Agents communicate in pairwise interactions with randomly assigned communication partners. Quality is measured in two ways: the number of interactions to complete the task and the number of states per agent. We present protocols for the majority problem that allow for a trade-off between these two measures. Compared to the only other trade-off result , we improve the number of interactions by almost a linear factor. Furthermore, our protocols can be made uniform (working correctly without any information on the population size n), yielding the first uniform majority protocols that stabilize in a subquadratic number of interactions.Population protocols are a model for distributed computing that is focused on simplicity and robustness. A system of  Majority is a fundamental problem in distributed computing. There are n different agents, each with one of two opinions, say A and B and the goal is to agree on the opinion with the larger support. This problem occurs when all elements of a distributed system have to reach consensus on the value of some parameter which reflects the prevailing opinion what this value should be. Because of its importance, the majority problem is frequently used as a case study in analysis and comparison of strengths and limitations of various models of distributed computing.In this article we consider the interaction. States are mapped to outputs by a problem-specific output function. In the case of the majority problem, one can think of an agent’s output as being A or B, indicating which opinion the agent believes to be the majority.The population model was introduced by Angluin et al. , 5 as a parallel time, which divides the number of interactions by n to account for the inherent parallelism of the system. In order to avoid confusion, we stick to the actual number of interactions throughout the article.The quality of a population protocol is measured in terms of the number of interactions (the runtime) and the number of states per agent required to “successfully compute” the desired output. The number of interactions is sometimes expressed in stable—i.e., no possible future transition can change the agents’ output. However, runtime notions differ in when this strict guarantee must be achieved. A natural definition is to measure the number t of interactions after which the system is in such a stable state with correct output. This notion is used in most recent publications, especially for lower bounds  always happens by the time when the computation stabilizes (on the correct output). The stabilization may, however, be reached later, sometimes much later, than convergence.One may wonder what the advantage in measuring the convergence time instead of the stabilization time may be. In  the authuniformity, in the sense that a single algorithm should be designed to work for populations of any size. Due to the simplicity of transition-based algorithms and the uniformity, uniform population protocols are well suited to model real-world systems that consist of many but comparatively simple agents, like a flock of birds or large sensor networks aggregating information . In both scenarios the agents’ computational power is bounded and the algorithms should not depend on the number of agents.A desirable feature of population protocols is The underlying theme of this article is to exhibit trade-offs in population protocols between the running time and the required number of states, highlighting methods which help achieving fast stability (in addition to convergence) and uniformity of protocols.absolute biasA and B, respectively. In the following we state the results for the tightest case when Our protocols for the majority problem in the population model provide an integer parameter converges with high probability in T interactions and uses stabilizes with high probability in T interactions but uses We present two hybrid exact majority protocols, both having a runtime of  Theorem . The oth Theorem .s, we provide a uniform version of the second of the above two majority protocols. This protocol has essentially the same guarantee for the stabilization time. However, with high probability it uses For a constant All protocols above except for the uniform one need knowledge of finite-state machines), for this protocol we adopt a generalized model [Our first result is a comparatively simple protocol that, with high probability Theorem . While ted model  in whichs, our majority results underline an important difference between stabilization and convergence. While the We highlight a few implications of the above results. For a constant  Theorem  is asympentclass1pt{minima Theorem  shows thWhen choosing ented in  are the When choosing 2 states .s, our uniform protocol that stabilizes in For a constant parameter phase clock from [junta of factor of additive term of the same order. See Sect. An import ingredient for our results is an improvement to the ock from —a distriThe original population model was introduced by Angluin et al. , 5, assuabsolute biasAngluin et al.  present s et al.  show thaVojnovic , who promentclass2pt{minimTo achieve fast runtime, Mocquard et al.  considerrotocols , 20 on trotocols  on the oAlistarh et al.  show a lh et al.  further A recent series of papers , 13, 11 The subquadratic-time protocols for majority presented in , 11, 21 In parallel to our work, Kosowski and Uznanski  recentlyWith the only exception of , all majn agents, in the following also referred to as nodes. Nodes are assumed to be identical finite-state machines.u, v) is chosen independently and uniformly at random. Node u is called the initiator and node v is called the responder. Let u and v at the beginning of such an interaction. Both nodes observe each other’s state and update themselves according to a fixed, deterministic transition function of the form c that maps each node to its current state. This function c is called the configuration of the system at that time.Population protocols are a computational model for a distributed system consisting of target configurations, whose definition depends on the considered problem. It is not required, indeed not possible in this model, that nodes realize when a target configuration has been reached. Target configurations are specified via an output function of the form s to a (problem specific) output value o.Nodes try to reach and stay in a set of opinions). We seek a configuration in which all nodes agree on the opinion with the initially larger support. The absolute biass to an output We are interested in population protocols for the majority problem, where nodes start in one of two different states . We say a node in state x has loadx. The transition function is discrepancy after t interactions, which is the difference between the maximum and minimum load among all nodes, and set load balancing timementclasspt{minima For anylevel from a range of factor of The next protocol rapidly elects a non-empty junta of size at most col from , where ee (as in ) but alse , where the levelactivity bit For the level calculation, the state of a node is a tuple of the form  if it is the initiator and state  if it is the responder. Since the interacting nodes are chosen randomly, this simulates a fair coin toss to decide whether the node should remain active and move up to level 1, or whether it should become inactive.To describe the transition function, we distinguish between a node’s first interaction and any of its subsequent interactions. During its first interaction, a node u changes its state only if it is still active  of size at most Fix any constant n  gives\dn be sufficiently large with respect to a and Fix any two constants u that just reached level l. Node u is still active and will either become inactive or proceed to level t be u’s next interaction. u to proceed to level t is at most l. By a straightforward coupling argumentThe probability for Equation ) to get u is among the last l to level If Equation ) to get Fix an n be sufficiently large. For all Fix any integer constant a levels and get Note that Equipped with the auxiliary claims from Sect.  es Lemma  as well ax Lemma . Finallymentclass2pt{minimn be sufficiently large with respect to a. With probability at least Fix any constant n/3 nodes in state  after The probability that a given node does not interact in a given interaction is mentclass2pt{minimn/3 nodes in state ). Thus, the probability that any fixed node remains active during the next From that point on, the probability for any fixed node to become inactive during a given interaction is at least n be sufficiently large with respect to a. With probability at least Fix any constant l and that for Let For the upper bound on mentclass2pt{minimFor the lower bound on mentclass2pt{minimThe lemma’s statement follows via a union bound. n be sufficiently large with respect to a. Then we have the bound Fix any constant Define We first prove the bound on the runtime. Lemma mentclass2pt{minimFormJunta isThe number of states per node required for hours consisting of minutes. These clocks do not run at a consistent speed and are not fully synchronized . However, the clocks can be set up such that, with high probability, each of the first lasts simultaneously spend all nodes We adapt the phase clock implementation from [phase counterp can be expressed as hn, depending on the protocol using the phase clock). We also allow Distributed protocols often benefit from some form of synchronization. Phase clocks  are one ion from  to our nWe continue with a formal description of the phase clock implementation in Sect. entclass1pt{minimap, b). The phase countermarker bitWe assume We assume that the number of marked nodes lies in the interval Section The state of a node is a tuple of the form  and a responder in state u to update its state according to the following transition function (from [Consider an interaction between an initiator on (from ):13The rSynchronization Properties. Remember that the m denotes the number of minutes in an hour. We define the houru with phase counter p(t) after t interactions as rounds, which represents a period of interactions during which all nodes have the same hour. Let start of round i) denote the interaction during which the last node reaches hour i. Similarly, let end of round i) denote the interaction during which the first node reaches hour t for which all nodes u have, simultaneously, the same hour length of round i as stretch of round i as i denotes the time it takes for the first node to reach hour i. In particular, we always have We now define the notion of m, we can ensure that both the round length and stretch of the first A key property of the above phase clock construction is captured by the following lemma. It states that, by carefully choosing the phase clock parameter ation of  is givenn to be sufficiently large with respect to them. There is a constant parameter Let m depends on the involved constants a, c,  and a of the desired error probability.Note that in the above lemma, the constant parameter We briefly explain how the simplifying assumptions we made for the protocol description can be avoided.Computing a junta on the fly Our protocol description in Sect. FormJunta from Sect. mentclass2pt{miniml, a, b, f, p). The (sub-) tuple  is used as the state for the junta protocol and consists of the level p, b) is used for the phase clock protocol and consists of the phase counter b is used by both protocols. All nodes start in state  (with only the activity bit set) and execute protocol FormJunta on the (sub-) tuple . Once the finished bit f of a node is set it starts to execute the phase clock protocol from Sect.  p, b). We make two simple observations: FormJunta becomes marked . By Theorem all nodes start to execute mentclass2pt{minimmentclass2pt{minimFormJunta). However, by Theorem When mentclass2pt{minimIt follows that Lemma The state of a node is a tuple (We can change Unbounded state space The population protocol as described in Sect. p is unbounded. We can avoid this by performing any arithmetic on the phase counter modulo h cannot tell whether it has hour h or hour 0).i and another node is in hour Note that Lemma p done modulo c is the constant from Lemma Assume (see Eq.  but withCKh (Eq. ) during (A)haseC locku according to Equation (P Equation .u’s marker bit b is set (meaning u is a junta node).u’s finished bit f is set.u decreased at least once in absolute value (due to the modulo u reached a new hour the last time it updated the phase counter.u increased by at least 2 (this happens if the clocks are not properly synchronized).The following function calls to (B)m was chosen suitably and n is sufficiently large):u is currently in a different hour as v.u is currently in the same hour as v.u is currently in a smaller hour than v.u is currently in a larger hour than v.The following function calls to (C)h  or might run too fast  .Before the first node reaches hour We now bound the number of states each node requires. By Theorem s and h . As many majority protocols, it is based on a variant of the cancellation and doubling approach . These copies are distributed among the nodes via a load balancing mechanism. This approach allows us to reduce the number of required phases. We will prove the following theorem:Our protocol uses n nodes with initial absolute bias Let h.According to Theorem We now describe the protocol’s state space and its transition function . Afterward, we give the proof of Theorem State space The state of a node u consists of the states required for the FormJunta, cf. Section load valueu’s current opinion (sign) and its “magnitude” . It is initialized with either u’s initial opinion. The output function maps the state of a node to the sign of its load value. Thus, the majority guess of a node u is equal to s unless all nodes have already the same positive or negative sign. This allows us to cap the absolute load values at 3s  and v (responder). The nodes’ actions can be divided into three parts: synchronization, load explosion, and load balancing. During the synchronization, the u and responder v to update the states of u’s phase clock. During the load explosion, u uses the s. During the load balancing, the nodes use the phase clock’s h (Guarantee (C) in Sect.  u and v balance their loads only if The following observation follows from the fact that all phase clock function calls work correctly with probability 1 until the first node reaches hour u and v balance their loads in h, both u and v are guaranteed to be in the same hour.Whenever two nodes This observation will be of importance for our stable majority protocol presented in Sect. Total & scaled total load Let u after t interactions and u experienced after t interactions. Define the total loadscaled total loadt interactions asThe following lemma will be useful to show that, if u and v balance their loads in an interaction u and v we have t.Let t. For the second part, assume all nodes’ load values have the same sign after t interactions. Note that no load balancing action can change this, afterward. Moreover, the total scaled load t. The invariant for We are now ready to prove this section’s main result.i  implies that no node skips any hour i. But then, by definition of a round’s length, we have For entclass1pt{minimaentclass1pt{minimai experienced exactly i load explosions. This implies that, whenever two nodes balance their loads during the first We now condition on the high probability event that the above properties hold and consider the first Note that if s unless all nodes’ loads have the same sign. Indeed, if not all loads have the same sign at the end of a round, the discrepancy bound (Property (2)) implies that all loads have absolute load at most 2. After the load explosion in the following round the load will be at most 2s. This allows us to cap the absolute load values at 3s as described at the beginning of this section and, with high probability, the protocol outcome will not change.To bound the number of states, observe that - conditioned on the high probability event that the above properties hold -no absolute load value exceeds 2In this section, we present and analyze the protocol n nodes with initial absolute bias Let We now describe the protocol’s state space and its transition function . Afterward, we give the proof of Theorem u executes a slow but exact protocol BackupMajorityh rounds of our fast but possibly incorrect u’s phase counter is zero and u thinks that protocol Each node u stops finished biterror bitNode u reached hour h . This marks the end of the first h rounds. Stopping at this point ensures that any load balancing operation happens between two nodes in the same hour (Observation u has absolute load at least 3s after its firsti. Then it had absolute load at least 3 at the end of round The (first) finished bit is set for one of two reasons (Lines 6 to 7): (i) i experienced exactly i load explosions. This might cause The (first) error bit is also set for one of two reasons (Lines 8 to 9): (i) Two finished nodes whose loads have different signs interact with each other, in which case Since the backup protocol is exact, our protocol is exact if the error bit is set. A major part of the analysis is to show that it is also exact if none of the error bits is set. Moreover, we have to show that, with high probability, no error bit is set and the protocol stabilizes fast.BackupMajority and the 4 combinations of the bits Let us first bound the number of states per node. By Theorem (i)The phase clock does not start: That is, in FormJunta all nodes set their activity bit to 0 before reaching level (ii)The phase clock starts and some node sets its error bit: The error bit is spread via a one-way epidemic (Line 3). Thus, with probability 1 eventually all nodes set their error bit. From then on, they use the output of the backup protocol, yielding again (iii)The phase clock starts and no node ever sets its error bit: If the phase clock runs, FormJunta marks at least one node and, eventually, all nodes u set their finished bit i load explosions. Moreover, after interaction T the loads no longer change and all nodes have the same load signs . Thus, by Lemma T, such that Let It remains to prove the theorem’s runtime bounds. We first show that Next, we prove that i. Define first finish or error bit is set because of three possible reasons: (i)s after its first interaction in an hour (finished bit),a node had load at least 3(ii)a node’s phase counter overflowed (finished bit), or(iii)a node skipped an hour (error bit).In a similar way to the proof of Theorem To this end, let h (its phase counter overflows), so For all For all entclass1pt{minimaAs in the proof of Theorem Note that the finished bit is set when a node reaches hour entclass1pt{minimai load explosions. Thus, Lemma In the remainder we condition on the high probability event that the above properties hold. Since no node ever skips an hour, whenever two nodes in hour s after its first interaction in hour Thus, we have forever have the correct load sign. This is still the case after interaction In summary, with high probability, after mentclass2pt{minimCombining everything above via a union bound, this yields that, with high probability, n, and for each configuration Finally, we show that the stabilization time BackupMajority stabilizes within n) interactions in expectation [Protocol ectation .The one-way epidemic completes within K interactions is For each Let The phase clocks of all nodes have reached their limit ofviewing the clocks  as if there were kept running until reaching the limit: Configuration BackupMajority.In the former case (when there is a node with an error), one instance of one-way epidemic raises the error flag in all nodes in expected BackupMajority protocol are sufficient to stabilize Similarly, in the latter case, two consecutive one-way epidemics (the first one to make two nodes with different load signs meet and the second one to spread out the information about the error) and then the completion of the In both cases, the protocol There is at least one marked node (by protocolFormJunta), but there is still at least one node whose clock has not yet reached the limit of: The marked node with the largest clock value increases its clock within one instance of one-way epidemics. Thus within at most No node is marked (by protocolFormJunta): We consider two sub-cases: All nodes inFormJunta are inactive: No node is ever marked and the phase clock never starts. Thus, BackupMajority).There is at least one active node inFormJunta: If an active node is the initiator of an interaction, then it either increases its junta level or becomes inactive. Thus, when this node initiated In summary, we see that n) interactions in expectation. We use the following facts about the expected running time of some basic protocols: In this section, we present and analyze the protocol n nodes with initial absolute bias Let We now describe the protocol’s state space and its transition function . Afterward, we give the proof of Theorem BackupMajority .u executes protocol Nodes first execute a backup protocol u to a majority guess as follows: Use the output of the backup protocol if the phase counter of the phase clock is zero or if the counter The output function maps the state of a node BackupMajority, which is exact.The phase clock never starts (no node is selected into the underlying junta). Since in this case all counters remain zero forever, the output of the protocol u to increase its u initiates the interaction with a marked node as responder). This happens 600 times in a row with probability at least u reach BackupMajority, which is exact.The phase clock starts (meaning the junta is not empty). The probability for a node This shows that protocol BackupMajority requires only four states and the counters are bounded by the constant 600, the number of states per node is a constant factor times the number of states required by We first show that our protocol (i)Once the junta is established, the output of protocol (ii)BackupMajority, which stabilizes in BackupMajority.After By choosing parameters such that BackupMajority, that subprotocol also has the correct output. Together, this implies the desired bound on the convergence time.It remains to prove the convergence time bound for the mentclass2pt{minimTo see Observation (i), note that, once there is a non-empty junta of size at most mentclass2pt{minimn, and for each configuration All nodes have switched to the backup protocol: Then BackupMajority).Some node has not yet switched to the backup protocol: The time until all nodes switch to the backup protocol is dominated by the sum of n geometrically distributed random variables with parameter BackupMajority after at most In summary, wee see that n) interactions in expectation. It remains to prove the bound on the expected convergence time n. Protocols where nodes are restricted to a constant number of states are always uniform. But most of the newer protocols allow for a super-constant number of states and use some upper bounds on n, so they are not uniform. In particular, protocols that stop their computation once a counter reaches a value of Uniformity in population protocols means that a single algorithm is designed to work for populations of any size. In particular, nodes have no information on the population size Uniform population model For studying uniform population protocols whose state requirements increase with the population size n, the original model—which considers nodes as finite-state machines (FSM), see Sect. f(n) for a non-trivial function f, then the nodes and, thus, the protocol inherently depend on n and cannot be simply “deployed” in a population of different size.origin symbol. Tape 1 (read-only) is called the input tape and tape 2 (read-write) the working tape. One two-way infinite working tape is sufficient for us since it allows maintaining two unbounded variables, as required in our protocol. For protocols with more unbounded variables, similarly to [n).Doty and Eftekhari  introduclarly to  one can s is the maximum number of tape cells written by any node during the execution.At the beginning of any interaction, a node’s working tape is identical to its working tape at the end of the previous interaction. Whenever two nodes interact, they copy each other’s working tape onto their own input tape and restart their TM by entering a start TM-state . We define the number of states used during a protocol execution as n with high probability.Having the above formal model in mind, our description sticks with the standard population protocol terminology. In particular, we assume a suitable encoding of the nodes’ states using the alphabet Uniform majority One of the rare examples of a uniform protocol whose state requirements increase with n is the junta protocol from [FormJuntaUniform. Observe that our protocol FormJunta is not uniform, as nodes need to know FormJuntaUniform.col from ; we refeFormJunta, none of them is uniform. In fact, to the best of our knowledge, until now there was no exact, uniform majority protocol that would stabilize with high probability in Since our majority protocols from the previous sections use the non-uniform junta n nodes with initial absolute bias Let s must be constant .Note that, in order for It uses subprotocol FormJuntaUniform instead of FormJunta.The phase clock uses Using the original junta instead of ours has the drawback that nodes must remember their level from the junta calculation indefinitely. However, since protocol FormJunta is inherently non-uniform, this seems unavoidable when aiming for a uniform protocol.Protocol FormJuntaUniform, using slightly different wording and notation than in [For the sake of completeness, we give a brief description of  than in , in ordeDescription ofFormJunta, protocol FormJuntaUniform is based on the level calculation described in Sect.  l, an activity bit a, and the transition function described by Equation . If encountered by another node in the level calculation, a defeated node is treated as if it were in state , independent of its actual level counter l.mentclass2pt{minimPhase Clocks on different Levels & Reset. Compared to FormJunta, any (inactive) node starts with the belief of being in the junta until it becomes defeated by a node from a higher level. This ensures that the junta is never empty. However, when used in the phase clock protocol, there will be a large number of nodes in the junta for the first few interactions (until lower-level nodes become defeated), causing the phase clock to run too fast.l encounters a node running a phase clock on a higher level reset also triggers a reset of the protocol using the phase clock. In our case this is the majority protocol u this entails a reset of the bits u. This idea of phase clocks running on different levels and a corresponding reset was first proposed and used in [To avoid problems in the protocol relying on the synchronization of the phase clock, nodes now also use the level (from the junta protocol) in the phase clock protocol. This basically results in multiple phase clocks running on different levels. When a node running a phase clock on level  used in  for the n, meaning that it is uniform. The remaining proof is similar to that of Theorem FormJuntaUniform that the junta is never empty simplify the argumentation considerably. Also note that the exact phase counters guarantee that any load balancing action is always guaranteed to be done only between nodes in the same hour.First note that the protocol requires no knowledge of FormJuntaUniform. Case (ii) (phase clock starts and some error bit is set) is identical, since with probability 1 eventually all nodes set their error bit and use the output of the backup protocol. For Case (iii) (phase clock starts and no error bits are ever set) we again first show that all nodes finish with probability 1. However, in Theorem t when all nodes are in hour at least u with absolute load at least 3s. This yields the desired contradiction, since u would have set its finished bit at the beginning of its current hour. Once we know that all node finish with probability 1 in this case, the exactness follows again as in Case (iii) in the exactness proof of Theorem Let To prove that, with high probability, we have With these properties, the remaining argumentation from Theorem FormJuntaUniform in expected The proof for the bound on the expected stabilization time is also identical, by noting that all nodes complete protocol ractions .FormJunta by FormJuntaUniform increases the required number of states by a factor of additive term), since we now need to store the level indefinitely. Now, above we saw that, with high probability, all nodes finish after at most For the bound on the number of states, note that replacing the phase clock’s junta algorithm doubling and cancellation approach. They feature a parameter s that allows for a trade-off between runtime and memory per node.We analyzed three similar variants of a population protocol for the majority problem: lower bounds in which one can similarly see the effect on the running time of increasing or decreasing the number of states per node.A natural open question is to improve the bounds we provide. In particular, for uarantee . While iAnother open research question for population protocols deals with the phase clock introduced in . It is uOur results formally show that lower bounds for the stabilization time can be bypassed by considering the convergence time. Unfortunately, there are currently no strong lower bounds regarding the convergence time. As convergence time might be considered the more practical runtime notion, finding such lower bounds and tightening the corresponding upper bounds should be deemed a worthy but challenging task.


Richter’s hernia, also called a partial enterocele, involves a protrusion of peritoneum with subsequent strangulation or incarceration of only part of the lumen of the anti-mesenteric portion of the small bowel through a fascial defect. We report a rare presentation of incarcerated incisional Richter’s hernia of the cecum in a 39-year-old female. The patient presented with acute abdominal pain that gradually improved. Physical examination revealed right lower quadrant tenderness and nodularity just above an abdominoplasty scar. Subsequent computed tomography scan demonstrated a 1 cm by 1 cm hypovascular pocket arising from the cecum with protrusion into the anterior abdominal wall. The hernia was successfully repaired surgically with resolution of symptoms. It is essential for clinicians to be mindful of the diagnosis of Richter’s hernia on the differential for abdominal pain as the risk of detrimental outcomes increases with delayed surgical intervention. Richter’s hernia involves a protrusion of peritoneum with subsequent strangulation or incarceration of only part of the lumen of the anti-mesenteric portion of the small bowel through a fascial defect. As a result, the lumen is incompletely contained in the defect. Timely repair of a Richter’s hernia is instrumental as mortality increases with delays in surgical intervention due to the high risk of strangulation and necrosis. The insidious nature and late presentation of a Richter’s hernia, as well as its relative rarity, may lead to late diagnosis or misdiagnosis ,2.A 39-year-old female patient initially presented to the emergency department after a fall at home. Computed tomography (CT) imaging of the abdomen and pelvis at the emergency department did not reveal any major pathology. She visited the clinic a month later for persistent tenderness at the right lower quadrant of the abdomen. She denied any exacerbating or relieving factors. The patient additionally denied any associated symptoms, including fever, chest pain, shortness of breath, palpitations, dysuria, or known exposure to coronavirus disease 2019 (COVID-19). The patient has a past surgical history of an appendectomy, three cesarean sections with bilateral salpingo-oophorectomy, and abdominoplasty. Her family history was only remarkable for unspecified heart disease.At the clinic, physical examination revealed right lower quadrant tenderness and nodularity 2 cm above her abdominoplasty scar that was potentially overlapping the site of a previous appendectomy incision. The nodularity could not be easily reduced into the abdominal cavity. On closer review of the previous CT scan from her emergency room visit, there was a newly appreciated area of hypervascularity near the palpable nodularity Figure .Given the likelihood of a hematoma or a fascial defect at the site of the incision along with the concern that this suspected hernia was not reducible, urgent imaging was warranted to rule out incarceration or strangulation. Subsequent preoperative CT imaging revealed a 1 cm by 1 cm hypovascular pocket arising from the cecum and protruding into the anterior abdominal wall without any inflammatory changes Figure . Such fiShortly after, the patient appropriately underwent an exploratory laparoscopy. The abdomen was entered in the left upper quadrant and left lower quadrant with three optical trocars Figure . It was The patient did well postoperatively and was discharged home from the recovery unit. The pathology laboratory reported the patient’s specimen as demonstrating benign disease with nonspecific chronic inflammation. A repeat CT of the abdomen and pelvis demonstrated full resolution of the incisional Richter’s hernia with lack of pathology at the surgical sites Figure .The initial differential for abdominal pain is very broad, and thus a careful attention to patient’s history, particularly those of symptoms as well as prior operations, physical examination, laboratory studies, and imaging is required. This tailors the differential based on the likelihood of diagnosis in that patient .It is important to note that the patient’s physical findings of tenderness and nodularity at the right lower quadrant was near the same site as a prior appendectomy, even though her cutaneous scar was excised during the subsequent abdominoplasty. This context is highly suggestive of herniation given the inherent weakness to the fascia from a past incision. A diagnosis of hematoma would have been sufficient if there was no history of surgery at the site.Additionally, the inability to reduce the mass was highly suggestive of incarceration or strangulation. At this stage, CT imaging was able to demonstrate the likely diagnosis of a hernia given the protrusion of the cecum through the anterior abdominal wall. It is for this reason that imaging can be essential to confirming a diagnosis that requires prompt management. Given the clinical urgency of incarceration or strangulation of abdominal hernias, CT imaging is considered superior for both its accuracy in diagnosing the condition as well as its fast turnaround . In contIn a Richter’s hernia, the portion of the bowel that is herniated may have subsequent compromise of the vasculature, leading to infarction and potential peritonitis secondary to perforation . This tyThe symptoms of a Richter’s hernia include nausea, vomiting, and crampy abdominal pain. Symptoms are often subclinical with consequent late presentation. CT imaging is typically diagnostic, as in this case. It is instrumental to repair Richter’s hernias as they are at higher risk of progression to gangrene in comparison to other hernia types, consequently with increased risk of mortality. Repair typically involves a preperitoneal approach with laparotomy and potential bowel resection, in addition to repair of the fascial defect ,8. The dOur patient is a female who presents with a history of multiple abdominal or pelvic surgeries, increasing the index of suspicion for a hernia as the source of her prolonged abdominal discomfort on presentation to the clinic. The insidious nature and late presentation of a Richter’s hernia may lead to late diagnosis or misdiagnosis ,2. EvaluThe operative management of a Richter’s hernia depends on bowel viability. With viable bowel, no bowel resection would be needed, but ischemic bowel may require a resection in addition to repair of the defect in the fascia . If a RiThe patient did not report any concerns regarding this case at the time of writing this report. Despite its rarity, a Richter’s hernia is a more dangerous type of hernia with high risk of strangulation and necrosis. The authors urge clinicians to be mindful of this rare type of hernia for its high risk of mortality in patients.


In primary rhegmatogenous retinal detachment (RRD), the foveal attachment is an important prognostic factors for post-operative vision. When the fovea is obscured by the RRD, its attachment status is considered uncertain. Using a model of the reduced emmetropic and − 10 dioptre myopic eye and the physical properties of the detached retina, we aimed to mathematically ascertain if it is clinically possible for the fovea to be attached while it is obscured by the primary RRD. With the patient upright, a primary RRD due to a 12 o’clock break directly above the fovea was considered. Mathematically, once the trough of the RRD touches the visual axis the edge of the RRD nearest to fovea is The online version contains supplementary material available at 10.1186/s40942-022-00359-3. A 58 years old man was referred from to the vitreoretinal service from a peripheral hospital with an acute primary rhegmatogenous retinal detachment (RRD) in the left eye. The gentleman could only see hand movement in the left eye and the referring clinician concluded that because the retina was ‘overhanging’ and obscuring the fovea as shown in Fig. RRD is a common and potentially blinding condition with an incidence of approximately 1 in 10,000 of the population/year . RRD is Usually, clinical distinction between ‘macula-on’ RRD and ‘macula-off’ RRD is straightforward. However, sometimes the fovea is obscured by the detached retina ‘overhanging’ the fovea creating uncertainty regarding the status of the foveal attachment.Using physical principles, we investigate whether it is possible to have an attached fovea when it is obscured by a detached retina in RRD.We consider a clinical scenario there is a retinal break directly above the fovea causing an RRD as shown in the Fig. A catenary is described by the hyperbolic cosine function:Furthermore, we know that the superior ora serrata is approximately opic eye . We alsoThe patient is examined uprightThe vitreous is fully synchytic and syneretic and has the same viscosity and density as waterThe RRD has a fully stretched smooth surface freely hanging under the effect of gravity.When considering an idealised model of the eye, we make the following assumptions:In vivo, this model will be modified by corrugation of the detached retina reducing its arclength Fig. b. The exThe vertical distance between the visual axis and the RRD trough below the visual axis, when the edge of the RRD is at the fovea, is eye Fig. a and \doThe shortest distance between the fovea and the edge of the RRD when visual axis makes a tangent with lowest point of the RRD is eye Fig. b and \doeye Fig. d.In emmetropic eye if the first order superior arcade vessel is seen on the RRD, by definition, the fovea is off Fig. c and d. Using the assumptions above, we can calculate the following:The calculation details have been described in Additional file Usually, RRD makes a shallow angle  , and the08g/cm3) , we beliMoreover, the detached retina rarely makes a large angle with the RPE. In this regard, the assumptions of the model are rather generous. In vivo, even if the gravitationally lowest point of the RRD appears precisely on the visual axis Fig. b and d, In summary, if the fovea is obscured by an RRD in both emmetropic and myopic eyes, it is likely that the fovea is detached and it is a ‘macula-off’ RRD. The clinical implication of correctly differentiating between a ‘macula-on’ RRD and a ‘macula-off’ RRD is that the patient could be triaged appropriately.Additional file 1. A mathematical model of primary rhegmatogenous retinal detachment secondary to a 12 o’clock break.


While the second wave of the Covid-19 pandemic is keeping the world on tenterhooks, the last few months have also led to a new wave of cybercrime. The following article analyzes the background and manifestations of pandemic-related cybercrimes and shows how our criminal law systems are able to deal with current challenges in the age of the coronavirus. Various measures taken in countries around the globe to contain the spread of the novel coronavirus SARS-CoV‑2 Covid-19) have led to public lockdowns and other severe social restrictions. As a consequence, people have relocated many aspects of their private and professional lives out of the physical and into the digital realm. While the restrictions on public life are partly responsible for a decrease in various kinds of “real world” crime, those crimes’ digital cousins have flourished under the new environment. With the spread of the virus and a general sense of uncertainty and insecurity among many parts of the population, Covid-19-related attacks by cybercriminals have become something of a pandemic on their own. International law enforcement agencies such as Europol (e.g. ) and Int have ledWith various kinds of cyberattacks either on the rise or still going strong, this article seeks to shed some light on arguably the most common form of cybercrime during this pandemic: Covid-19-related “phishing” and “malware” attacks. In doing so, we will both take a look the different kinds of phishing methods perpetrators have come to use and examine the legal implications that come along with them.While the practical phenomena and legal issues discussed in this paper are primarily based on observations of the current cybercrime scene in Germany, analogous reports from many other countries indicate that law enforcement agencies around the world are confronted with very similar issues e.g. , 21), sh, sh21]),In forensic practice, cybercriminals have proven to be very adaptative to the new conditions generated by the Covid-19 pandemic. Out of their commonly known repertoire, two methods have proved to be especially suitable to exploit people’s good faith and their increased need for information and digital interaction during these chronically uncertain times: phishing and malware attacks.With respect to the phenomenon of phishing, there is a fundamental distinction to be made between the two different crime phases: The phase of data acquisition and the phase of data use.During the data acquisition phase, the perpetrators try to deceive the potential victim into revealing sensitive data, especially access data for online banking and other payment systems , either by email or via special websites . In doinOnce they follow the perpetrators’ request and transmit the desired data, the second phase begins with the perpetrators trying to use the data for financial gains. This can, for example, be achieved by selling the data, purchasing goods at the expense of the victim or—if the data in question grants access, e.g. to an online banking account—by transferring money from the victim’s account to a third-party account. In order to disguise their own identity, cybercriminals regularly use one of two strategies: They either employ another party to function as a (more or less) unsuspecting money laundering agent, who allows money to be transferred to their private bank accounts and then forward it to the perpetrators for a commission, or they use alias accounts opened under a fake identity  . In the As a variation of the phishing method described above, malware attacks use similar, yet technically often more sophisticated means, especially with respect to the phase of data acquisition. Malware attacks have in large part emerged as a direct reaction to the improvements of cybersecurity concepts within companies. In order to infect the company’s home network with the malware program, the perpetrators usually send fictitious emailsA characteristic feature of the methods described above is that the perpetrators seek to exploit human emotions and traits like trust, curiosity, respect for authority, fear or concerns in order to induce their potential victims to ignore any sense of doubt, precaution and other protective mechanisms .One example of this is the various methods specifically designed to exploit the financial concerns of many companies facing shutdowns and other restrictions with potentially devastating effects to their businesses. The state police in the German state of Lower Saxony for instance observed a significant number of phishing cases where cybercriminals claimed to be public officials in the service of the Federal Employment Agency, offering unsolicited assistance in applying for public emergency aid granted by the federal government to temporarily reimburse companies for the costs of wages owed to their employees. Under the pretext of requiring specific information in order to proceed with the application, the perpetrators requested different sets of critical data about the company and their employees to be sent to an email address under their control.4Similar to these phishing attacks, cybercriminals have also used malware-based approaches to gain access to data from people and companies in need of public emergency aid. One way in which this was done was sending emails in the name of a public bank known to be administering the distribution of the federal emergency aid funds for small businesses and self-employed entrepreneurs. The message claimed to inform the potential victim of a (supposed) legal duty to report any emergency aid received from the bank to the competent tax authorities using a  document attached to that email.5Another way in which the methods of cybercriminals have been adapted to the pandemic is to take advantage of people’s strong demand for information on the development of the pandemic itself. Some perpetrators for example offered potential victims so-called Coronavirus maps,8Another popular and quite effective kind of Covid-19-based cyberattack came in the form of so-called man-in-the-middle-attacks,While criminals have thus been quite successful in adapting to the pandemic and using it to their economic advantage, concerns by international law enforcement agencies about other corona-related cybercrimes with potentially more devastating effects seem to have not come true. In Germany at least, authorities have not detected any significant increase in attacks on critical infrastructures,As the experience of the last few months shows, cybercriminals have proven to be as adaptive as they are creative in exploiting the current crisis in their favor. Even though this situation naturally presents law enforcement authorities with some new practical challenges, the legal issues that arise can mostly be resolved adequately by relying on already existing laws and legal methods. The reason for this is that although the thematic focal point of many cyberattacks may have shifted, the perpetrators overwhelmingly fall back on their familiar methodological repertoire when implementing them. For some years now, legislators and courts in most countries have already found appropriate ways to deal with this repertoire under criminal law. The following analysis will exemplify the general legal classification of phishing and malware attacks as the two main forms of pandemic-related cybercrime under German criminal law. However, the basic considerations of this analysis can also be applied to many other legal systems, in particular those of civil law countries whose codified criminal law bears resemblance to the German Criminal Code (StGB).As explained above,The data acquisition phase does not in itself constitute a “fraud” or “computer fraud” (sec. 263 and 263a StGB), since the victim does not suffer any direct financial losses at this point.If the crime reaches the second phase, the perpetrators usually commit “computer fraud” under sec. 263a StGB , e.g. if17Similar to phishing, the criminal law assessment of malware attacks is based on two basic factual phases: The first phase of sending and installing the malware program and the second phase of the program being executed. In the first phase, the perpetrators are generally liable to prosecution for “computer sabotage” (sec. 303b (1) no. 2 StGB). Once the program has been installed, further criminal liability depends on its specific mode of operation: If it has been programmed to lock the infected system in order to demand a ransom under the threat of serious harm, e.g. to delete all personal data , such an act will fall under the definition of “blackmail” according to sec. 253 (1) StGB,The last months of the pandemic have shown how incredibly fast cybercriminals were able to adapt to the new circumstances and use them to their advantage. The main reason for this pandemic of cybercrimes appears to be the fact that the perpetrators’ traditional toolbox, in particular phishing and malware methods, proved to be perfectly suited to being reconfigured and applied to a global situation that is as novel as it is unsettling for the population and which has caused widespread emotional and economic distress to many people, often up to the point of threatening their entire livelihoods. With more people spending more time in the digital world, the number of potential victims and opportunities to prey on these victims has also increased. On the bright side, however, cybercrimes and viruses seem to differ in one important way: While potential mutations to Covid-19 always pose a serious threat to our current vaccines and other defense mechanisms become more or less obsolete, adaptations of cybercrimes tend to rely on the same methods prosecutors have come to know and expect. Thus, most countries’ criminal law systems are prepared to deal with the new kinds of cybercrimes and will most likely continue to do so in the future. Covid-19 may be a catalyst for cybercrime, but the legal antidote has luckily already been developed within our existing criminal codes. The real issue with this antidote—much like with its medical counterpart—is its efficient distribution in practice, since cybercriminals are notoriously hard to pin down if they are smart enough to make proper use of the digital world’s various tools to cover their tracks. Thus, prosecutors will have to carefully watch the ways in which cybercriminals adapt to the new environment and constantly stay alert in order to monitor this aspect of the current pandemic as well as possible.


To investigate laypeople and dental professionals' opinions of the aesthetic outcome from implant therapy (IT) and orthodontic space closure (SC) for missing maxillary lateral incisors.n = 26), laypeople 50–70 years of age (n = 26) and orthodontists (n = 25). The assessors viewed photographs of 44 different cases treated with IT or SC, and made an evaluation of the aesthetics.Evaluation was performed by three groups: laypeople 20–30 years of age (p = 0.000). The orthodontists preferred the aesthetics of the dentition in the SC cases (p = 0.042). The young laypeople, compared to the older laypeople and orthodontists, were more dissatisfied with the color of the tooth replacing the missing lateral incisor in SC cases (p = 0.043).The gingival color adjacent to the replaced tooth was rated as having better aesthetics in the SC cases (The color of the gingiva adjacent to the implant‐supported crowns had a lower aesthetic rating than the SC group. Laypeople rated both treatments as equally good. The orthodontists had a slight preference for the aesthetics in the SC cases. There are no differences in the duration of the treatment when comparing SC or orthodontic redistribution of space in the dental arch for prosthetic replacements  using the same digital camera in a predetermined sequence, lighting, and position. The images were not manipulated. The photographs were presented to the assessors printed on paper to ensure that the colors were similar to all the assessors.The subjects were recruited at the Institute for Postgraduate Dental Education, Jönköping, Sweden, all patients who had undergone IT due to missing one or two maxillary lateral incisors and treated between 2001 and 2008 were asked to participate in the present study. This group was matched with a group treated with orthodontic SC, with respect to diagnosis and sex. Patients with cleft lip and palate or oligodontia were excluded.The implant installations were performed before the patients were 25 years old, between the years 2001 and 2008. The surgical procedure was performed at the Departments of Oral and Maxillofacial Surgery or Periodontology by eight different specialists and the prosthetic treatment was performed by seven different specialists in prosthodontics. Forty‐one patients with IT met these criteria, 20 males and 21 females. The photographs were taken at least 5 years after the prosthetic therapy and the patients were 24 years and 7 months to 33 years and 8 months old.The patients were contacted by letter and were informed about the study. Twenty‐two of these subjects agreed to participate in the study, 8 males and 14 females born between 1979 and 1989.The group treated with SC consisted of 22 patients, born between 1984 and 1994, registered at the Department of Orthodontics between 2002 and 2004. They were matched to the implant group according to diagnosis, sex and number. The treatment planning and the treatments were performed at the Department of Orthodontics by 11 different orthodontists. The patients in this group (SC) had been treated with fixed orthodontic appliances. These patients were contacted by letter and informed about the study; all of them agreed to participate. The photographs were taken at least 5 years after the orthodontic treatment was finished, at ages between 20 years and 6 months and 30 years and 8 months.In the IT group there were 7 cases with bilateral aplasia of the maxillary lateral incisor and 15 cases with unilateral aplasia. In the SC group there were 12 patients with bilateral aplasia of the maxillary lateral incisor and 10 patients with unilateral aplasia.All patients who agreed to participate in this study were included in the assessments. No case was excluded due to poor aesthetic outcome, in order to mirror a standard treatment result in a society with subsidized orthodontic treatment.Twenty‐five orthodontists in the south of Sweden were asked to participate in the aesthetic evaluation of the photographs. One hundred and fifty‐five randomly selected laypersons, 20–30 years of age, were asked to participate in the aesthetic evaluation, 30 of these agreed to participate and we received answers from 26 of them. Furthermore, 75 randomly selected laypersons, 50–70 years of age, were asked to participate in the evaluation, 28 of these agreed to participate and answers were received from 26 of them. The sample of laypeople was randomly selected from the population and selected from the Swedish Central Population Registry at Statistics Sweden.Before the assessors were presented with the different cases, they were informed that the pictures they were to assess were of patients with one or two missing anterior teeth in the maxilla and that the patients had been treated with prosthodontics or SC. They were not informed which case had been treated with which treatment method.The photographs of the cases were printed and were sent to the laypeople and orthodontists for them to assess at home. The tooth or teeth to be assessed were clearly marked. The photographs were in no special order and the photographs of each case were to be assessed on a questionnaire consisting of seven questions, see 2.1U test was used for the statistical analyses within the groups of assessors but between treatments. Bonferroni correction was applied for multiple tests. Pearson's correlation test was used to analyze correlations of questions number one and seven within the assessors for each case. To be able to detect a 10‐point deviation between rating groups with power of 0.8 and alpha 0.05, 23 individuals were needed in each group of assessors based on the variation in aesthetic evaluation using VAS . The sample size regarding the number of cases was calculated based on the possibility of detecting a difference of 35% in rating outcome of crown and gingival appearance, with a significance level of 0.05 and power of 80%. The power analysis showed that 22 cases in each group were sufficient. The data was not normally distributed so an independent‐samples Kruskal–Wallis test was used for the statistical analyses between the groups of assessors and a Mann–Whitney 3In total, 77 responses with evaluations of the 44 cases were received. Twenty‐six of them were from the younger laypeople, 26 were from the older laypeople, and the remaining 25 were from the orthodontists. The response rate, out of those who agreed to participate, was 86.7%, for the young laypeople, 92.6% for the older laypeople, and 100% for the orthodontists.p = 0.000) , Table p = 0.012), Table There was no difference in the ratings of the midline between the IT and the SC groups. The orthodontists preferred the aesthetics of the dentition as a whole in the SC cases compared to the cases treated with IT p = 0.04, Table pp = 0.043), Table Comparing the three groups when evaluating the cases with SC, the younger laypeople were more dissatisfied with the color of the tooth replacing the missing maxillary lateral incisor than the orthodontists p = 0.04, Table 3p = 0.000) and they were more satisfied with the midline compared to the laypeople (p = 0.000), Table There was also a difference between the three groups of assessors when assessing the color of the gingiva in the SC cases. The orthodontists were more satisfied with the color of the gingiva adjacent to the replacing tooth compared to the laypeople . No assessments were excluded due to inadequate consistency.The seventh question on the questionnaire was a multiple‐choice question meant to validate the answer to question number one, thus verifying the intra‐examiner reliability within the assessors. When comparing all the assessors, the correlation was The case rated with the best dental aesthetics among the cases treated with SC was a case with a unilateral agenesis of the maxillary lateral incisor and no restorations or whitening had been performed owing to the space‐closing therapy Figure . Among t4Three groups of people evaluated photographs of 44 cases with aplasia of one or both maxillary lateral incisors, either replaced by an implant‐supported crown or by SC. The cases were treated at least 5 years before evaluation. The three groups of assessors consisted of two age groups of laypeople and one group of orthodontists. The number of assessors in each group was judged to be sufficient to give a reliable result based on the findings by Kiekens et al. , who fouWe found that the color of the gingiva adjacent to the implants was rated lower than the color of the gingiva adjacent to natural teeth. This finding is in agreement of Batra et al. who found that a pigmentation of the gingiva is considered to have a negative impact on the aesthetics according to laypeople (Batra et al., The overall ratings shows no major differences between these two methods to treat patients with missing maxillary lateral incisors, which is in disagreement with Armbruster et al.  and QadrAn asymmetry in the dental arch is more conspicuous than a general irregularity (Ma et al., We found that orthodontists rated the midline as better than the laypeople in both groups. This could be due to laypeople being more doubtful of the concept of the midline and therefore chose to place their mark in the middle of the VAS. The number of cases with bilateral aplasia of the maxillary lateral incisor and cases with unilateral aplasia differed between the IT group and the SC group. Despite this, the ratings of the midline did not differ between the IT group and the SC group when all assessors were included. Previous studies have found that orthodontists are less tolerant to discrepancies in the dental midline than laypeople (Pinho et al., Robertsson et al.  studied Patients treated due to missing one or two maxillary lateral incisors are likely to have equally attractive smiles as persons with a complete dentition (De‐Marchi et al., In the present study there are a few shortcomings. One of them is the presence of soft tissue in the pictures, which can influence the ratings concerning the dentition in its entirety. The use of color photographs of the lower third of the face has been widely used in previous studies Brisman,  and photAnother limitation is that this is a quantitative study relying on group values, which can conceal individual opinions.The questionnaire was comprehensive and time‐consuming, which made retest evaluations almost impossible, and we chose to use a confirming question instead.The aspect of time in this study should be seen as something both positive and negative. The negative aspect is that since the IT was performed, implant materials have been developed that allow clinicians to choose more aesthetic materials. The positive aspect is that we aim to give our patients a lifelong treatment and to do that we need to have long‐term follow‐ups.5The color of the gingiva adjacent to the implant‐supported crowns had a lower aesthetic rating than the SC group. Laypeople rated both treatments as equally good. The orthodontists had a slight preference for the aesthetics in the SC cases. The young laypeople were slightly more dissatisfied with the color of the tooth in the SC group compared to the older laypeople and the orthodontists.The authors declare that they have no conflict of interest.Cecilia Hedmo planned the research, collected and analyzed data, and wrote the paper. Rune Josefsson planned the research, collected and analyzed data, and wrote the paper. Eva Lindsten planned the research, analyzed data, and wrote the paper.AppendixS1: Supporting InformationClick here for additional data file.


This clarification is not the result of an error on the part of the journal:A. brasiliensis:⚬A. brasiliensis. Therefore, it could be stated that this biosurfactant extract possesses good fungicidal properties to be used as a preservative in foods storage under refrigeration conditions”.“It was observed that the biosurfactant extract under evaluation showed a fungicidal capacity at low concentrations of biosurfactant, 0.33 mg/mL, at 4 °C against A short explanation should be included in the Results and Discussion section after the discussion of The Foods Editorial Office would like to apologize for any inconvenience caused to the readers by these changes. The changes do not affect the scientific results. The published version will be updated on the article webpage, with a reference to this Correction.


The rise in temperatures, the acidification of the oceans and the prolonged periods of draught represent significant problems for much of life on earth and pose an immediate challenge for almost all aspects of humanity's immediate future.The entire globe is currently experiencing increasing temperatures and more extreme weather events caused by the continuous rise in the atmospheric levels of CO2 on global warming is normally attributed to the Swedish physical chemist Svante Arrhenius (1859–1927), whom under the impetus of the contemporary Swedish geologist Arvid Högbom (1857–1940), was seeking an explanation for the low temperatures during the ice ages. To this end, Arrhenius performed what he later called “tedious calculations” demonstrating that atmospheric CO2 retains the heat emanating from the ground, and hence, also retains part of the energy that would otherwise have been lost as radiation.1 Being akin to the situation in a greenhouse, this mechanism was soon to be known as the “greenhouse effect.”The causal influence of increased atmospheric CO2 it is less-appreciated that the Nobel laurate in physiology August Krogh (1874–1949) contributed by extraordinarily precise measurements of atmospheric CO2, as well as quantitative assessment of the buffering by the oceans. The measurements were performed during an expedition to Greenland in the summer of 1902 and in his 1904 publications, Krogh points to the importance of fossils fuels in determining atmospheric CO2.4While much has been written about the role of Arrhenius, and how his work was based on the earlier theoretical insight from Joseph Fourier, John Tyndall, and Claude Pouillet,6 He came under the tutelage of Christian Bohr at the Physiological Laboratory at Copenhagen University in 1899, and quickly embarked on studies of gas exchange in insect larvae and frogs. To this end, he developed and refined methods for very precise measurements of CO2 and oxygen in water, blood, and air samples. These techniques were conceived by what Krogh called “visual thinking” in a later autobiographical paper,7 and provided an unmatched precision that enabled the description of the influence of CO2 on blood oxygen binding8 as well as the realization that gas exchange across respiratory epithelia, including the human lung, occurs by diffusion alone.9August Krogh was undoubtedly one of the most influential physiologists in the early 20th century.Peru, Krogh shifted his attention to determining whether CO2 was indeed in equilibrium between fresh and sea waters and the atmosphere. In the introduction to the synthesizing paper from 1904, Krogh remarked: “The results I attained with regard to the least mentioned question appeared to me to be so interesting that I devoted most of my time to it during my stay in Greenland, and I have since my return pursued it further.”4In 1902, the year before he defended his doctoral thesis, Krogh was invited to participate in a scientific expedition to Greenland, where he intended to study the respiratory gas exchange of the organisms of the Arctic sea using his newly developed equipment. However, soon after leaving Copenhagen on the sailing vessel 2 tension in air was slightly higher than that of the waters and surmized that the atmospheric CO2 tension was on the rise. At that time, the explanations for a rise in atmospheric CO2 were far from clear, but Krogh reasoned that combustion of fossil fuels was the only credible explanation. He reasoned that “9 ton of carbonic acid or rather more than 1/1000 of the quantity present in the atmosphere. In the geological insignificant period of 1000 years the per-centage of carbonic acid could therefore be doubled by this course alone, if other factors remained unchangedThe combustion of coal by man is an ever increasing factor that has in recent years reached very considerable magnitude—The worlds production of coal amounted in 1902 to 700 million tons giving by combustion 2.6×10.”4While at sea and while travelling up and down the west coast of Greenland, Krogh ascertained with very high precision that the CO2 was actually on the rise. He was also amongst the first scientists to acknowledge that the ocean serves as a crucial CO2 buffer and he performed similar “back of the envelope,” but very reasonable, calculations showing that the oceans could absorb vast amounts of the CO2 produced. What Krogh did not predict, however, was the almost explosive rise in fossil fuel combustion that would ensue over the coming century, where we currently emit more than 35×109 ton CO2 per year.Krogh's measurements provided some of the first conclusive evidence that atmospheric CO


As part of a research project aimed at evaluating a hospital‐based adolescent transition programme, we asked ourselves what is known about the ethical and methodological challenges of research involving adolescent patients as co‐researchers. The aim of our review was to summarize empirical evidence and identify knowledge gaps about the involvement of young patients as co‐researchers.We conducted a scoping review through searches in MEDLINE, EMBASE, PsychINFO, AMED.We found reports of young patients being actively engaged as co‐researchers in any stage of a research project, although commonly they were not involved in every stage. Including young patients as co‐researchers is resource demanding and time‐consuming. Involving young patients as co‐researchers contributes to the fulfilment of their right to participation and may improve the relevance of research. Benefits for the young co‐researcher include empowerment, skills building and raised self‐esteem. Few authors go into detail about ethical considerations when involving young co‐researchers. None of the included articles discuss legal considerations.is and what it entails in this context.No lists of recommendations are given, but recommendations can be deduced from the articles. There is need for time, funding and flexibility when including young patients as co‐researchers. Knowledge gaps concern legal and ethical dilemmas of including a vulnerable group as co‐researchers. More reflection is needed about what meaningful participation This review is part of a research project where the hospital youth council has been involved in discussions of focus area and methods. It was a question of empty ritual versus real power to influence.[…] this kind of research remains far from standard’.social conditions for dialogue between patients and researcher are not given and should be sought for’. In other words, in most cases the process does not spontaneously unfold. In addition, there are differences in the terminology used about participation and a need to clarify what concepts one uses. AbmaWe are seeing an increasing focus on patient participation in health research, with an increasing number of national guidelines being issued, for example in the UK,patients are co‐researchers. Children up to 18 years of age should be given the right to participate in research within the overall framework of human rights. Furthermore, the ladder of participation has been adapted also to children's participation.et alet alet alAs part of a research project evaluating a hospital‐based adolescent transition programme,patients as co‐researchers. By co‐research, we refer to full participation in the research team, that is levels 4 and 5 of the simplified participation ladder.The aim of this review was to summarize empirical evidence and identify knowledge gaps about the involvement of young 2To explore these topics, we performed a scoping review following the outline of the process as described by Peterson et al2.1A search of databases Table  was perfet alIn stage two of the process, the first and last author screened the abstracts. The following inclusion criteria were used: articles concerning patients 12 years and above; patients are co‐researchers; the research topic is in relation to health care and health systems research; describes 4th and 5th level of participation according to the participation ladder as amended by Teunissen and referenced by de Wit The characteristics of the included articles are presented in Table 2.2et al, it helps to put the results in context.Although not mandatory for a scoping review, we chose to perform a quality assessment of the included articles. As argued by van Schelven 2.3An extraction grid was developed. Categories were created to reflect the research questions and the stages of a research process. The extraction grid was tested by the first and last authors. Amendments were made before application to the included articles. An integrated form of analysis was performed33.1et alNone of our 14 articles were included in the review by Haijes & van Thiel.3.2Quality assessment scores ranged between 0.65 and 0.95. All scores but three were over 0.8, which indicates a high quality of reporting for most articles Table .3.3preparatory phase, there is focus on research training. Van Staa et alThere is evidence of adolescent co‐researchers being involved in all six stages of research Table , but onlrecruitment phase. This is the area with least reported participation. These articles report on co‐creation of the recruitment toolet aldesign phase including project development meetings,data collection phase. Young researchers were involved in data collection such as interviews or focus groups.Four articles report on involvement in the analysis phase. Some report briefly that the analysis was done by the team,continuous review of all work’et alet alet aldissemination stage, some simply state that dissemination was done ‘as determined by the team’All articles report on involvement in the 3.4et alet alet alThe reasoning behind choice of methods revolves around the importance of involvement to better meet the needs of adolescentset alFew authors go into detail on ethical considerations. Most state that an ethics committee evaluated their project. Safeguarding confidentiality and informed consent is touched upon. Lincoln 3.5et alet alBenefits for the young co‐researchers, the professional researcher and the research process are discussed. Arguments used are primarily for the research itself, including relevance of questions, methods and findings, as well as recruitment.3.6No article draws up a list of recommendations, but the importance of clarifying roles has been emphasized,4patients participating at level 4 or 5 according to the simplified ladder of participation. We identified only 14 articles matching the inclusion criteria. We find it interesting that we do find evidence of young co‐researchers being part of all stages of the research process. This is a similar to findings reported in the review by Van Schelven et alThis is the first scoping review of adolescent 4.1et al argue ‘Ethical issues to be considered when carrying out research or service evaluation with children may include: power relationships, consent issues, confidentiality and dissemination of results’.It is our opinion that the included articles would have benefited from a more in‐depth portrayal of ethical challenges of involving young co‐researchers. This would have provided important learning points for other researchers. Most mention that they have gone through ethics committee approval, but with one exception,et al who developed a self‐care plan for the young co‐researchers' well‐being.A related issue is research ethics boards' project assessment. How can we ensure that they properly assess not just the research process the co‐researchers are to be involved in, but also the role as co‐researcher? Researchers wanting to involve young co‐researchers should consider and have to account for ethical questions suitable for their projects. This was exemplified by Lincoln 4.2not aim for the top rung of the ladder, for example if aiming for the 5th level of the ladder entails leaving the adolescents on their own in a research process where they should have support? Jones argues that a healthy work environment is the responsibility of the adult researcher.It is not appropriate that some children feel that they must always only follow the initiative of others any more than it is good for any child to feel that they should always be a leader’.The discussion on ethics ties into reflections we should have when utilizing the ladder of participation. It is easy to get the impression that participation at the higher rungs of the ladder should be preferred. However, is this really the case? Hart states that his adaptation of the participation ladder, contrary to how it has been used by some, was not meant to be an evaluation tool. He intended his adaptation to be used as a way of portraying different forms of participation and as a point of departure for reflexion.4.3A question that arises after reviewing the included articles is what is relevant or adequate research training for young co‐researchers? It has been suggested by others that fully involving children as co‐researchers can be problematic due to their lack of the theoretical knowledge needed.People understand the world in different ways but we do not know what those differences are until we have an opportunity to share and discuss’.It is important to keep in mind that research training is not the only way to tackle the issue of power balance in the research group. This could also be addressed through creating spaces and opportunities for meaningful participation,The question remains what all this entails for the training of young co‐researchers. We find the example of Jones4.4et al report in their review that some of their included articles problematize that lack of research experience can lead to lack of depth in for instance interviews.et al included in this review.Van Schelven 4.5someone follow each step? And if adolescents are invited to full participation, are well informed, but opt for a lower level of participation such as being consulted, does that mean that the project is not truly participatory? We argue that research would benefit from a mix of forms of participation.Finally, what can be learned from the included articles is the need for flexibility. The need for flexibility was also pointed out by van Schelven and colleagues.4.6et alOne limitation of the review could be specific databases indexing that may have prevented us from identifying relevant publications. Van Schelven 5is and what it entails in the context of research. Proceeding in line with a child/human rights framework could form the basis for adopted recommendations on how to involve young co‐researchers, as well as the development of self‐care plans. Research ethics boards need the competencies to properly assess not just the research process that the adolescents are to be involved in, but their role as co‐researchers.No lists of recommendations are given in the included articles, but recommendations can be deduced from the articles. There is need for time, funding and flexibility when including young patients as co‐researchers. We would argue that research involving adolescents co‐researchers should be based on and would benefit from proceeding in line with a rights based framework. We miss a more thorough discussion of legal and ethical dilemmas when including a particularly vulnerable group as co‐researchers. More reflection is needed about what meaningful participation The authors declare no conflict of interest.Kjersti J. Ø. Fløtten designed and conducted the work presented in this paper, analysed the data and wrote the paper. Isabelle Aujoulat designed the work, analysed the data and has co‐written the final version of the manuscript. Ana Isabelle F. Guerreiro has participated in the design, offered advice on data collection and has co‐written the final manuscript. Ilaria Simonelli has participated in the design, offered advice on data collection and has co‐written the final manuscript. Anne Lee Solevåg participated in the design, offered advice on data collection and has co‐written the final version of the manuscript.


Sensory adaptation is a useful tool to identify the links between perceptual effects and neural mechanisms. Even though motion adaptation is one of the earliest and most documented aftereffects, few studies have investigated the perception of direction and speed of the aftereffect at the same time, that is the perceived velocity. Using a novel experimental paradigm, we simultaneously recorded the perceived direction and speed of leftward or rightward moving random dots before and after adaptation. For the adapting stimulus, we chose a horizontally-oriented broadband grating moving upward behind a circular aperture. Because of the aperture problem, the interpretation of this stimulus is ambiguous, being consistent with multiple velocities, and yet it is systematically perceived as moving at a single direction and speed. Here we ask whether the visual system adapts to the multiple velocities of the adaptor or to just the single perceived velocity. Our results show a strong repulsion aftereffect, away from the adapting velocity (downward and slower), that increases gradually for faster test stimuli as long as these stimuli include some velocities that match some of the ambiguous ones of the adaptor. In summary, the visual system seems to adapt to the multiple velocities of an ambiguous stimulus even though a single velocity is perceived. Our findings can be well described by a computational model that assumes a joint encoding of direction and speed and that includes an extended adaptation component that can represent all the possible velocities of the ambiguous stimulus. The motion of a one-dimensional visual feature seen through a small aperture, such as the small receptive field of a velocity tuned channel , is ambiVisual adaptation produces perceptual aftereffects that have been extensively documented over the last decades and provides a glimpse into the properties of the underlying neural mechanisms of perception. For example, prolonged exposure to a motion stimulus leads to strong, illusory perceptual biases in the perceived velocity (speed and direction) of static  took part in the experiment, and they all had normal or corrected-to-normal vision. All participants were naive with regard to the purpose of the study, and they gave informed written consent in accordance with the local ethics committee and the Declaration of Helsinki.unidirectional), and RDK that moved over a range of directions . The stimuli moved at eight different speeds  or were stationary (0°/s)  spatial broadband drifting gratings (frequency range from 1/3 cycles/deg to 2 cycles/deg) moving upward at 6°/s with phases randomized on each trial. The adapting stimuli were shown inside a circular aperture of 10° in diameter. The test stimuli consisted of two types, random dot kinematograms (RDK) that moved coherently in one direction (y (0°/s) . The tesThe experimental procedure is illustrated in The test stimuli were presented in blocks of 40 trials, two trials per speed condition plus six trials with random speeds and directions slightly above or below leftward or rightward. These random trials were added to introduce ambiguity in the stimulus’ motion direction. Participants did three blocks for each type of test stimulus, first without adaptation, and then with adaptation in a single session. In total, there were six trials per participant for each of the 68 different conditions. At the beginning of the session, participants did at least 100 practice trials to get familiar with the task. In those trials, the speed and direction of test stimuli were randomized across the whole range of possible velocities and feedback was provided at the end of the trial in the form of a green Gaussian blob that indicated the correct speed and direction of the presented dots along with the participant's estimation as a red small dot. No adapting stimuli were shown during training. During the main part of the experiment, no feedback was given to participants.v′ = 2arctan(v/r0), where v is the original speed in deg/s, v′ is the transformed speed, and r0 = 10. This closed space allows us to use a representation that is similar to a normalized logarithm but with the added benefit that it includes zero speed. The same space was used during participants’ estimation of the stimulus’ velocity, that is, the length of the estimation bar did not have a linear relation with speed. We run pilot sessions with both a linear and nonlinear relation, and we found that a nonlinear relation appears more natural to participants and better balances motor errors across speeds.Similar to We implemented a neural model to simulate the effect of adapting to an ambiguous moving stimulus. The model is based on the one proposed by x0i, y0i] is the central horizontal (x0i) and vertical (y0i) velocity of the channel,α is the maximum activity (“firing rate”) of the channel. Channels are homogeneously distributed in the horizontal-vertical closed velocity space, from −3 to 3 in units determined by the “arctan” transform defined above, in steps of 0.25 units for a total of 625 channels. As an illustration of the unit used for speed, 3 corresponds to 141°/sec whereas 1 to 5.46°/sec. A cartoon example of the network grid is shown in The network is composed of velocity channels φ, which are represented by bivariate Gaussian distributions:wst1 is the strength of the gain reduction andAdaptation to a stimulus is simulated in two stages. First, there is a nondirectional gain reduction that affects only fast or slow speed channels depending on the speed of the adapting stimulus. For slow speed adaptation, the gain reduction is expressed as a bivariate Gaussian function wst2 is the strength of the gain reduction andβ parameter for the horizontal variance. We will see below that this parameter is chosen to be very large, thereby creating a directional adaptation that applies to all potential velocities of the ambiguous stimulus .There are two types of input stimuli. The uni-directional dots are assumed to be a bivariate Gaussian centered at the stimulus physical velocity and variancen to each stimulus s with Poisson variability based on the original φi or altered Each channel produces a response xv(s) and vertical yv(s) velocities are calculated from the population average, so that:The horizontal wst1, σst1, wst2, σst2.The model is run for 1000 iterations to produce averaged curves of expected biases at different stimulus horizontal velocities. A summary of the model's parameters is shown in Participants reported the perceived velocity of leftwards or rightwards moving dots before and after adapting to an upwards moving grating. P < 0.01. These are indicated by the small horizontal lines above each stimulus speed in F(1100) = 21.11, P < 0.001)). Overall, the biases appear to follow a similar pattern for both types of stimuli.The differences between the two types of stimuli can be better evaluated in P < 0.01. In particular, there is a significant effect for all but one speed faster than the adapted speed (five of six). For example, at the highest rightward speed of 10.5°/sec, there was a significant effect of stimulus type on estimation biases (F(1100) = 10.29, P = 0.002). For speeds slower than the adapted speed, only two of nine differences are significant at P < 0.01. Finally, of interest are the almost identical biases to stationary stimuli for both types of stimuli. Obviously, the two stimuli are identical because there is no movement, so the similarity suggests a quite robust effect. All responses by individual participants decomposed to their horizontal and vertical components are plotted in The experimental results seem to validate our hypothesis that the upwards velocities of the fast multidirectional stimuli are strongly affected by the adapting stimulus even though they are further away in velocity space. To explain our findings, we implemented a computational model of velocity channels and simulated the effect of adaptation to an upward moving stimulus (see Methods). β = 1), the model fails to correctly predict the pattern of vertical velocity biases at fast speed (Our goal with the proposed model is not to precisely predict the absolute speed estimates of participants but rather to explain the changes in speed estimates between before and after adaptation. Our preadaptation data present some biases toward slow speeds that may have a number of possible origins, including a sensory prior for slow speeds, a motor bias because of the reproduction method we used, or a response bias reflecting a regression to the mean. We assume that these biases are independent of the adaptation, thereby allowing us to focus on the differences in estimates between before and after adaptation. st speed .Finally, we tested a version of the model with only the directional stage but with an “extended” gain profile where the bandwidth of velocity channels increases with speed. We used the energy model by Because of the aperture problem, a drifting grating will activate multiple neurons sensitive to different combinations of directions and speeds. In this report, we have shown that when such a moving stimulus is presented, observers adapt to all the velocities consistent with this ambiguous stimulus. When tested with stimuli moving in a single direction, participants exhibited uniformly weak biases towards downward directions, but when tested with stimuli that contained multiple directions, they exhibited increasingly stronger downward biases for increasing speeds. Intuitively, this can be explained by looking at the expansion of a multidirectional stimulus in velocity space . As the A number of studies have looked at the effects of motion adaptation at multiple levels of the visual hierarchy . Scarfe The ambiguous adapting stimuli were broadband drifting gratings, whereas the test stimuli were random dot kinematograms. Ideally, adapting and test stimuli should have the same spatiotemporal frequency spectra in order to minimize the dimensions in which the stimuli differ. However, it would be very difficult to precisely control the different velocities of a multi-directional test stimulus composed of multiple superimposed drifting gratings. It has been shown that integration of stimuli of multiple spatiotemporal frequencies can affect the perceived speed for stimuli that share the same veridical speed  or diffeWe implemented a computational model with two stages of adaptation. First, a nondirectional gain reduction is applied that affects fast speed channels. Second, a directional gain reduction is applied that affects channels close to the adapting velocities but extends infinitely into faster horizontal velocities at the same vertical velocity. To find signatures of the two mechanisms, both must be involved in the experimental task. In experiments where participants only report the speed or the direction of a stimulus, it is difficult to isolate the effects of these two potential mechanisms. Neurophysiological studies have found that neurons in the visual cortex change their response profiles after adaptation . These c