟩)/
+√
+2 in thermal equilibrium ˆρ = P(|0, 0⟩⟨0, 0| +
+e−ℏωa/kBT |e⟩⟨e|) [60]. When one measures the number
+of photons and the outcome the b-mode is |1⟩, W =
+xℏωa work is extracted in the cavity a-mode.
+Here,
+x = e−ℏωa/kBT is the classical probability for realizing
+the two-mode system in the excited state |e⟩ and it is
+equal to the occupation ¯n at low T , i.e., ¯n = x. This re-
+sult is the same with the max entanglement (ξ = 1) case
+of EIWE. In this example, too, all thermodynamical en-
+ergy present in the a-mode is converted into directional
+energy (work). In this case, however, the work extrac-
+tion (the same amount) is subject to the measurement of
+the b-mode in the excited state. In EIWE, in difference,
+any measurement outcome extracts this amount of work.
+We also examine the work associated to the (iii) sym-
+metrization (max) entanglement [57] as a third exam-
+ple.
+In a recent study, we investigate the work ex-
+tracted by identical particles in a system of N total
+number of symmetrized bosons. The extracted work by
+
+3
+(N − 1) particles is calculated when one of the (ran-
+dom) particles is measured in the excited state, of en-
+ergy ωeg [61]. In parallel with the previous cases, (i) and
+(ii), the extracted work, by pushing the board located
+within the condensate region, comes out as W3 = xℏωeg
+at low T . Here, x = e−ℏωeg/kBT is the probability for
+one of the N particles to occupy the excited state ei-
+ther before the measurement or after the rethermaliza-
+tion of the condensate.
+Similarly, (xℏωeg) is the en-
+tire thermal energy of the condensate at equilibrium.
+The lowermost excited state of such a condensate is
+the Dicke state |N, 1⟩ [62], where a single-particle ex-
+citation is symmetrically distributed among N bosons,
+|N, 1⟩ = (|e, g, g, ...⟩+|g, e, g, ...⟩...+|g, ...g, e⟩)/
+√
+N. This
+is a maximally entangled state with respect to any one of
+the particles. ωeg is the level-spacing between the excited
+|e⟩ and ground |g⟩ states of a single particle.
+We observe that the amount of extracted work, one
+more time, is equal to the complete thermal energy of the
+condensate (system) at thermal equilibrium. This takes
+place again for a max entangled state, |N, 1⟩. We can take
+the excited state, e.g., as the motional states of a Bose-
+Einstein condensate with ℏωeg = h2/2mL2 [63]. Then,
+the thermal energy of the condensate can be converted
+into the directional (kinetic) energy of a board placed in
+the condensate region. Here, again, the presented value
+of the extracted work is subject to the realization of the
+measured-particle in the excited state. The investigation
+of this symmetrization problem has further importance
+as entanglement of symmetrized many-body states and
+nonclassicality of photonic states are intimately related.
+A Dicke (many-body) state becomes a Fock (photon)
+state when N → ∞ [64–66]. Similarly, separable coher-
+ent atomic states become the photonic coherent states in
+the same limit.
+Summary and Discussions
+Summary.— We reinvestigate an already studied phe-
+nomenon –work extraction from an entangled system via
+measurement backaction [20, 21]– when nature itself per-
+forms the measurements. This is when there is no intel-
+lectual being (such as a human, daemon, or an ancilla) is
+present for the measurement; but the monitoring is con-
+ducted by the environment/nature itself. In this case,
+measurement-basis becomes the coherent states [40–47].
+This fixes the measurement strength to λ = 1. The state
+of the a-mode, in which work-extraction is carried out, is
+already independent from the outcome of the b-mode [28–
+31] as long as Gaussian states and measurements are con-
+cerned [28–31]. (The measurement performed by the en-
+vironment is a Gaussian one as the measurement-basis
+is coherent states.) Therefore, in total, the nature itself
+assigns a particular amount of work/energy to a given
+degree of entanglement.
+Entanglement converts a disordered (randomly moving
+particles, microscopic) form of energy into an ordered
+form where the molecules of the board move along the
+same direction, i.e., macroscopic motion. The letter is
+referred as the mechanical energy, or we can tell that it
+is the KE.
+We find that at low T , the directional energy associated
+with the entanglement is the “total thermal energy” times
+the degree of the entanglement for a two-mode Gaussian
+state: W = ξ(r)Uther. Here, ξ(r) = [1 − 2/(1 + cosh 2r)]
+increases with the entanglement and gets closer to ξ =
+1 around the max entanglement.
+r is the squeezing
+strength. That is, all thermal energy can be converted
+into directional energy for a maximally entangled Gaus-
+sian state. Similarly, the entire thermal energy can be
+converted into directional energy also for (ii) max entan-
+gled number state (|0, 1⟩ + |1, 0⟩)/
+√
+2 and for (iii) sym-
+metrization entanglement of identical bosons [57]. How-
+ever, the conversion in (ii) and (iii) is subject to the re-
+alization of the measured mode/particle in the excited
+state; while in (i) EIWE any measurement outcome ex-
+tract that amount of work.
+Squeezing and potential energy.— In this section, we
+would like to introduce a correspondence also between
+potential mechanical energy and single-mode nonclassi-
+cality, SMNc, (e.g., squeezing). Entanglement and SMNc
+are two different types of nonclassicalities (quantumness).
+The two not only can be converted into each other via
+passive optical elements, such as a beam splitter (BS),
+but they also satisfy a conservation-like relations [67–
+69]. When a cavity mode is squeezed, it cannot be con-
+verted into work directly.
+This is because, squeezing
+keeps the entropy unchanged.
+However, the situation
+changes when the squeezed cavity field leaks out through
+the mirror(s). The interaction between the cavity and the
+output modes �
+k(gkˆb†
+kˆa + H.c.) is in the form of a BS
+interaction. Thus, the cavity and the output modes get
+entangled [70]. Environmental monitoring on the output
+modes ˆbk [71] makes the cavity extract work. We can also
+view the process as the potential mechanical energy (as-
+sociated with squeezing) transforms into the kinetic me-
+chanical energy (associated with entanglement).
+Connection with a recent study.— As a final but impor-
+tant point, we indicate that the present research actually
+investigates the results of a recent study [72]. Rigorous
+calculations, employing the standard methods, clearly
+show an intriguing phenomenon in an optical cavity. On-
+set of entanglement exhibits itself in the response func-
+tions of the optical cavity. At this point, nonanalytic-
+ity of the response function moves into the upper-half of
+the complex frequency plane (UH-CFP). One needs to
+avoid this incident from implying the violation of causal-
+ity.
+Fortunately, surveys [73–75] show that a nonana-
+lyticity in the UH-CFP does not imply the violation of
+causality if there exists a small curvature in the back-
+ground. In the present study, the total curvature of the
+cavity (a-mode) increases as the disordered thermal en-
+ergy is converted into ordered directional kinetic energy
+of the board. That is, it indeed increases with the amount
+of entanglement.
+
+4
+Acknowledgements
+We gratefully thank Vural Gökmen for the motiva-
+tional support, Bayram Tekin for the scientific support,
+Wojciech H. Zurek for letting us know his influential
+work [40] and Alessio Serafini for his support on the
+continuous-variable quantum information. We acknowl-
+edge the fund TUBITAK-1001 Grant No. 121F141.
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+[50] Or for a Swinger pair creation (mass-energy conversion)
+process taking place over a critical electric field.
+[51] We can tell that the result is T -independent except a
+factor ¯n = 1/(eℏωa/kBT − 1) → e−ℏωa/kBT which stands
+for the probability of finding one of the two modes in the
+excited state. (¯nℏωa) is already the energy present in the
+a-mode.
+[52] See Supplemental Material.
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+ergy” at equilibrium, because this is the grand canonical
+mean energy which is (classical) probabilistic. The en-
+ergy present inside the cavity after the rethermalization
+is also probabilistic. Conservation of energy tells us the
+following. If the energy realized inside the cavity after
+the rethermalization assigns one of the classical probable
+ones, the extracted work needs to be equal to that value.
+[55] Please note that energy of a maximum entangled two-
+mode Gaussian state approaches to infinite. Here, we
+confine ourselves to a regime where squeezing rate r ≪
+ℏωa/kBT . The latter is typically ∼ 100 for optical modes
+at room temperature. See the discussion above Eq. (S11)
+in the SM [52].
+[56] Please note that this statement is valid at any tempera-
+ture.
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+which actually have the same energy. We do this for the
+sake of a reasonable comparison only.
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+a single particle in a condensate of N identical particles
+is not a straightforward process. It necessitates certain
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+monotone
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+not
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+ture 452, 72 (2008).
+
+7
+I.
+SUPPLEMENTARY MATERIAL
+FOR
+ENVIRONMENTAL-INDUCED WORK
+EXTRACTION (EIWE)
+In this supplementary material (SM), we first ob-
+tain the environmentally extracted in a two-mode (TM)
+squeezed thermal (Gaussian) state in Sec. 1. We show
+that it is in the form W
+= ξ(r) × (¯nℏω) at low-
+temperatures (T ) —e.g., room temperature for optical
+modes.
+Here, ξ(r), given in Eq. (??), quantifies the
+strength of the entanglement. We use von Neumann en-
+tropy (SV ) in our calculations, in difference to Ref. [20].
+Second, in Sec. 2, we show that the same form for the
+work extraction, i.e., W = ξ(r) × (¯nℏω), appears also for
+other TM Gaussian states.
+1. EIWE for two-mode squeezed thermal state
+Initially, before the measurement, both modes, a and b,
+are in thermal equilibrium with an environment at tem-
+perature T . When one carries out a Gaussian measure-
+ment on the b-mode, the enropy of the b-mode reduces
+below the one for the thermal equilibrium. When the a-
+mode re-thermalizes with the environment it performs a
+work in the amount of [27]
+W = kBT (S(ther)
+V
+− S(meas)
+V
+).
+(1)
+Here, S(meas)
+V
+is the reduced entropy of the a-mode after
+the measurement in the b-mode is carried out. S(ther)
+V
+is
+the entropy of the a-mode after the rethermalization of
+the mode.
+Entropy of a Gaussian state can be determined by its
+covariance matrix, which includes the noise elements of
+the modes. Covariance matrix of a biparitite Gaussian
+state can be cast in the form [? ]
+σab =
+�σa cab
+cT
+ab σb
+�
+(2)
+via local symplectic transformation Sp(2, R) ⊕ Sp(2, R)-
+i.e., transformations altering neither of the entropy or
+entanglement. Here, σa = diag(a, a) and σb = diag(b, b)
+are the reduced covariance matrices of the a and b
+modes, respectively. cab = diag(c1, c2) refers to corre-
+lations/entanglement between the two mode. For a sym-
+metrically squeezed two-mode thermal state, i.e., both
+modes used to be in thermal equilibrium with the same
+T in the squeezing process, b = a and c2 = −c1 = −c.
+The state into which the a-mode collapses is indepen-
+dent from the outcome of the b-mode measurement as
+long as a Gaussian measurement is carried out [28–31].
+The covariance matrix of the a-mode after the measure-
+ment becomes
+σπb
+a = σa − cab (σb + γπb)−1 cT
+ab.
+(3)
+Here, γπb = R(φ) diag(λ/2, λ−1/2) R(φ)T refers to the
+covariance matrix associated with a Gaussian operation
+(measurment) [28–31]. λ is the measurement strength.
+For a Gaussian measurement having the coherent states
+as a basis, λ = 1 and γπb = diag(1/2, 1/2) independent
+of the rotations R(φ) in the a-mode, i.e., ˆa → ˆaeiφ.
+The entropy of a Gaussian state is determined solely
+by purity, µ =
+1
+2n√
+Detσ, which takes the form
+SV = 1 − µ
+2µ
+ln
+�1 + µ
+1 − µ
+�
+− ln
+� 2µ
+1 + µ
+�
+(4)
+for a single-mode state [? ].
+The purity of the a-mode, after the b-mode measure-
+ment, can be obtained as
+µ1 ≡ µ(meas) =
+a + 1/2
+2(a2 − c2 + a/2).
+(5)
+For a TM squeezed thermal state,
+a = (¯n + 1
+2) cosh(2r),
+(6)
+c = (¯n + 1
+2) sinh(2r),
+(7)
+the purity becomes
+µ1 =
+a + 1/2
+2(¯n + 1/2)2 + a,
+(8)
+where ¯n = (eℏωa/kBT − 1)−1 is the occupation of the a-
+mode, which becomes ¯n → e−ℏωa/kBT at low T— e.g.,
+the room temperature for optical modes of resonance ωa.
+r is the two-mode squeezing strength with which entan-
+glement increases [53].
+¯n is extremely small at low T regime. So, it is
+µ1 ∼= 1 −
+2¯n
+a + 1/2.
+(9)
+Then, the entropy can be approximately written as,
+S(meas)
+V
+∼=
+¯n
+a + 1/2[ln(2) − ln(2¯n) + ln(a + 1
+2)]
+(10)
+−
+2¯n
+a + 1/2,
+where a = (¯n + 1/2) cosh(2r). The last term originates
+from the ln
+�
+2µ
+1+µ
+�
+term given in Eq. (S4).
+In the square brackets, in Eq. (S10), ln(¯n) = ℏωa
+kBT ≫ 1
+and ln(a + 1/2) ∼= ln(cosh(2r)). Assuming that the squ-
+uezing rate is much smaller than
+ℏωa
+kBT , which is about
+∼100 at the room temperatures, i.e., r ≪
+ℏωa
+kBT , the en-
+tropy takes the form
+S(meas)
+V
+∼=
+2¯n
+1 + cosh(2r)
+ℏωa
+kBT ,
+(11)
+where the last term in Eq. (S10) is also neglected.
+
+8
+Some time after the measurement, a-mode rethermal-
+izes with the enviroment and at equilibrium its purity
+becomes
+µ2 ≡ µ(ther) =
+1
+1 + 2¯n.
+(12)
+The entropy at equilibrium can similarly calculated as
+S(ther)
+V
+∼= ¯n ℏωa
+kBT .
+(13)
+Therefore, the extracted work becomes
+W =
+�
+1 −
+2
+1 + cosh(2r)
+�
+¯nℏωa,
+(14)
+where the kBT term in Eq. (S1) is canceled by 1/kBT
+appearing in (S11) and (S13).
+2. EIWE for other Gaussian states
+We derived the simple form for the extracted work
+W = ξ(r)(¯nℏωa) for the TM squeezed thermal states.
+Now, we aim to show that a similar form appears also for
+other Gassian states given by the covariance matrix (S2).
+When the b-mode is measured by the enviroment, the
+a-mode collapses to a covariance matrix with determi-
+nant
+detσπb
+a = (a2 − c2
+1 + a/2)(a2 − c2
+2 + a/2)
+(a + 1/2)2
+.
+(15)
+The entropy after the measurement is similarly µ(meas) =
+1
+2√
+detσ
+πb
+a
+[76]. For c1 = −c2 = c, µ(meas) becomes the
+purity given in Eq. (S5).
+As we aim to show that the extracted work has a “form”
+similar to W = ξ · (¯nℏωa) also for other Gaussian states,
+we express Eq. (S15) in terms of Sp(4, R) invariants
+detσ = (a2 − c2
+1)(a2 − c2
+2),
+(16)
+∆ = 2(a2 + c1c2),
+(17)
+where detσ is the determinant of the two-mode state
+before the measurement.
+We do this because any one
+of the two-mode Gaussian states, Eq. (S2), can be ob-
+tained from Sp(4, R) transformations of the TM squeezed
+thermal state Eq. (15) of Ref. [76]. The determinant in
+Eq. (S15) can be expressed as
+detσπb
+a
+= (a2 − c2
+1)(a2 − c2
+2) + a
+2 (2a2 − c2
+1 − c2
+2)
+( 1
+2 + a)2
+,
+(18)
+where the first term is the Sp(4, R) invariant detσ. In
+the second term, I = 2a2 − c2
+1 − c2
+2 can be expressed in
+terms of Sp(4, R) invariants (Please note that a is only
+local Sp(2, R) invariant) as
+detσπb = (a2I + ∆2/4 − ∆a2)
+(a + 1
+2)2
+.
+(19)
+We note that detσ = (˜a2
+TMS − ˜c2)2 and ∆ = 2(˜a2 − ˜c2)
+for the TM squeezed thermal states and they are Sp(4, R)
+invariant. Thus, Eq. (S19) can be recast as
+(˜a2 − ˜c2)2 = a2I + (˜a2 − ˜c2)2 − ∆a2.
+(20)
+Please note that, in this section, we use tilde symbol for
+the covariance matrix elements of the TM squeezed ther-
+mal states in order to distinguish them from the variable
+a given in the general matrix given in Eq. (S2).
+Cancellation in Eq. (S20) results
+I = 2a2 − c2
+1 − c2 = ∆.
+(21)
+Using this in Eq. (S19), we obtain the expression
+µ(meas) =
+a + 1/2
+2(z + a/2),
+(22)
+where z = ˜a2−˜c2 = (¯n+1/2)2. Please note that Eq. (S22)
+is in the same form with Eq. (S5) from which we obtain
+the extracted work
+W = ξ (��nℏωc).
+(23)
+Thus, above we showed that Eq. (S23) is a general form
+for the extracted work from a symmetric TM Gaussian
+state.
+

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