In the Abstract and the Introduction sections, “infection-free” should be corrected to “uninfected.”R0=(kδ/pu)x0, R0=(kδ/pu)x0 .”“We can denote the basic reproduction number of the HIV virus for the model (3) as E0 near;” should be corrected to “Proof. We consider linear system of the model (3) at E0.”The sentence “Proof. We consider linear system of the model (3) in R0 < 1, pu − kδx0 ≠ 0. Therefore, λ=0 is not root of (9). If (8) has pure imaginary root λ=iω(ω > 0) for some τ > 0, substituting it into (8) and separating the real and imaginary parts,” should be corrected to “When R0 < 1, pu − kδx0 ≠ 0. Therefore, λ=0 is not root of (9). If (9) has pure imaginary root λ=iω(ω > 0) for some τ > 0, substituting it into (9) and separating the real and imaginary parts.”The sentence “When τkn)( > 0 such that (23) has pure imaginary λ=iωk,” should be corrected to “Then, we get the corresponding τkn)( > 0 such that (19) has pure imaginary λ=iωk.”The sentence “Then, we get the corresponding b2ω2+b32=(ω3 − a2ω)2+(a1ω2 − a3)2” should be corrected to “From (22), we obtain b22ω2+b32=(ω3 − a2ω)2+(a1ω2 − a3)2.”The sentence “From (19), we obtain Equation (L) should be corrected as follows:In the Local and Global Stability of the Equilibria section, there were the following errors:(iii) In the Simulations and Conclusions section, the sentence “By direct calculations, we get that (19) has a positive root;” should be corrected to “By direct calculations, we get that (24) has a positive root.”(iv) The formats of references 2, 14, 25, 26, 27, and 29 were updated.In the article titled “Modeling Inhibitory Effect on the Growth of Uninfected T Cells Caused by Infected T Cells: Stability and Hopf Bifurcation” , there wThis has been corrected in place.


The authors are retracting this article  after anFigure 1C, upper panel: areas of the image labelled as “Mock” and of that shown as “Scramble” are duplicated.Figure 1D: the image labelled as “Scramble, 24 h” and that shown as “siNFAT5, 48 h” are duplicated.Figure 3A: areas of the image labelled as “Invaded, siCtrl” and that shown as “Before Scraping, siS100A4” are duplicated.


VaccinationCoverageforSelectedVaccinesandExemptionRatesAmongChildreninKindergarten—UnitedStates,2017–18SchoolYear,” on page 1121, an incorrect figure was published. The corrected figure follows.In the report “


The number given as “2.3%” should have been “2.6%.” This article has been corrected.1In the Original Investigation titled “Antipsychotic Treatment Among Youths With Attention-Deficit/Hyperactivity Disorder,”


Scientific Reports 10.1038/s41598-018-33381-3, published online 11 October 2018Correction to: This Article contains an incomplete equation in the Methods section, under the subheading ‘Statistical models’.ς, τγ and τρ. fς was further multiplied by φ and 1 − φ to obtain the proportion of variance explained by υ and v, respectively”.“The proportion of marginal variance explained by each component of the RR and lethality models was given by:should read:ς, τγ and τρ. fς was further multiplied by φ and 1 − φ to obtain the proportion of variance explained by υ and v, respectively”.“The proportion of marginal variance explained by each component of the RR and lethality models was given by:


Figure 5, and we have corrected data in the result sections. First, in the results section , there is an error in the legend box. (Column 1 as “Period”), the label “1” should be amended a “1=1989–93”. It is not, “1=1994–93”There were three errors in Figure ), there are errors in the legend box (Column 1 as “Period”), the label “1” should be amended a “1=1989–93”. It is not, “1=1994–93”. The column 2 as “Number”, the number for “1=1989–93”, “2=1994–98”, “3=1999–03”, “4=2004–08”, “5=2009–13” should be amended as 564, 557, 904, 857, and 817, respectively.Second, in the results section , the Survival curve was duplicated from Figure , it should be changed to the correct figure.Third, in the results section (page 203, Figure 5 is provided below. The authors apologize for these errors.The correct


Muscle protein synthesis and muscle net balance plateau after moderate protein ingestion in adults. However, it has been suggested that there is no practical limit to the anabolic response of whole-body net balance to dietary protein. Moreover, limited research has addressed the anabolic response to dietary protein in adolescents. The present study determined whether whole-body net balance plateaued in response to increasing protein intakes during post-exercise recovery and whether there were age- and/or sex-related dimorphisms in the anabolic response.− 1·h− 1) as crystalline amino acids modeled after egg protein. Steady-state protein kinetics were modeled noninvasively with oral L-[1-13C]phenylalanine. Breath and urine samples were taken at plateau to determine phenylalanine oxidation and flux (estimate of protein breakdown), respectively. Whole-body net balance was determined by the difference between protein synthesis (flux – oxidation) and protein breakdown. Total amino acid oxidation was estimated from the ratio of urinary urea/creatinine.Thirteen adults  and 14 adolescents  performed ~ 1 h variable intensity exercise  prior to ingesting hourly mixed meals that provided a variable amount of protein , indicating an anabolic plateau. Net balance was maximized at ~ 0.15, 0.12, 0.12, and 0.11 g protein·kg− 1·h− 1 in M, F, AM, and AF, respectively. When collapsed across age, the y-intercept  was greater (overlapping CI did not contain zero) in adolescents vs. adults. Urea/creatinine excretion increased linearly  across the range of protein intakes. At plateau, net balance was greater (P < 0.05) in AM vs. M.Mixed model biphasic linear regression explained a greater proportion of net balance variance than linear regression (all, Our data suggest there is a practical limit to the anabolic response to protein ingestion within a mixed meal and that higher intakes lead to deamination and oxidation of excess amino acids. Consistent with a need to support lean mass growth, adolescents appear to have greater anabolic sensitivity and a greater capacity to assimilate dietary amino acids than adults. With a target of n = 42 unique intakes, adults were randomly assigned to consume a protein intake from seven pre-defined ranges . However, due to time constraints and scheduling conflicts participants completed a minimum of five and a maximum of eight trials. In addition, based on an interim analysis, some participants were additionally randomized to intakes 0.188 g/kg/h to better define the breakpoint. With a target of n = 42 unique intakes, adolescents were randomly assigned to consume a protein intake from six pre-defined ranges  with all participants completing six trials. Adult female participants completed trials during the predicted luteal phase, which was defined as the second half of the menstrual cycle. Following an overnight fast, participants consumed a protein-free liquid carbohydrate beverage (1 g·kg− 1) as a 1:1 ratio of maltodextrin  and sports drink powder  before reporting to the laboratory. The purpose of the beverage was to help replenish overnight fasted losses of liver glycogen and to provide exogenous carbohydrate to fuel the exercise stimulus. Approximately 1 h after carbohydrate ingestion, participants completed the LIST according to a previously described exercise protocol [Each participant performed 5–8 metabolic trials consisting of two components: a modified version of the Loughborough Intermittent Shuttle Test (LIST) and a subsequent 8-h postprandial period with a variable protein intake  and fat (0.08–0.18 g·kg− 1·h− 1) with 0.42 g·kg− 1·h− 1 of carbohydrate per meal. Protein was provided as crystalline amino acids  modeled after the composition of egg protein [− 1 Na13CO2 and 1.86 mg·kg− 1 L-[1-13C]phenylalanine  [− 1·h− 1 L-[1-13C]phenylalanine to maintain isotopic steady state and a constant oral infusion of phenylalanine. Excess tyrosine (3.33 mg·kg− 1·h− 1) was provided to ensure metabolic partitioning of the phenylalanine carboxyl group into either synthesis or oxidation [13C]phenylalanine has been previously shown to produce the isotopic steady-state condition required for amino acid flux and oxidation determination within an 8-h postprandial period phenylalanine enrichment by liquid chromatography-tandem mass spectrometry, respectively, as previously described phenylalanine produced isotopic steady states in urine, plasma, and breath CO2 similar to steady states produced by intravenous phenylalanine tracer models [13C]phenylalanine tracer yields similar enrichments of [13C] phenylalanine in urine as compared to plasma, which subsequently obviates the necessity of concomitant blood sampling [13C]phenylalanine is an acceptable oral tracer in studies of amino acid flux and oxidation using urinary (rather than plasma) tracer enrichment, which makes it ideally suited for use in vulnerable populations such as children and adolescents.Oral amino acid tracers have been used to model whole-body protein metabolism noninvasively in adults  and adolr models , 33. Fursampling . Thus, t13C]phenylalanine in urine (as an estimate of plasma enrichment) [13CO2 breath enrichment was calculated using standard equations developed by Matthews et al. [− 1·h− 1) was measured from L-[1-13C]phenylalanine tracer dilution in the urinary pool at isotopic steady state and calculated as follows:The isotopic enrichment of L-[1-ichment)  and 13COs et al.  and Roses et al. . Phenyla13C] phenylalanine ingested (μmol·kg− 1·h− 1); I is the rate of L-phenylalanine ingested (μmol·kg− 1·h− 1); Ei and Eu are the isotopic enrichments as mole fractions of the test drink and urinary phenylalanine, respectively, at isotopic plateau.Where i is the rate of L-phenylalanine enrichment as follows:The rate of phenylalanine oxidation  was calculated as the difference between protein synthesis and protein breakdown.Protein breakdown was estimated by the rate of appearance of phenylalanine in the urine and protein synthesis was calculated as the difference between phenylalanine flux and oxidation (see above) . Whole-br2 of the linear regression mixed model (described below) was compared to the r2 of the biphasic linear regression mixed model (described below), with the highest r2 identifying the preferred model. The linear mixed model used participants as a random variable using PROC MIXED  to determine the r2. If the data conformed to a biphasic model, breakpoint analysis of the net balance data using a biphasic linear regression mixed model  was performed in agreement with previous studies [To determine whether the relationship between protein intake and whole-body net balance was better explained by linear or biphasic regression, the  studies , 36, 37 whereby the null hypothesis was rejected if the interval did not contain zero . To estaP ≤ 0.05.whereby the null hypothesis was rejected if the interval did not contain zero. Linear correlation  was used to identify a relationship between urea/creatinine excretion and protein intake. The whole-body net balance breakpoint was identified for each analysis and a two-way ANOVA  was used to identify main effects of age, sex, and age × sex interactions for the mean net balance values above the breakpoint to assess between-group differences. Where significant interactions were identified, a Bonferroni corrected t-test  was used to compare net balance means above the breakpoint. Significance was set at − 1·h− 1) and adolescents (0.12 ± 0.01 g·kg− 1·h− 1). The y-intercept  was greater  in AM (1.2 ± 0.06 mg·kg− 1·h− 1) when compared to M (0.46 ± 0.09 mg·kg− 1·h− 1), F (0.35 ± 0.07 mg·kg− 1·h− 1), and AF (0.67 ± 0.05 mg·kg− 1·h− 1). When collapsed across age, adolescents (0.87 ± 0.04 mg·kg− 1·h− 1) had a greater (overlapping CI did not contain zero) y-intercept vs. adults (0.44 ± 0.07 mg·kg− 1·h− 1).Mixed model biphasic linear regression explained a greater proportion of net balance variance than linear regression Table , indicat− 1·h− 1 in M, F, AM, and AF, respectively, with no differences  in the breakpoint intake. When collapsed across age, the breakpoint protein intake was not different (overlapping CI contained zero) between adults (0.16 ± 0.03 g·kgFFM− 1·h− 1) and adolescents (0.13 ± 0.02 g·kgFFM− 1·h− 1). The y-intercept was greater  in AM (1.3 ± 0.08 mg·kgFFM− 1·h− 1) when compared to M (0.54 ± 0.1 mg·kgFFM− 1·h− 1), F (0.33 ± 0.06 mg·kgFFM− 1·h− 1), and AF (0.77 ± 0.04 mg·kgFFM− 1·h− 1). When collapsed across age, adolescents (1.1 ± 0.05 mg·kgFFM− 1·h− 1) had greater (overlapping CI did not contain zero) y-intercept vs. adults (0.45 ± 0.08 mg·kgFFM− 1·h− 1).When normalized to FFM, whole-body net balance increased up to a plateau in all groups at a protein intake corresponding to 0.17 ± 0.05, 0.14 ± 0.04, 0.14 ± 0.04, and 0.12 ± 0.02 g·kgFFMr ≥ 0.76; P < 0.01) with protein intake in all groups  increased linearly  and an age × sex interaction (P = 0.04) for whole-body net balance at plateau  had a greater (P < 0.05) net balance at plateau than M (~ 1.9 mg·kg− 1·h− 1) with no further differences (P > 0.05) between groups. When normalized to FFM, there was a main effect age (P = 0.04), sex (P = 0.02), and an age × sex interaction (P < 0.01) for whole-body net balance at plateau  had a greater (P < 0.05) net balance at plateau than M (~ 2.2 mg·kgFFM− 1·h− 1) with no further differences (P > 0.05) between groups.There was a main effect of age  alone [− 1 in isolation [− 1 [− 1 [− 1·h− 1, respectively, and would fall within the intake ranges (0.02–0.25 g·kg− 1·h− 1) in the present study. With such a limited number of protein intakes, which in our hands would represent single intakes below and above the breakpoint, respectively, we argue that it is challenging to draw clear inferences as to the potential saturation of whole-body net balance. In contrast, the repeated design of the present IAAO study permitted us to investigate a greater dynamic range of protein intakes that resulted in a clear breakpoint in whole-body net balance in all groups studied. Given that our adult males plateaued after the ingestion of ~ 0.15 g·kg− 1·h− 1, it is possible that had others [− 1 bolus or 0.23 g·kg− 1·h− 1) a similar plateau in whole-body anabolism would have been apparent. Alternatively, bolus protein ingestion more robustly stimulates whole-body protein synthesis [The post-exercise consumption of dietary protein is important for recovery and growth  as it would help replenish exercise-induced amino acid oxidative losses  and prov1) alone –12. This1) alone . This thsolation  or when solation , whereasion [− 1 , 11 and − 1 [− 1  when con− 1 [− 1 , 11 and − 1 [− 1 . Metabol− 1 [− 1 –46 would− 1 [− 1 , 48 suchynthesis , 50, whiynthesis , and whoynthesis  are suppynthesis , we beliynthesis , would pIt has been suggested that the level at which protein intake becomes excessive is reflected by an increase in amino acid oxidation with graded protein intakes . When di− 1 (~ 96 mg·h− 1) in females and 3.8 g·d− 1 (~ 150 mg·h− 1) in males during this rapid growth period [During the pubertal growth spurt, a healthy normally developing child has a ~ 3-fold increase in growth velocity compared to the pre-pubertal period (< 10 y) , wherebyh period . Moreoveh period  and may h period , which sh period . Given th period , we specUnlike resistance exercise, variable intensity exercise is generally not associated with marked growth or muscle hypertrophy in weight-stable adults . In contWe report that whole-body net balance plateaued in response to increasing protein intakes after a bout of variable intensity exercise, which is in agreement with our hypothesis and suggests that the anabolic response to post-exercise protein ingestion has a practical limit in both adults and adolescents. Importantly, we did not observe any statistical differences in the breakpoint between sexes within an age group, suggesting that postprandial protein recommendations to maximize whole-body net balance after exercise are primarily influenced by total body and fat-free mass. In further agreement with our hypothesis, the greater whole-body net balance at very low and optimal protein intakes in adolescents compared to adults highlight that active youth during the pubertal growth spurt have both a greater anabolic sensitivity and anabolic potential to mixed meal dietary protein ingestion during recovery from exercise.


Nature Communications 10.1038/s41467-019-11595-x, published online 9 August 2019.Correction to: aAssuming a tangential velocity of 150–300 km−1 from the proper motions of ref. 22’ The correct version states ‘150–300 km s−1’ in place of ‘150–300 km−1’.The original version of this Article contained an error in last sentence of the legend to Table 4, which incorrectly read ‘−6 M⊙ y−1)’, ‘−5 M⊙ y−1)’ and ‘−4 M⊙ y−1 km−1)’, instead of the correct ‘−6 M⊙ yr−1)’, ‘−5 M⊙ yr−1)’ and ‘−4 M⊙ yr−1 km s−1)’, respectively.The original version also contained errors in Table 5, in which the headings of the fifth, sixth and seventh rows incorrectly read, ‘aAssuming a tangential velocity of 150–300 km−1 from the proper motions of ref. 22’ The correct version states ‘150–300 km s−1’ in place of ‘150–300 km−1’.The last sentence of the legend to Table 5 originally incorrectly read ‘S5.8 = 0.794 ± 0.03 mJy, ν = 5.8 GHz, vj = 600 ± 100 km−1, θ0 = 52.3° ± 4.4°, D = 2.2 kpc, T = 10,000 K, the resulting ionised mass-loss rate is 1.81 ± 0.33 × 10−6 M⊙ yr−1, consistent with ref. 22.’ The correct version states ‘vj = 600 ± 100 km s−1’ in place of ‘vj = 600 ± 100 km−1’.The original version also contained an error in the third sentence of the first paragraph of the ‘Ionised mass-loss rate on source’ section of the Methods, which incorrectly read ‘From our observations we obtain the following parameters for core B: This has been corrected in both the PDF and HTML versions of the Article.


The “universality” refers to the fact that this result is model‐free, that is, not dependent on an underlying stochastic process. We extend Cover's theorem to the setting of stochastic portfolio theory: the market portfolio is taken as the numéraire, and the rebalancing rule need not be constant anymore but may depend on the current state of the stock market. By fixing a stochastic model of the stock market this model‐free result is complemented by a comparison with the numéraire portfolio. Roughly speaking, under appropriate assumptions the asymptotic growth rate coincides for the three approaches mentioned in the title of this paper. We present results in both discrete and continuous time.Cover's celebrated theorem states that the long‐run yield of a properly chosen “universal” portfolio is almost as good as that of the best  This enables us to analyze strategies which depend on the market weights, and the performance of relative wealth with respect to the market portfolio.In this paper, we work under the setting of SPT. Namely, the market portfolio is taken as the benchmark, or “numéraire,” so that the primary assets are the market weights which take values in the open 1.1.1M‐Lipschitz portfolio maps denoted by LM. Each element of LM maps the market weights to long‐only portfolio weights in Δ¯d t=0∞ the relative wealth process corresponding to a portfolio strategy(πt)t=1∞, we are interested in comparing the asymptotic growth ratesbest retrospectively chosen portfolio at time T in the class L:=⋃M=1∞LM ;the analog of Cover's universal portfolio whose relative wealth process (Vt(ν))t=0∞ is defined in L with full support on each LM);the log‐optimal portfolio among the class of long‐only strategies, whose relative wealth process is denoted by (V^t)t=0∞.the Denoting by Theorem 1.1(μt)t=0∞ be a time‐homogenous ergodic Markov process in discrete time describing the dynamics of the market weights. ThenLet The first two portfolios can be compared in a model‐free way  is asymptotically as good as the best one chosen with hindsight, and the log‐optimal portfolio constructed with full knowledge of the underlying process.Intuitively, this theorem says that a suitable 1.1.2functionally generated portfolios  is correctly specified, model parameters cannot be estimated precisely and always come with a confidence interval. So, in practice, the As for the original theorems of Cover and Jamshidian, a valid criticism is of course that we only establish asymptotic equality on a first‐order log‐return basis. As such, a lot of important information is lost in the limit. However, one cannot expect to obtain any information on higher order terms unless further quantitative assumptions are made on the considered models. Cover's aim and also the goal of the present paper is to be as model‐free as possible.The remainder of the paper is organized as follows. In Section 2For expositional simplicity, time is discrete in this section.2.1Cover's insight reveals that the “wisdom of hindsight” does not give significant advantages over a properly chosen “universal” portfolio constructed using only historical and current prices of the assets. The relevant optimality criterion here is the asymptotic growth rate of the portfolio.T∈N and think of an investor who at time T looks back which stock she should have bought at time t=0 . There is an obvious solution: pick i∈{1,⋯,d} which maximizes the normalized logarithmic returnt=0 instead of t=T. Here is the remedy T which tends to zero as T→∞. Hence this buy‐and‐hold portfolio, which corresponds to a universal portfolio in the sense of Cover, has asymptotically the same normalized logarithmic return as the—only retrospectively known—best performing stock.Let us sketch this—at first glance surprising—result in a particularly easy setting ∈Δ¯d, that is, bj≥0 and ∑j=1dbj=1. The value of the corresponding constant rebalanced portfolio (Vt(b))t=0∞ starting at V0(b)=1 is defined by holding throughout the proportion bj of the current wealth in stock j, so that V0(b)=1 ands=((stj)j=1d)t=0∞⊂d of the stocks.Instead of these “pure” investments, Cover considered a more ambitious setting, namely, all T and define the quantity VT∗ bys=t=0T. Again, the idea is that, with hindsight, that is, knowing t=0T, one considers the best weight b∈Δ¯d which attains the maximum (VT∗)T=0∞ as T→∞, uniformly over all price paths. For this reason, the portfolio is said to be universal. In order to do so, let ν be a probability measure on Δ¯d which replaces the previous uniform distribution over the d stocks. The universal portfolio is built by investing at time 0 the portion dν(b) of initial capital in the constant rebalanced portfolio V(b) and by subsequently following the constant rebalanced portfolio process (Vt(b))t=0T. The explicit formula for the wealth isVt(b) is defined by universal portfolio is given by the wealth‐weighted averageFix again Theorem 2.1Δ¯d with full support. Thenall trajectories s=t=0∞ for which there are constants 0<c≤C<∞ such thatCover, : Let ν bLet us now recall Cover's celebrated result:Remark 2.2 distribution on Δd  be the set of probability measures on Δ¯d. For each μ∈M1(Δ¯d), consider the value ∫Δ¯dVT(b)(s)dμ(b) of the mixture portfolio with initial measure μ. Note that the constant rebalanced portfolio VT(b) corresponds to the case where μ is the point mass at b. It is easy to see thatVT∗(s) is defined by Let 2.2 denote the market capitalizations of the stocks rather than their prices. Then we define the vector of market weights∈Δd bymarket portfolio  as the numéraire t=0∞, expressed in units of the market portfolio and starting at V0=1, is obtained by the following recursive relation:(πt)t=1∞, where the portfolio weight πt is used over the time interval . In this paper, all trading strategies are fully invested in the equity market, that is, the portfolio weights sum to 1 for all t. In particular, the strategies do not lend or borrow money. Henceforth, all wealth processes are measured in units of the market portfolio.The portfolio maps. These are (Borel) measurable functionsμt= the weights (π(μt)=(π1(μt),⋯,πd(μt)) according to which an agent distributes current wealth among the d stocks at time t. The constant rebalanced portfolio strategies considered by Cover correspond to the constant functions π:Δd→Δ¯d.We will focus on trading strategies defined by (deterministic) constant rebalanced portfolios to certain families of portfolio maps. First, we note that Cover's and Jamshidian's definition of a universal portfolio as in G denote some appropriate space of portfolio maps, B(G) its Borel σ‐algebra and ν some probability measure on G.Definition 2.4). Then, the corresponding universal portfolio at time t is given by the wealth‐weighted averageLet ν be a probability measure on In this paper, we extend Cover's theory of πν is given byFrom 2.3S=t=0∞ and the corresponding relative market capitalizations μ=t=0∞ are now assumed to be stochastic processes defined on a filtered probability space t=0∞,P).To define the log‐optimal portfolio, we consider a probabilistic setting. The stock price process T, this portfolio is by definition the maximizer of the expected logarithmic growth rate(πt)t=1T. Under mild assumptions on the process a unique optimizer exists; see, for example, Becherer  has the form πt=π(μt−1), where π:Δd↦Δ¯d as in π^.To connect the log‐optimal portfolio with universal portfolios in the sense of Definition The Markovian assumption can be motivated by the stability of capital distributions of equity markets t=0∞ with values in Δd. We consider as far as possible a model‐free approach, but will introduce a probabilistic setting when the log‐optimal portfolio is involved.Throughout this section, we work in discrete time and assume that the market weights are described by a 3.1We start by defining rigorously, in the present setting, the three portfolios introduced in Sections 3.1.1T a strategy which is optimal within a certain class of strategies, in our case portfolio maps π:Δd→Δ¯d. A moment's reflection reveals that it does not make sense to allow to choose among all measurable functions π:Δd→Δ¯d. Indeed, there is no restriction to choose π such that π(μt)=ej(t), where j(t)∈{1,⋯,d} maximizes μt+1j/μtj. This is asking for too much clairvoyance and does not allow for meaningful results −π(y)∥1≤L∥x−y∥1, x,y∈Δd.For However, it does make sense  to restrict to more regular trading strategies. In particular, we work with the following set of Remark 3.2LM of M‐Lipschitz functions π:Δd→Δ¯M−1d is a compact metric space with respect to the topology of uniform convergence induced by the norm ∥π∥∞=sup{∥π(x)∥1:x∈Δd}.The set Remark 3.3Instead of Lipschitz functions we could just as well consider other compact function spaces, for example, Hölder spaces equipped with a proper norm. This is done in the context of functionally generated portfolios in Section Definition 3.4(μt)t=0T∈(Δd)T+1, we defineFor a given trajectory The retrospectively chosen best performing portfolio among the above Lipschitz maps is defined as follows:π↦VTπ, there exists an optimizer π∗,M∈LM (not necessarily unique) such that VT∗,M=VTπ∗,M, thus the sup above can be replaced by max .By compactness (see Remark 3.1.2predictable process πM=(πtM)t=1∞, that is, one which depends only on the history of the market weights, such that the performance of (VtπM)t=0∞ is asymptotically as good as that of t=0∞. This can be achieved by the universal portfolio introduced in Definition G is now LM as in Definition LM is a compact metric space, we may find a (Borel) probability measure ν on  with full support; this will be essential for establishing an analog to Theorem Our aim is to find a 3.1.3μ=(μt)t=0∞ is a time‐homogeneous Markov process  log‐optimal portfolio, we assume that )x∈Δd, that is, for all Borel sets A⊆Δ¯d we have P[μt+1∈A|Ft]=ϱ. For further notions concerning ergodic Markov processes, we refer to Eberle  of μt+1. We therefore choose π^(x)∈Δ¯d as the maximizerπ^(·) can be chosen to be measurable . For x∈Δd, define the number L(x) as the value of the optimization problem π(x)=x (which corresponds to the market portfolio) we clearly have L(x)≥0 for each x∈Δd. We obtain the a.s. relationV^=(V^t)t=0∞ denotes the long‐only log‐optimal wealth process Vπ^ defined by the portfolio map π^ via Assumption 3.6Using the above notation we assume thatThe Theorem 3.7μ0∈Δd,L1.Under Assumptions π:Δd→Δ¯d be any measurable portfolio map such that0, thatL1.More generally, let Applying Birkhoff's ergodic theorem for discrete time Markov processes t=0∞ in Δd, we haveFix We are now ready to compare the asymptotic performance of the three approaches. We first establish an analog of Theorem LM is compact and ν has full support, it is not difficult to see that for any η>0, there exists δ>0 such that every η‐neighborhood of a point π∈LM has ν‐measure bigger than δ.The inequality “≥” is obvious. For the reverse inequality, we follow the argument of Blum and Kalai . As LM i(μt)t=0∞ in Δd be given. For a fixed time T, let π∗,M∈LM be an optimizer of πM∈LM with ∥πM−π∗,M∥∞<η, that is, such that, for every x∈Δd we have ∥πM(x)−π∗,M(x)∥1=∑j=1d|πM(x)j−π∗,M(x)j|<η.Let a trajectory η>0 small enough so that α=ηMd<1 and define, for x∈Δd,π∼ maps Δd into Δ¯d.Choose Using ε>0. Choosing η>0 sufficiently small we can make α=ηMd small enough such that the final term is bigger than −ε. Summing up, we have∥πM−π∗,M∥∞<η.Fix B=Bη the ∥·∥∞‐ball with radius η in LM which has ν‐measure at least δ>0, where δ only depends on η. As each element πM of B satisfies T to infinity and letting ε to zero. □Denote by LM for a fixed M. In Corollary L=⋃MLM.Theorem 3.10Ω=(Δd)N be the canonical path space equipped with its natural filtration and a probability measure P. Define μ=(μt)t=0∞ to be the canonical process, that is, μt(ω)=ωt, which takes values in Δd and satisfies Assumptions M>0 be a fixed Lipschitz constant for the space LM. Consider the following objects that are defined for each trajectory (μt)t=0∞:(i)T∈N the portfolio map π∗,M∈LM as well as the corresponding wealth VT∗,M:=VTπ∗,M as in Define for each (ii)LM with full support and consider the wealth process of the universal portfolio (VtM(ν))t=0∞ as of Fix a probability measure ν on (iii)π∈LM by(V^tM)t=0∞=(Vtπ^M)t=0∞ via Define the log‐optimal portfolio among the portfolio maps Then, we have P‐a.s.Lπ is given in all trajectories (μt)t=0∞ in Δd.Let Note that in Theorem π^M is well‐defined; simply use the compactness of LM with respect to ∥·∥∞ t=0∞ in Δd was shown in Theorem That the first equality in T∈N, we obviously haveP‐a.s.(V^tM)t=0∞ as the log‐optimizer within the class LM, we haveLM.For each fixed 1Tlog(VTM(ν)) is bounded from below, see, e.g., lim infT→∞1Tlog(VTM(ν)) is P‐a.s. constant and equal to limT→∞1Tlog(V^TM). This completes the proof of the theorem. □Combining now M to infinity in the following way. For M=1,2,3,⋯ choose a measure νM on LM with full support. Define ν=∑M=1∞2−MνM and the wealth of the universal portfolio V(ν) as in L=⋃M=1∞LM. Recall that (V^t)t=0∞ is the wealth process of the (long‐only) log‐optimal portfolio Corollary 3.11P‐a.s.L is defined in Under the assumptions of Theorem Next we will send M→∞ in VT(ν)≥2−MVTM(νM) for every M, so we have by Theorem MLetting P‐a.s. As by Lemma (Vt(ν)V^t)t=0∞ is a nonnegative supermartingale, it converges P‐a.s. to a finite limit as t→∞. This in turn implies □Now the corollary is proved ifLemma 3.12(Vt(ν)V^t)t=0∞ is a nonnegative supermartingale.The process π:Δd→Δ¯d, (VtπV^t)t=0∞ is a nonnegative supermartingale. Indeed, by Lemma (Vt(ν)V^t)t=0∞,□First note that for any Lemma 3.13π^ be given by π:Δd→Δ¯d and every x∈Δd,Let Here we establish the supermartingale property used in the previous proof.α∈ and define πα=απ+(1−α)π^. Then by the (long only) log‐optimality of π^ we have for every x∈Δdπα≥(1−α)π^. By Fatou's lemma, we therefore have□We proceed as in the proof of Becherer  denotes the Hölder space of 2‐times continuously differentiable functions from Δ¯d→R whose derivatives are α‐Hölder continuous. That is,k denoting a multi‐index in N2. For α=0, the second term in this norm is left away. Note that G is only defined on the simplex Δd. In order that the partial derivatives are well‐defined, we assume that each G is extended to an open neighborhood of Δd such that G(x)=G(x′), where x′ is the orthogonal projection of x onto Δd. The choice of the extension is irrelevant.We consider the following set of concave functions. For some fixed Lemma 4.1M,α>0, the set GM,α is compact with respect to ∥·∥C2,0.For any Here is an analytical lemma whose proof is given in the appendix.GM,α, we associate now the set of functionally generated portfolios FGM,α in the spirit of Fernholz n=1∞ of  along (Tn) in the sense of Föllmer  and πG be defined as in (μt)t≥0 be a continuous path satisfying Assumption VπG satisfiesg(dt)=−12G(μt)∑i,jDijG(μt)dt.Let Using 4.2We again consider (i) the best retrospectively chosen portfolio, (ii) the universal portfolio, and (iii) the log‐optimal portfolio. To define the log‐optimal portfolio, we will restrict to a specific stochastic model introduced in Section 4.2.1FGM,α and a given continuous path (μt)t≥0 satisfying Assumption M,α>0 fixed, we defineWe consider the set of functionally generated portfolios Lemma 4.4T,M,α>0 be fixed and (μt)t≥0 be a continuous path satisfying Assumption G↦VTG where VTG is given by G↦VTG is continuous from  to R.Let We first prove that an optimizer exists by establishing the following continuity property whose proof can be found in the appendix.Proposition 4.5T be fixed and (μt)t≥0 be a continuous path satisfying Assumption VT∗,M,α be defined by GT∗∈GM,αand in turn a portfolio πT∗ generated by GT∗ such thatLet  as shown in Lemma □This is simply a consequence of continuity as proved in Lemma 4.2.2m be a Borel probability measure on . Consider the mapπG is given by  a Borel probability measure ν via the pushforward ν=F∗m. As in Definition Remark 4.6More precisely, we need to verify that the universal portfolio still allows for pathwise integration and that the value of the portfolio  is given by the right‐hand side of To define the analog of Cover's/Jamshidian's portfolio in the present setting, let 4.2.3μ=t≥0 follows a time‐homogeneous Markovian Itô diffusion, defined on t≥0,P) with values in Δd, given by· denotes the matrix square root, W is a d‐dimensional Brownian motion, λ is a Borel measurable function from Δd→Rd, and c is a Borel measurable function from Δd→S+d, satisfyingΔd. Note that (μt)t≥0 given by structure condition λ(μs)ds). This structural condition characterizes the condition of “no unbounded profit with bounded risk” (NUPBR) in the case of continuous semimartingales .In this setting, the proportions of current (relative) wealth invested in each of the assets are described by processes π in the following set:Vπ and Vθ for π,θ∈Π. Using for everyπ∈Π if we choose θ∈Π such thatθ/μ to ordinary portfolio weights via Fernholz and Karatzas ]≤E[log(V^T)] for all π∈Π.Next we consider the log‐optimal portfolio defined by Karatzas , the genBy all strategies in Π. In the sequel, we shall mainly consider suprema taken over smaller sets, in particular over FGM,α. Note that in this case the optimizer will still be a function of the market weights due to the Markov property of (μt)t≥0.So far we have optimized over Proposition 4.7(μt)t≥0 be of the form G, that is,Let In this context, let us also answer the question of when the log‐optimal portfolio is functionally generated. This is needed to relate its asymptotic growth rate to the one of the best retrospectively chosen portfolio and the universal portfolio.□The assertion follows from expression 4.2.4Assumption 4.8Δd.The process μ as given in limT→∞1TlogVTπ. For the precise notion of ergodicity in continuous time, we refer to Eberle (With this assumption we derive an expression of the asymptotic growth rate o Eberle .Hd.Theorem 4.9(i)π:Δd→Hd be any (ϱ‐measurable) portfolio map such that0, thatLet (ii)L:=12∫Δdλ⊤(x)c(x)λ(x)ϱ(dx)<∞. Then, for ϱ‐a.e. starting value μ0, it holds thatAssume that Under Assumption In the following theorem, we consider portfolio maps which are not necessarily long‐only, but can take values in the hyperplane Lemma 4.10M be a continuous local martingale such thatlimT→∞1TMT=0, P‐a.s.Let The proof of Theorem Proof of Theorem 4.9logVTπ reads as(loglogT)/T, therefore yields Condition P‐a.s. ) and thus assertion (i).Let us start by proving statement (i). By logV^T simplifies to□Concerning statement (ii), note from 4.3Theorem 4.11M,α>0 be fixed and let (μt)t≥0 be a continuous path satisfying Assumption i∈{1,…,d}m on GM,α with full support and set ν=F∗m with F defined in V∗,M,α and VM,α(ν) are defined in Let As in discrete time, we will establish asymptotic equality of the growth rates of all three portfolio types introduced in Section m has full support and GM,α is compact, we have that, for η>0 there exists some δ>0, such that every η‐neighborhood of a point G∈GM,α has m‐measure bigger than δ.The inequality “≥” is obvious. For the converse inequality, we proceed similarly as in the previous section (using only generating functions). As T≥1 and denote by GT∗ the optimizer as of Proposition G such ∥G−GT∗∥C2,0≤η. Then it follows from ε>0 and note that by assumption T↦1TT on T can be bounded by some constant. Therefore, we can choose η sufficiently small such that KT≤ε for all T≥1.Let B=Bη(GT∗) the ∥·∥C2,0‐ball with radius η in GM,α which has m‐measure at least δ>0, where δ only depends on η. We then may estimate using Jensen's inequality and T→∞ for any given ε (which determines η and in turn δ) yields the assertion. □Denote by FGM,α and suppose henceforth that (μt)t≥0 is of the form V^M,α by V^M,α=Vπ^M,α, whenever π^M,α is well‐defined. Asπ^M,α as optimizer for all T>0, V^M,α corresponds to the log‐optimal portfolio among functionally generated portfolios with generating function in GM,α.Theorem 4.12M,α>0 be fixed and let (μt)t≥0 be a stochastic process of the form m on GM,α with full support and set ν=F∗m with F defined in V^TM,α denotes the log‐optimal portfolio among FGM,α‐maps defined via V∗,M,α and VM,α(ν) are defined pathwise in Let To compare the asymptotic performance with that of the log‐optimal portfolio, we optimize over portfolio maps in π^M,α is well‐defined. Indeed, the map to R. This together with compactness of GM,α with respect to ∥·∥2,0 imply the well‐definedness of π^M,α.We first note that G imply the assumptions of the ergodic theorem t≥0 as log‐optimizer within the class FGM,αVTM,α(ν) is given by FGM,α.Due to P‐a.s. constant and equal to limT→∞1Tlog. Hence the assertion is proved. □Combining now M on α. Setting α=1M we choose for M=1,2,3,⋯ a measure mM on GM,1M with full support. Define m=∑M=1∞2−MmM and the process V(ν) byAs in the previous section, we can formulate a result not depending explicitly on the constant Corollary 4.13(μt)t≥0 be a stochastic process of form c satisfy G^∈C2(Δ¯d). Then we have P‐a.s.Let In order to compare the performance with the one of the global log‐optimal portfolio, whenever it is functionally generated, we combine the above results with Proposition L is well‐defined due to ε>0, there exists some M>0 and some function G∈GM,1M such thatG↦VG as asserted in Lemma G∈GM,1M close enough with respect to the ∥·∥C2,0 to the optimizing function G^∈C2(Δ¯d) whose generated portfolio yields V^ due to (Vt(ν)V^t)t≥0 is a nonnegative supermartingale. It converges P‐a.s. to a finite limit as t→∞. This in turn implies □Note first that C2‐functionally generated portfolios without imposing the drift condition in Proposition VTfun the wealth process of the log‐optimal portfolio among concave C2‐functionally generated portfolios, that is, πfun is defined as in arg max over all concave C2‐functionally generated portfolios.Corollary 4.14(μt)t≥0 be a stochastic process of form Let Finally, a similar result can be obtained by restricting the log‐optimal portfolio to the class of lim inf because the supermartingale argument from the proof of Corollary □The proof is the same as the first part of Corollary


This article explores how probabilistic programming can be used to simulate quantum correlations in an EPR experimental setting. Probabilistic programs are based on standard probability which cannot produce quantum correlations. In order to address this limitation, a hypergraph formalism was programmed which both expresses the measurement contexts of the EPR experimental design as well as associated constraints. Four contemporary open source probabilistic programming frameworks were used to simulate an EPR experiment in order to shed light on their relative effectiveness from both qualitative and quantitative dimensions. We found that all four probabilistic languages successfully simulated quantum correlations. Detailed analysis revealed that no language was clearly superior across all dimensions, however, the comparison does highlight aspects that can be considered when using probabilistic programs to simulate experiments in quantum physics. Probabilistic models are used in a broad swathe of disciplines ranging from the social and behavioural sciences, biology, the physical and computational sciences, to name but a few. At their very core, probabilistic models are defined in terms of random variables, which range over a set of outcomes that are subject to chance. For example, a measurement on a quantum system is a random variable. By performing the measurement, we record the outcome as the value of the random variable. Repeated measurements on the same preparation allow determining the probability of each outcome. Probabilistic programming offers a convenient way to express probabilistic models by unifying techniques from conventional programming such as modularity, imperative or functional specification, as well as the representation and use of uncertain knowledge. A variety of probabilistic programming languages (PPLs) have been proposed , i ∈ {0, 1}. In a similar way, variables B0 and B1 are introduced. Therefore, the preceding four pairwise distributions can be represented as the the grid of sixteen probabilities depicted in The basis of the EPR experiment is two systems The goal of an EPR experiment is to empirically determine whether quantum particles are entangled. We will not go into the details of what entanglement is, but rather focus on showing how statistical correlations between variables determine the presence of entanglement. Entanglement is determined if any of the following inequalities is violated.A, B) of quantum particles to joint measurements. More specifically, each such pair is measured in one of the four measurement conditions represented by the grid of probabilities depicted in For historical reasons, the set of four inequalities have become known as the Clauser-Horn-Shimony-Holt (CHSH) inequalities . The datA0, A1, B0, B1, the maximum value that can be computed by any of the inequalities happens to be 2. This is why the boundary of violation in the inequalities is 2 as it demarcates the boundary which standard statistical correlations cannot transcend. This fact presents a challenge for a PPL, which is based on standard probability theory. How can a PPL be developed to simulate non-classical quantum correlations?The maximum possible violation of the CHSH inequalities is 4, i.e., three pairs of variables are maximally correlated (= 1) and the fourth is maximally anti-correlated (= -1). However, if the experiment is modelled by a joint probability distribution across the four variables P, e.g., entangled quantum particles, is to be studied. An experimental design is devised in which P is examined in the four experimental conditions called “measurement contexts”. A measurement context Mi, 1 ≤ i ≤ 4 is designed to study P from a particular experimental perspective. For example, one measurement context corresponds to X = 0 and Y = 1 which yields probabilities over the four possible outcomes of joint measurements of A and B. We will denote these outcomes as {00|01, 01|01, 10|01, 11|01}. For example, 00|01 denotes the outcome A = 0, B = 0 in the measurement context M2 = {X = 0, Y = 1}.P. Composition offers the distinct advantage of allowing experimental designs to be theoretically underpinned by hypergraphs in a modular way , ], , ]],   , ], , ]]]The next step involves producing four hyperedges to represent the four explicit joint measurement contexts on both systems. Two variables are given to assist with computing this result.edge_aH[0]:edge_bH[1]: Thereafter, each hyperedge is defined as the combined sets produced by the following expression.fr_edge = vertex_aedge_a:vertex_bedge_b: fr_edge.append   fr_edges.append(fr_edge)m) are all members of set M, where M are the possible measurement choices for the scenario (in this case two), the second (n) being the other possible measurement choice, and the last variable o being all edges from the hypergraph associated with the measurement choice.The last step involves calculating the hyperedges of both systems as dependent on the other. To achieve this programmatically, three variables are declared. The first :mc_i =1-mc) edgeH[mc]: o, in some selected hypergraph, a second variable j is declared as two possible values. For each possibility, a hyperedge is then defined as the variable k, being the combination of all vertices in o given to a function that produces the hyperedge.For each j:fr_edge =  i): s, q, r, u, t, and v.The mentioned function computes the hyperedge by declaring single-use variables edge_b = H[mc_i][i]vertex_a = edge[(i-j)]vertex_b = edge_b[0]vertex_c = edge_b[1]vertices_a = [vertex_a[0], vertex_b[0], vertex_a[1], vertex_b[1]] vertices_b = [vertex_a[0], vertex_c[0], vertex_a[1], vertex_c[1]] Thereafter, a set is constructed by use of its variables, and portions of the desired hyperedges are iteratively returned.Upon calculation of the last step of the process, the hyperedges corresponding to the measurement choices of both EPR systems as dependent on the other are produced, totaling the necessary constraints described in binary format.fr_edge.append   fr_edge.append   fr_edges.append(fr_edge) fr_edgesa, b, x, and y that were previously mentioned as specifying one of the 16 possible vertices. These values are restricted to binary outcomes by means of specifying Bernoulli distributions for which the sampler runs the process. The experiment is fixed such that each vertex has an equal possibility of being sampled as any other vertex, and results may only be discounted if they do not comply with specified input correlations. Upon selecting a vertex at a step in the iterative process, the array index associated with the binary representation is incremented by one, via the use of the vertex mapping function. Simultaneously, another array representing the hyperedges of the FR product is also incremented by one at all indexes associated with the hyperedges containing the said vertex. The iterative process only exits when the sum of the global distribution is equivalent to the desired number of iterations. Thereafter, each vertex is normalised by the sum of the values in the corresponding array of hyperedges in which its associated vertex is contained, and is multiplied by 3 (for reflection of the number of associated hyperedges). A visualisation of the hyperedges associated with the vertex at index 00|00 of the global distribution can be seen in To compute the four pairwise distributions at the basis of the EPR experiment :hyperedges = foulis_randall_product hyperedges_tallies = zeros(12) global_distribution = zeros(16)  < N: with pm.Model:  pm.Uniform   pm.Bernoulli   pm.Bernoulli   pm.Bernoulli   pm.Bernoulli   S = pm.sample   c = S.get_values   a = S.get_values   b = S.get_values   x = S.get_values   y = S.get_values   i:  :   edgeget_hyperedges:     hyperedges_tallies[edge] += 1     global_distribution[    get_vertex] += 1     z =  a, b, x, yproduct: summed_tally = ))   global_distribution[get_vertex] /= summed_tally  global_distribution *= 3 global_distribution Given below in numpyzeros, array, fliplr,itertoolsproductpymc3 as pmget_vertex:((x*8)+(y*4))+(b+(a*2)) get_hyperedges:l =  idx, e(H): ne:  l.append(idx)   l foulis_randall_product:fr_edges =  H = , ], , ]], , ], , ]]]    edge_aH[0]: edge_bH[1]:  fr_edge =    vertex_aedge_a:   vertex_bedge_b:    fr_edge.append      fr_edges.append(fr_edge)   mc: mc_i = abs(1-mc)  edgeH[mc]:  j:   fr_edge =     i):    edge_b = H[mc_i][i]     vertex_a = edge[(i-j)]     vertex_b = edge_b[0]     vertex_c = edge_b[1]     vertices_a = [     vertex_a[0], vertex_b[0],      vertex_a[1], vertex_b[1]]      vertices_b = [     vertex_a[0], vertex_c[0],      vertex_a[1], vertex_c[1]]      fr_edge.append      fr_edge.append      fr_edges.append(fr_edge)    fr_edges generate_global_distribution:hyperedges = foulis_randall_product hyperedges_tallies = zeros(12) global_distribution = zeros(16)  < N: with pm.Model:  pm.Uniform   pm.Bernoulli   pm.Bernoulli   pm.Bernoulli   pm.Bernoulli   S = pm.sample   c = S.get_values   a = S.get_values   b = S.get_values   x = S.get_values   y = S.get_values   i:  :   edgeget_hyperedges:     hyperedges_tallies[edge] += 1     global_distribution[    get_vertex] += 1     z =  a, b, x, yproduct: summed_tally = ))   global_distribution[get_vertex] /= summed_tally  global_distribution *= 3 global_distribution @model macro. To obtain randomly sampled non-negative values for a Bernoulli distribution, the model requires the declaration of a uniform Beta prior, invoked with the Beta method. Then Bernoulli distributions are declared with a Bernoulli method, once again accompanied by probabilities describing sampling biases for later generated distributions, as well as a Uniform distribution.Like PyMC3, Turing.jl isolates operations on random variables to a single model with the use of the generate_global_distribution method of sample function invokes the model, a step-method, as well as the number of desired iterations. In this case, the Sequential Monte Carlo sampling (abbreviated to SMC) has been applied.In the foulis_randall_product method, before returning the result.To obtain the trace of a distribution in the model, the output must be indexed with the precession of a colon. In the example, the results are retrieved, tallied, and normalised by means of the TuringDistributionsfr_edgesArray{Array{Array{Float64}}}(0) H,],,]], ,],,]]]     i1(H[1])[1] j1(H[2])[1]  fr_edgeArray{Array{Float64}}(0)   k1(H[1][i])[1]   l1(H[1][j])[1]    append!             append!      mc12 mc_i(3mc)  k1(H[mc])[1]  j12   fr_edge = Array{Array{Float64}}(0)    i1(H[mc][k])[1]    edge_bH[mc_i][i]     vertex_aH[mc][k][(ij)1]     vertex_bedge_b[1]     vertex_cedge_b[2]     vertices_a[vertex_a[1], vertex_b[1],     vertex_a[2], vertex_b[2]]        vertices_b[vertex_a[1], vertex_c[1],     vertex_a[2], vertex_c[2]]        this_edge_bArray{Float64}(0)     append!        append!            append!          fr_edges ((x8)(y4))(b(a2))1 l i1(H)[1]   append!      l @modelz a(0.5) b(0.5) x(0.5) y(0.5) c hyperedges hyperedges_tallies(12) global_distribution(16)  < N r)  ar  br  xr  yr  cr  i1N  (c[i] < constraints[x[i]1][y[i]1][a[i]1][b[i]1])   I        associated_hyperedges    j1(associated_hyperedges)[1]    hyperedges_tallies[     associated_hyperedges[j]]1          global_distribution1           a01, b01, x01, y01 summed_amount0  I      associated_hyperedges  edge_index1(associated_hyperedges)[1]  summed_amounthyperedges_tallies[edge_index]     global_distributionsummed_amount   global_distribution3 Flip method, generating a Bernoulli distribution on which the If method can then associate the results to desired values, or perpetuation of other methods.To achieve a joint-probability distribution on a measurement context of random variables, Figaro’s syntactic elements reveal fundamental differences in its approach. A class is the advised object for the purpose of declaring a model. For each random variable in the model, a probability bias is applied to Figaro’s GenerateGlobalDistribution method, it can be seen that after initialising the FR product , the sampling process is called by means of start, stop, and kill chains applied on the algorithm object. On the preceding line, the MetropolisHastings method implies the Metropolis-Hastings step-method will be used for the sampling process, and the outcomes of the model class will be considered. For more complex experiments, the ProposalScheme may be modified, however not in this case. The sampleFromPosterior sub-method chained to calls on each variable compile the required distributions on execution. The take sub-method chained to the sampling methods are used to declare the number of outcomes retrieved from the sampler. This aspect is consequent of sampler delivering results via Stream primitives, a resource-efficient consideration ensuring that only required data is evaluated. Thereafter, the proceeding code tallies the indexes of the globalDistribution array, and normalises the results.In com.cra.figaro.algorithm.sampling._com.cra.figaro.language._com.cra.figaro.library.compound.Ifcom.cra.figaro.library.atomic.continuous.UniformMain {: Array[Array[Array]] = { foulisRandallEdges = Array[Array[Array]]   hypergraphs = Array, Array),   Array, Array)),       Array, Array),   Array, Array)))       (edgeA <- hypergraphs) {  (edgeB <- hypergraphs) {   foulisRandallEdge = Array[Array]    (vertexA <- edgeA) {    (vertexB <- edgeB) {     foulisRandallEdge ++= Array)       }      }     foulisRandallEdges ++= Array     }    }   (measurementChoice <-to){   measurementChoiceInverse =– measurementChoice    (edge <- hypergraphs(measurementChoice)) {    (j <-to) {     foulisRandallEdge = Array[Array]      (i <- edge.indices) {      edgeB = hypergraphs(measurementChoiceInverse)(i)       vertexA = edge(Math.abs(i-j))       vertexB = edgeB       vertexC = edgeB       verticesA = Array                      verticesB = Array                      foulisRandallEdge ++= Array(       Array(        verticesA(measurementChoice),         verticesA(measurementChoiceInverse),         verticesA(measurementChoice+),         verticesA(measurementChoiceInverse+)))         foulisRandallEdge ++= Array(       Array(        verticesB(measurementChoice),         verticesB(measurementChoiceInverse),         verticesB(measurementChoice+),         verticesB(measurementChoiceInverse+)))         }      foulisRandallEdges ++= Array      }     }    }   foulisRandallEdges   } Model { = Array[Element]  (i <-to){  : += If   }  : += Uniform  } := { ((x*)+(y*))+(b+(a*))  } : Array = { l = Array  (i <- H.indices) {  (H(i).deep.contains(n.deep)) {   l: += i    }   }  l  } : = {  hyperedges = FoulisRandallProduct  hyperedgesTallies = Array.fill{}  globalDistribution = Array.fill{}   {  model =Model   algorithm = MetropolisHastings   algorithm.start   algorithm.stop   algorithm.kill   a = algorithm.sampleFromPosterior.take(N).toArray    b = algorithm.sampleFromPosterior.take(N).toArray    x = algorithm.sampleFromPosterior.take(N).toArray    y = algorithm.sampleFromPosterior.take(N).toArray    c = algorithm.sampleFromPosterior.take(N).toArray    (i <-until N) {   val x_x = x(i).toInt   val y_y = y(i).toInt   val a_a = a(i).toInt   val b_b = b(i).toInt   (c(i) < constraints(x_x)(y_y)(a_a)(b_b)) {   )) {     hyperedgesTallies(edge) +=      }    globalDistribution) +=    }   }  }  { summedAmount =  associatedHyperedges = GetHyperedges)   (edgeIndex <- associatedHyperedges.indices) {  summedAmount += hyperedgesTallies(edgeIndex)   }  globalDistribution) =  globalDistribution) / summedAmount   }  globalDistribution  } }def containers; or can be flexibly given implicitly, encouraging the use of stochastic functions to specify probabilistic models. Inside a container, probabilities of random variables are specified first. As Pyro is built upon PyTorch, explicit values match PyTorch types, in this case resulting in Tensor type values.Pyro’s context management is integrated into Python’s generate_global_distribution method, Pyro’s atomic sampling capabilities allow for the requirement of fewer syntactic constructs to communicate the sampling process. Each sample call accepts a distribution, in this case either Bernoulli or Uniform. Upon sampling, the outcomes are tallied and normalised, before presenting the result.As can be seen in pyrosampletorchTensortorch.autogradVariablenumpyzeros, array, fliplr,itertoolsproductpyro.distributionsBernoulli, Uniformget_vertex:((x*8)+(y*4))+(b+(a*2)) get_hyperedges:l =  idx, e(H): ne:  l.append(idx)   l foulis_randall_product:fr_edges =  H = , ], , ]], , ], , ]]]    edge_aH[0]: edge_bH[1]:  fr_edge =    vertex_aedge_a:   vertex_bedge_b:    fr_edge.append      fr_edges.append(fr_edge)   mc: mc_i =(1-mc)  edgeH[mc]:  j:   fr_edge =     i):    edge_b = H[mc_i][i]     vertex_a = edge[(i-j)]     vertex_b = edge_b[0]     vertex_c = edge_b[1]     vertices_a = [     vertex_a[0], vertex_b[0],      vertex_a[1], vertex_b[1]]      vertices_b = [     vertex_a[0], vertex_c[0],      vertex_a[1], vertex_c[1]]      fr_edge.append      fr_edge.append      fr_edges.append(fr_edge)    fr_edges generate_global_distribution:hyperedges = foulis_randall_product hyperedges_tallies = zeros(12) global_distribution = zeros(16)  < N: a =))))  b =))))  x =))))  y =))))  value =),  Variable(Tensor([1.0])))))   :  edgeget_hyperedges:   hyperedges_tallies[edge] += 1    global_distribution[get_vertex] += 1   z =  a, b, x, yproduct: summed_tally = ))   global_distribution[get_vertex] /= summed_tally  global_distribution *= 3 global_distribution A and B in the four measurement contexts are specified. For example, the following code fragments A and B to be maximally correlated in three measurement contexts and maximally anti-correlated in the fourth. With these input correlations, maximum violation of the CHSH inequalities would be expected, essentially simulating a PR box ,         ]         p = generate_global_distributionconstraints, ],, ]],          , ],          , ]]]          pconstraints = Array, Array), Array, Array)),     Array, Array), Array, Array)))     P = GenerateGlobalDistributionFor each of the four PPLs, code is specified equality:f1: ((2 * (p[v1] + p[v2])) − 1)  f2: (p[v1] + p[v2]) − (p[v3] + p[v4])  delta = 0.5 *  − f1) +  − f1) +   − f1) +  − f1))  2 * (1 + delta) >=) + ) +  ) + ))  tests =   ((2(p[v1]p[v2]))1)    (p[v1]p[v2])(p[v3]p[v4])   delta0.5)) +  )))  (2(1delta)) >=)) +  )))  tests =  (v1:v2:v3:v4: Int):= {(v1:v2:):= { Math.abs((* (p(v1) + p(v2))) −)  } (v1:v2:v3:v4:):= { (p(v1) + p(v2)) − (p(v3) + p(v4))  } delta =* ( (f1 − f1) + (f1 − f1) +  (f1 − f1) + (f1 − f1))  (* (+ delta)) >= Math.abs( (v1*f2) + (v2*f2) +  (v3*f2) + (v4*f2))  }tests = ArrayUpon conducting simulations using the input correlations given above, the predicted maximum violation of the CHSH inequalities were observed for all four PPLs specified above.We conducted several experiments to compare the numerical accuracy and execution time of the different implementations.For all PPLs, statistical outputs confirming the success of the FR product  are given with an acceptable margin of error. While this is consequent of more than a single factor, it is perceived that the largest contributor to accuracy is the computation of random values for each PPL. What can be observed is that, with larger sample sizes (bearing more perfectly random distributions), that the margin of error decreases, as can be seen below. This is typical, as the experiment design’s normalisation process is dependent on the even spread of tallies across the global distribution, and more specifically, the degree to which the sampler is random. It should also be considered that the various PPLs apply data types that round values for a loss of statistical precision where it may serve meaning. For example, while a single value may lose a minute portion of its whole beyond the decimal point (due to automatic rounding), when calculating a handful of these values per iteration of some few thousand iterations, the difference becomes observable.From observing the results in Another statistic that has been observed is the compilation time of each PPL, which typically increases with number of iterations. For relativity of results, it should be noted that all non-accelerated tests of this kind were executed within a Bash execution terminal, on a Macintosh operating system, bearing a 45nm “Penryn” 2.4 GHz Intel “Core 2 Duo” processor, and 4 GB of SDRAM, whereas all accelerated tests were executed within a bash execution terminal of Amazon Web Services Linux (2nd distribution). The specification of the ‘Elastic-Compute Cloud’ on which the Linux distribution executed was a 2018 “p2.xlarge” 2.7 GHz Intel Broadwell processor, with 61 GB of SDRAM. The instance also provides an NVIDIA GK210 GPU multi-vCPU (count of 4) processor, and 12 GB of GPU RAM.In Recall that the challenge posed was how to develop probabilistic program which can simulate quantum correlations in an EPR experiment. The solution adopted was to program a hypergraph formalism to underpin the simulations. This formalism is modular where the FR product of the modules is used to impose the no-signalling constraint. In execution, all four PPLs successfully simulated an EPR experiment producing quantum correlations. Therefore, we conclude that the hypergraph formalism has been shown to be a promising basis for such simulations. In addition, the hypergraph formalism is also rendered into program syntax in a fairly straightforward way. However, the formalism does pose a challenge with respect to the accessibility criterion of the PPLs. The challenge is due to an inherent ambiguity present in the composite hypergraph produced by the FR product, which has 12 edges in the EPR experiment. Four of those edges represent “actual” measurement contexts  for the case of the experimentation. Thus it wouldn’t be perceived that this is a determinant factor.In experimentation, we found that Pyro provided the syntactic constructs needed to neatly describe its processes in fewer procedures than those of the others. While PyMC3’s origins in Python also made it an expressive alternative, the excessive nomenclature surrounding the declarations of methods and data-types for both Turing.jl and Figaro convoluted their descriptions. While in comparison to the other PPLs, Figaro’s accuracy is inferior, it could be argued that the sample iterations describe an experimental setting that does not consider improving float precision. Coupled with the trend of Figaro’s improvement in its number of iterations, and the measure of accuracy between PPLs may converge. The same cannot be said for the time complexity of the EPR experimentation, where it was observed that PyMC3’s compilation grew substantially with the number of sample iterations being executed. Still, in instances where accuracy is a key factor and limitations are perceived in Pyro’s functionality, PyMC3 would be the suitable alternative.Probabilistic programming offers new possibilities for quantum physicists to specify and simulate experiments, such as the EPR experiment illustrated in this article. This is particularly relevant for experiments requiring advanced statistical inference on unknown parameters, especially in the case of techniques that involve large amounts of data and computational time. Furthermore, probabilistic machine learning models that are conveniently expressed in probabilistic programming languages can advance our understanding of the underlying physics of the experiments.It is important to note that the benefits of probabilistic programming are not restricted to experiments involving the analysis of quantum correlations. Since any probabilistic programming language is based on random variables, we can ask the question what exactly is a random variable in quantum physics. Focusing on a single measurement context, due to the normalization constraint, we can think of the measurement context as a  probability distribution over random variables, which describes the measurement outcomes. The probability distribution is a normalized measure over the sigma algebra defined by the outcomes. This measure is defined via Born’s rule, that is, the quantum state is embedded in the measure. An EPR experiment is essentially a state preparation protocol where deterministic operations are embedded in the quantum state (the unitary operations leading to its preparation), followed by the measurement, which results in stochastic outcomes. More generally, we can think of a larger system where we only measure a subsystem. This leads to quantum channels, which are described by completely positive trace preserving maps. A quantum channel, however, must be deterministic in the sequence of events, and, for instance, a measurement choice at a later time step cannot depend on the outcome of a previous measurement. We must factor in such classical and quantum memory effects, as well as the potentially indeterminate causal order of events. The quantum comb , 23 or p


Arthonia ulleungdoensis Lee & Hur is described as a new lichen species from South Korea. The new species is distinguishable from Arthonia ruana A. Massal. by its large, rounded and non-punctiform apothecia, taller apothecial section, asci with fewer spores, and larger and permanently colorless spores. Molecular analyses employing mitochondrial small subunit (mtSSU) and RNA polymerase subunit II (RPB2) sequences strongly support Arthonia ulleungdoensis as a distinct species in the genus Arthonia. Overall, 22 Arthonia species are currently recorded in South Korea. A surrogate key is provided to assist in the identification of all 10 taxa of Arthonia/Arthothelium with muriform spores in Northeast Asia.  Arthonia is one of the least explored species in lichen taxonomy although the genus is comprised of about five hundred species worldwide [Arthonia and related genera is unclear, and many Arthonia species have synonyms for Arthothelium or other genera. Molecular phylogeny also showed that Arthonia species are arranged over genera and even families [Arthonia is the ‘unstable structure’ of the species. Many species in Arthonia are highly pioneering. They disperse rapidly but become sterile with a deformed structure after spore discharge and dispersal in a short term. Therefore, although Arthonia species are encountered in the field, they are useless for analysis in many cases by the old, unfilled and barren ascomata with no ascus and spore. The third reason for the demanding genus is a ‘poor description’ of previous references as old descriptions are insufficient to explain specific characteristics of microlichens in anatomy, such as genus Arthonia particularly. The deficient depiction in the past is due mainly to the poor quality of microscopes, and still discourages a comparison with other species/specimens.The genus orldwide . One of families . AnotherArthonia received much less attention in Asia compared to Europe and America, several Asian countries have actively studied Arthonia. In particular, in Asia, India has explored the genus the most since 2000 and dynamic studies such as local detection, new species/records discovery, diversity assessment and ethnolichenological survey for Arthonia were reported [Arthonia species which were detected locally [Arthonia since 2013 and many new species and records, including lichenized and lichenicolous fungi, were discovered [Arthonia species were reported recently from China [Although the genus reported ,4,5,6. T locally ,8,9. Souscovered ,11,12. Fom China ,15.Arthonia. A concentrated field study was achieved in Ulleung Island and Pohang, which are an eastern island and seashore region of South Korea, during the summer of 2017  and a compound microscope  and imaged using a software program  and an Axiocam ERc 5s camera  mounted on a Zeiss scope A1 microscope . The ascospores were investigated at 1000× magnification in water. The length and width of the ascospores were measured and the range of spore sizes was shown with averages, standard deviation, and number of measured spores. Thin-layer chromatography (TLC) was performed using solvent systems A and C according to standard methods . Hand-cut sections of ascomata or thallus from all collected specimens were prepared for DNA isolation and DNA was extracted in line with the manufacturer’s instructions . Two-way PCR amplification for the mitochondrial small subunit (mtSSU) and RNA polymerase subunit II (RPB2) genes was achieved using Bioneer’s AccuPower PCR Premix  in 20-μL tubes and primers, mrSSU1 and mrSSU3R  and fRPBAll mtSSU and RPB2 sequences were aligned and edited manually using ClustalW in Bioedit . The bootstrap values were obtained in RAxML GUI 1.5 beta   using thArthonia/Arthothelium taxa to show the position of the new species in molecular phylogeny. The integrated tree shows that the new species is classified in the genus Arthonia . Arthonia ulleungdoensis exhibits important characteristics of the genus Arthonia, particularly A. sect. Arthonia, such as maculate ascomata, olive-brown ascomatal pigments, persistently colorless spores, and photophilous [Arthonia ulleungdoensis is identified as a unique species in both morphology and molecular phylogeny, being positioned in the genus Arthonia.A concatenated tree was produced from two different loci, which involved overall 70 sequences (35 sequences for each mtSSU and RPB2) . This coArthonia . Arthoniophilous , althougArthonia ulleungdoensis B.G.Lee & J.-S.Hur sp. nov. .Type: South Korea, Gyeongsangbuk-do, Ulleung-gun, Ulleung-eup, Ulleung forest trail, 37° 31.20′ N; 130° 54.29′ E, 300 m alt., on n = 6); bark layer hyaline, 10–15 μm; apothecial section 100–130 μm thick; epihymenium brown to dark brown, 15–20 μm high; hymenium hyaline to light brown, 50–60 μm high, oil droplets near base of hymenium or in hypothecium; hypothecium brown to dirty brown, 25–35 μm high; paraphysoids septate, anticlinally arranged, 1–1.5 μm wide, somewhat branched at tips; tips swollen and pigmented, 1.5–2 μm wide. Asci wide to narrow clavate, 2-, 4- or 6-spored, 32–49 × 20–31 μm (n = 5); ascospores hyaline, muriform, 4- to 8-transverse and 0- to 3-longitudinal septa, without a gelatinous sheath, old spores with dark septum, 20.5–31 × 9–13 μm (length n = 28). Pycnidia not detected.Thallus corticolous, crustose, hypophloedal, white to gray, without bleaching; epinecral layer 15–18 μm; medulla indistinct; photobiont trentepohlioid, forming a distinct algal layer, parallel to the substratum, between the epinecral layer and the brown bark layer or under apothecia, often in top layer of the brown bark, 45–54 μm thick, cells irregular, globose to angular, single or in chains, 4.5–6 × 6–9 μm, looking somewhat shrunken and supposed to be a little larger originally. Apothecia rounded, erumpent, black, epruinose, with or without epinecral bark layer on the surface of apothecia, 0.79–2.63 × 0.72–1.97 mm .Acer takesimense. It is currently known by a specimen in the type collection . Distribution and ecology: This species occur on the bark of Etymology: This species epithet is named for the collection locality, South Korea.Arthonia ulleungdoensis can be confused with Arthonia ruana in having black and epruinose apothecia, brown epihymenium, hyaline to light brown hymenium, brown hypothecium, K+ olive green reaction on an apothecial section, and the presence of a trentepohlioid alga [vs. up to 1.6 mm), a taller apothecial section (100–130 μm vs. 70–95 μm), taller hymenium (50–60 μm vs. 35–50 μm), 2- to 6-spored asci (vs. 8-spored asci), and larger spores (20.5–31 × 9–13 μm vs. 15–26 × 7–10.5 μm) without changing their colors when mature (vs. brown old spores) [Arthothelium norvegicum in having a white thallus, black and rounded apothecia without pruina, brownish epihymenium and hypothecium, colorless hymenium, K+ greenish apothecial section reaction, muriform spores, and the presence of a trentepohlioid alga [vs. 0.3–0.6 mm diam.), smaller spores (20.5–31 × 9–13 μm vs. 29–36 × 12–15 μm), and colorless spores when mature (vs. brown old spores) [Arthothelium fecundum Zahlbr. in having black, rounded and epruinose apothecia, dark-colored epihymenium and hypothecium, permanently colorless spores, similar ascospore size (20.5–31 × 9–13 μm vs. 25–27 × 9–11 μm), and the presence of a trentepohlioid alga [vs. up to 0.7 mm diam.), immersed thallus , no prothallus , 2- to 6-spored asci (vs. 8-spored asci), and I+ light blue hymenium (vs. I+ reddish hymenium) reaction [Notes: oid alga . However spores) . The newoid alga . However spores) . The newArthonia/Arthothelium with muriform spores from Northeast Asia (10 taxa): This key includes all Arthonia/Arthothelium species with muriform spores in Korea (4), Japan (4) and China (3). However, only one species, Arthothelium japonicum, is excluded from the key because its ascomata represent a perithecioid type [Arthonia/ Arthothelium species.Key to the species in oid type  and may Protococcus ………………………………………………………….Arthothelium collosporum1. Photobiont Trentepohlia ………………………………………………………………………… 2- Photobiont absent or 2. Hypothecium dark-colored ………………………………………………………………………………. 3- Hypothecium colorless or pale brownish ………………………………………………………………... 5Arthothelium feduncum3. Ascospores over 10 transversely-septate ……………………………………….- Ascospores less than 10 transversely-septate …………………………………………………………… 4Arthonia ulleungdoensis4. Ascospores permanently colorless but with dark septum when mature, 20.5–31 × 9–13 μm, 2–6-spored asci ……………………………………………………………………………- Ascospores brownish when mature, 15–26 × 7–10 um, 8-spored asci……………………………………………………………………….Arthonia ruana (Arthothelium dispersum)Arthothelium punctatum5. Ascospores up to 10 transversely-septate ……………………………………..- Ascospores less than 10 transversely-septate …………………………………………………………… 6Arthothelium spectabile6. Thallus usually delimited by a brown line ………………………………………- Thallus effuse without a delimited line ………………………………………………………………….. 7Arthothelium pertenerum7. Ascospores smaller, 16–20 × 7 μm …………………………………………….. - Ascospores larger, over 20 × 10 μm ………………………………………………………………………. 88. Ascospores 19–27 × 10–15 μm, photobiont not seen, hypothecium colorlessArthothelium scandinavicum…………………………………………………………………………………….Trentepohlia, hypothecium colorless or pale brownish ……………………………………………………………………Arthothelium scandinavicum var. japonicum- Ascospores 28–36 × 12–14 μm, photobiont


An implicit Euler finite‐volume scheme for a degenerate cross‐diffusion system describing the ion transport through biological membranes is proposed. The strongly coupled equations for the ion concentrations include drift terms involving the electric potential, which is coupled to the concentrations through the Poisson equation. The cross‐diffusion system possesses a formal gradient‐flow structure revealing nonstandard degeneracies, which lead to considerable mathematical difficulties. The finite‐volume scheme is based on two‐point flux approximations with “double” upwind mobilities. The existence of solutions to the fully discrete scheme is proved. When the particles are not distinguishable and the dynamics is driven by cross diffusion only, it is shown that the scheme preserves the structure of the equations like nonnegativity, upper bounds, and entropy dissipation. The degeneracy is overcome by proving a new discrete Aubin–Lions lemma of “degenerate” type. Numerical simulations of a calcium‐selective ion channel in two space dimensions show that the scheme is efficient even in the general case of ion transport. On a macroscopic level, the transport can be described by nonlinear partial differential equations for the ion concentrations  and the surrounding electric potential. A classical model for ion transport are the Poisson–Nernst–Planck equations ui and fluxes ith ion species is governed by the equationsi = 1, …, n, where u0=1−∑i=1nui is the concentration (volume fraction) of the electro‐neutral solvent, Di > 0 is a diffusion coefficient, β > 0 is the  inverse thermal voltage, and ith species. Observe that we assumed Einstein's relation which says that the quotient of the diffusion and mobility coefficients is constant, and we call this constant 1/β. The electric potential is determined by the Poisson equationλ2 is the  permittivity constant and f = f(x) is a permanent background charge density. Equations In the general case, the evolution of the concentrations ∂Ω is supposed to consist of two parts, the insulating part ΓN, on which no‐flux boundary conditions are prescribed, and the union ΓD of boundary contacts with external reservoirs, on which the concentrations are fixed. The electric potential is prescribed at the electrodes on ΓD. This leads to the mixed Dirichlet–Neumann boundary conditionsu‾i1≤i≤n and Φ‾ can be defined on the whole domain Ω. Finally, we prescribe the initial conditionsIn order to match experimental conditions, the boundary Aij(u)), defined by Aij(u) = Diui for i ≠ j and Aii(u) = Di(u0 + ui) is not symmetric and not positive definite. It was shown in Burger and coworkers u =  and u0=1−∑i=1nui, such that Bii = Diu0ui, Bij = 0 for i ≠ j provide a diagonal positive definite matrix, and wj are the entropy variables, defined byThe main mathematical difficulties of Equations ∂h/∂ui.We refer to , Lem. 7 u, Φ) ↦ w to entropy variables can be inverted, giving u = u withThe entropy structure of ui is positive and bounded from above, that is,Then C > 0 depends on the Dirichlet boundary data. Because ofu01/2ui and u01/2. Since u0 may vanish locally, this does not give gradient bounds for ui, which expresses the degenerate nature of the cross‐diffusion system. As a consequence, the flux has to be formulated in the terms of gradients of u01/2ui and u01/2 only, namelyThis yields The challenge is to derive a discrete version of this formulation. It turns out that Di are the same and the drift term vanishes). This suggests to use the entropy variables as the unknowns, as it was done in our previous work ui for the numerical discretization.Our aim is to design a numerical approximation of Vi = ∇u0 − βziu0∇Φ have the structure ∇v + vF, where ∇v is the diffusion term and vF is the drift term. We discretize V by using a two‐point flux aproximation with “double” upwind mobilities.We propose a backward Euler scheme in time and a finite‐volume scheme in space, based on two‐point approximations. The key observation for the numerical discretization is that the fluxes can be written on each cell in a “double” drift‐diffusion form, that is, both nonnegativity, which follows from a discrete minimum principle argument. It is well known that the maximum priciple generally does not hold for systems of equations. Therefore, it is not a surprise that the upper bound comes only at a price: We need to assume that the all diffusion coefficients Di are the same. Under this assumption, u0=1−∑i=1nui solves a drift‐diffusion equation for which the (discrete) maximum principle can be applied. In order to prove that the scheme satisfies an entropy‐dissipation inequality and also to complete the convergence analysis of the scheme successfully, we need a stronger additional assumption: We assume that the drift terms, and therefore the coupling with the Poisson equation, can be neglected. This means that our main results are obtained for a simplified degenerate cross‐diffusion system, no more corresponding to the initial ion transport model but still of mathematical interest. Nevertheless, the scheme we propose can be applied to the full ion transport model, and this is done in the last section of this paper.Under certain assumptions, the structure of the equations is preserved on the discrete level. Because of the drift‐diffusion structure, we are able to prove that the scheme preserves the ∂Ω. Mixed Dirichlet–Neumann boundary conditions could be prescribed as well, but the proofs would become even more technical. The main results are as follows.Di = D for all i, we prove the existence of solutions to the fully discrete numerical scheme  is an open, bounded, polygonal domain with ∂Ω = ΓD ∪ ΓN ∈ C0, 1, ΓD ∩ ΓN = Ø.Domain: H2T > 0, Di > 0, β > 0, and i = 1, …, n.Parameters: H3Background charge: H4uiI∈L∞Ω, u‾i∈H1Ω satisfy uiI≥0, u‾i≥0 and 1−∑i=1nuiI≥0, 1−∑i=1nu‾i≥0 in Ω for i = 1, …, n, and Φ‾∈H1Ω∩L∞Ω.Initial and boundary data: We summarize our general hypotheses on the data:A1∂Ω = ΓN, that is, we impose no‐flux boundary conditions on the whole boundary.A2Di = D > 0 for i = 1, …, n.The diffusion constants are equal, A3The drift terms are set to zero, Φ ≡ 0.For our main results, we need additional technical assumptions:Remark 1Di = D for all i, summing i = 1, …, n gives(Discussion of the assumptions). Assumption (A1) is supposed for simplicity only. Mixed Dirichlet–Neumann boundary conditions can be included in the analysis . For simplicity, we consider a uniform time discretization with time step △t > 0, and we set tk = k△t for k = 1, …, N, where T > 0, t = T/N. The domain Ω is discretized by a regular and admissible triangulation in the sense of [𝒯 of open polygonal convex subsets of Ω , a family xKK∈𝒯 associated to the cells. The admissibility assumption implies that the straight line between two centers of neighboring cells xKxL‾ is orthogonal to the edge σ = K|L between two cells K and L. The condition is satisfied by, for instance, triangular meshes whose triangles have angles smaller than π/2 [For the definition of the numerical scheme for ℰext=ℰextD∪ℰextN. For given K∈𝒯, we define the set K, which is the union of internal edges and edges on the Dirichlet or Neumann boundary, and we set We assume that the family of edges h𝒯=supdiamK:K∈𝒯. For σ = K|L, we denote by dσ = d the Euclidean distance between xK and xL, while for σ = d. For a given edge m(σ) denotes the Lebesgue measure of σ.The size of the mesh is defined by ζ > 0 such that for all K∈𝒯 and We impose a regularity assumption on the mesh: There exists This hypothesis is needed to apply discrete functional inequalities see , 17 and ℋ𝒯 of piecewise constant functions is defined byIt remains to introduce suitable function spaces for the numerical discretization. The space H1 norm on this space is given byThe (squared) discrete H−1 norm is the dual norm with respect to the L2 scalar product,The discrete Thenℋ𝒯,△t of piecewise constant in time functions with values in ℋ𝒯,L2) normFinally, we introduce the space ui∈ℋ𝒯 with values u‾i,σ on the Dirichlet boundary . Then we introduceFor the numerical scheme, we introduce some further definitions. Let σ = K|L, requiring that they vanish on the Neumann boundary edges, σ∈ℰext,KN. Then the discrete integration‐by‐parts formula becomes for u∈ℋ𝒯The numerical fluxes ∂Ω = ΓN, this formula simplifies toWhen 2.2K∈𝒯 and edges u0,KI=1−∑i=1nui,KI and u‾0,σ=1−∑i=1nu‾i,σ.We need to approximate the initial, boundary, and given functions on the elements K∈𝒯, k ∈{1, …, N}, i = 1, …, n, and ui,Kk−1≥0 be given. Then the values ui,Kk are determined by the implicit Euler schemeℱi,K,σk are given by the upwind schemeτσ is defined in 𝒱i,K,σk is the “drift part” of the flux,i = 1, …, n. Observe that we employed a double upwinding: one related to the electric potential, defining u^0,σ,ik, and another one related to the drift part of the flux, 𝒱i,K,σk. The potential is computed viaThe numerical scheme is as follows. Let u‾i,σ and Φ‾σ for σ∈ℰextD.We recall that the numerical boundary conditions are given by ui,𝒯,△t, Φ𝒯,△t the functions in ℋ𝒯,△t associated to the values ui,Kk and ΦKk, respectively. Moreover, when dealing with a sequence of meshes 𝒯mm and a sequence of time steps (△tm)m, we set ui,m=ui,𝒯m,△tm, Φm=Φ𝒯m,△tm.Remark 2u0,Kk and u0,σk are defined in ui,σk simplifies toWhen Assumptions (A1)–(A3) hold, the numerical scheme simplifies toui,σk, the upwinding value does not depend on i anymore such thatIn the definition of ∑i=1nui,σk from below in the proof of the discrete entropy inequality; see This property is needed to control the sum This formulation is needed in the convergence analysis.We denote by 2.3Theorem 1Let (H1)–(H4) and (A2) hold. Then there exists a solution  to Schemesatisfyinguk∈𝒟‾and, if the initial data lie in𝒟, uk∈𝒟. If additionally Assumptions (A1) and (A3) hold, the solution is unique.(Existence and uniqueness of solutions). Since our scheme is implicit and nonlinear, the existence of an approximate solution is nontrivial. Therefore, our first result concerns the well‐posedness of the numerical scheme.u0k=1−∑i=1nuik is nonnegative. Indeed, summing i = 1, ….n, we obtainAssumption (A2) is needed to show that ∑i=1nDiui,Kk=D1−u0,Kk, and we can apply the discrete minimum principle, which then implies an uik. This bound allows us to apply a topological degree argument; see u0k, which solves a discrete nonlinear equation, and then to show the uniqueness of uik for i = 1, …, n by introducing a semimetric d for two solutions uk=u1k…unk and vk=v1k…vnk and showing that it is monotone in k, such that a discrete Gronwall argument implies that uk = vk.Under Assumption (A2), it follows that Theorem 2Let Assumptions (H1)–(H4) and (A1)–(A3) hold. Then the solution to Schemeconstructed in theoremsatisfies the discrete entropy inequalitywith the discrete entropyand the discrete entropy production. mptions H–(H4) andThe second result shows that the scheme preserves a discrete version of the entropy inequality.∑i=1nui,σk. In the continuous case, this sum equals 1 − u0. On the discrete level, this identity cannot be expected since the value of ui,σk depends on the upwinding value; see ∑i=1nui,σk≥1−u0,σk; see Section k is the discrete counterpart of Assumption (A3) is required to estimate the expression Theorem 3Let (H1)–(H4) and (A1)–(A3) hold and let𝒯mand (△tm) be sequences of admissible meshes and time steps, respectively, such thath𝒯m→0and △tm → 0 asLet  be the solution toconstructed in theoremThen there exist functions u0, u =  satisfyinguxt∈𝒟‾,where u is a weak solution towith ΓN = ∂Ω), that is, for allϕ∈C0∞)and i = 1, …, n,(Convergence of the approximate solution). Let H–(H4) andThe main result of this paper is the convergence of the approximate solutions to a solution to the continuous cross‐diffusion system.u0,m1/2. The difficult part is to show the strong convergence of u0,m1/2ui,m, since there is no control on the discrete gradient of ui, m. The idea is to apply a discrete Aubin–Lions lemma of “degenerate” type, proved in lemma The compactness of the concentrations follows from the discrete gradient estimates derived from the entropy inequality 33.1i = 1, …, n. We show that this truncation is, in fact, not needed if the initial data are nonnegative. In the following let (H1)–(H4) hold.Lemma 4uik). Let  be a solution toThenui,Kk≥0for allK∈𝒯, k ∈{1, …, N}, and i = 1, …, n. IfuiI>0andu‾i>0then alsoui,Kk>0for allK∈𝒯, k ∈{1, …, N}.. Let Assumption (A2) hold and let  be a solution toandThenu0,Kk≥0for allK∈𝒯, k ∈ {1, …, N}. Ifu0I>0andu‾i>0then alsou0,Kk>0for allK∈𝒯, k ∈ {1, …, N}. hold. Then schemehas a solution  which satisfiesuKk∈𝒟‾for allK∈𝒯and. k = 0, we have uK0∈𝒟‾ by assumption. The function Φ0 is uniquely determined by scheme uk − 1, Φk − 1) with uKk−1∈𝒟‾. Let n and the number of cells K∈𝒯. For given K∈𝒯 and i = 1, …, n, we define the function uK0, , ui, σ, uσ0, , and u^0,σi are defined in F=Fi,Ki=1,…,n,K∈𝒯. Then u↦Fi,Ku0=mKui,K−ui,Kk−1/△t is affine.The function F = 0 satisfies u∈𝒟 or F = 0 with ρ ∈  satisfies We have proved above that any solution to F = 0 has the unique solution u = uk − 1 and consequently, The equation We argue by induction. For uk to F = 0 satisfying uk∈𝒟‾. Hence, uk solves the original scheme We infer the existence of a solution Lemmas 3.2The proof of Theorem Step 1: uniqueness for u0. If k = 0, the solution is uniquely determined by the initial condition. Assume that u0k−1 is given. Thanks to Assumptions (A2) and (A3), the sum of i = 1, …, n gives an equation for u0k=1−∑i=1nuik :u0 and v0 be two solutions to the previous equation and set w0: = u0 − v0. Then w0 solvesLet wK0, /D, sum over K∈𝒯, and use discrete integration by parts We multiply this equation by y|y| − z|z|)(y − z) ≥ 0 for y, z ↦ z|z|, that the third term is nonnegative, too. Consequently, the three terms must vanish and this implies that wK0,  = 0 for all K∈𝒯. This shows the uniqueness for u0.The first two terms on the right‐hand side are clearly nonnegative. We infer from the elementary inequality  = 0, we find after resolving the recursion thatSince ui,Kℓ are nonnegative and bounded by 1 for all K∈𝒯, for all ℓ ≥ 0 and for all 1 ≤ i ≤ n, it is clear that ∑ℓ=1kϵS3ℓ→0 when ϵ → 0. Then, we may perform the limit ϵ → 0 in the previous inequality yielding dϵ → 0. A Taylor expansion as in Zamponi and Jüngel [dϵukvk≥18∑K∈𝒯mK∑i=1nui,Kk−vi,Kk2. We infer that uk = vk, finishing the proof.As the densities 44.1Proof of Theorem 2logui,Kk,ϵ/u0,Kk,ϵ, where ui,Kk,ϵ≔ui,Kk+ϵ for i = 0, …, n. The regularization is necessary to avoid issues when the concentrations vanish. After this multiplication, we sum the equations over i = 1, …, n and K∈𝒯 and use discrete integration by parts to obtainThe idea is to multiply h(u) − h(v) ≤ h'(u)(u − v) for all u, The convexity of y, z > 0. Then, by the Cauchy–Schwarz inequality,In order to estimate the remaining terms, we recall two elementary inequalities. Let Inequality u0,Kk=1−∑i=1nui,Kk in A2 to find thatWe use the definition of B1 by using the abbreviation ui,σk,ϵ=ui,σk+ϵ:We rewrite B11. Indeed, if u0,Kk≤u0,Lk, we have ui,σk=ui,Kk and we use the first inequality in u0,Kk>u0,Lk then ui,σk=ui,Lk and we employ the second inequality in We apply inequality B2. In view of Assumption (A3), Equation Finally, we consider A2 such thatThe last expression cancels with A0, A1, B1, and A2 + B2, we deduce from Putting together the estimates for ϵ → 0, we infer that Since the right‐hand side converges to zero as First, we prove 4.2h𝒯 and time step △t. The scheme provides uniform v∈ℋ𝒯,△t byFor the proof of the convergence result, we need estimates uniform in the mesh size Lemma 7Let (H1)–(H4) and (A1)–(A3) hold. The solution u to schemesatisfies the following uniform estimates:where the constant C > 0 is independent of the mesh𝒯and time step size △t.(A priori estimates). k = 1, …, N to obtainWe claim that estimates t = 0 is bounded independently of the discretization, we infer immediately the bound for u01/2 in ℋ𝒯,△t. For the bound on u01/2ui in ℋ𝒯,△t, we observe thatSince the entropy at time ui,Therefore, together with the k = 0, …, N and using the estimates from the entropy inequality, we achieve the bound on u01/2ui.Then, summing over ϕ∈ℋ𝒯 be such that ϕ1,𝒯=1 and let k ∈ {1, …, N} and i ∈ {1, …, n}. We multiply the Scheme K and we sum over K∈𝒯. Using successively discrete integration by parts, the rewriting of the numerical fluxes ui, we computeIt remains to prove estimate i = 1, …, n,Δt−1u0k−u0k−1=−Δt−1∑i=1nuik−uik−1 follows from those for i = 1, …, n, completing the proof.This shows that, for 5In this section, we establish the convergence of the sequence of approximate solutions, constructed in theorem 5.1TKL as the cell with the vertexes xK, xL and those of σ. For TKσ as the cell with vertex xK and those of σ. Then Ω can be decomposed as𝒩K denotes the set of neighboring cells of K. The discrete gradient ∇𝒯,△tv on ΩT: = Ω ×  for piecewise constant functions v∈ℋ𝒯,△t is defined bynKL denotes the unit normal on σ = K|L oriented from K to L. To simplify the notation, we set ∇m≔∇𝒯m,△tm. The solution to the approximate scheme um0, , um1, , …, un, m.Lemma 8There exist functionsandsuch that, possibly for subsequences, as m → ∞,um0, ) is uniformly bounded in ℋ𝒯,△t. Indeed, by the First, we claim that , as defined in lemma We finish the proof of theorem ϕ∈C0∞) and let suppϕ⊂Ω‾×0Nm−1△tm . For the limit, we follow the strategy used, for instance, in Chainais‐Hillairet and coworkers Let The convergence results of lemma ϕKk=ϕxKtk, we multiply Scheme △tmϕKk−1 and sum over K∈𝒯m and k = 1, …, Nm. Thenm from now on to simplify the notation,Next, setting Fi0(m) − Fi(m) → 0 as i = 1, 2, 3. Then, because of F10(m) + DF20(m) − DF30(m) → 0, which finishes the proof. We start by verifying that F10(m) − F1(m) → 0. For this, we rewrite F1(m) and F10(m), using ϕKN=0:The aim is to show that ϕ and the uniform ui, we find thatIn view of the regularity of u0,σk1/2=u0,Kk1/2+u0,σk1/2−u0,Kk1/2, i.e.Using discrete integration by parts, the second integral becomesF20(m) = G1(m) + G2(m), whereFurthermore, we write F21(m) − G1(m) → 0, F22(m) → 0, and G2(m) → 0. This implies thatThe aim is to show that ϕ, by taking the mean value over TKL,C > 0 only depends on ϕ. It yieldsF21(m) − G1(m)| → 0 as First we notice that, due to the admissibility of the mesh and the regularity of F22(m) and G2(m). To this end, we remark that dσ≤h𝒯 and hence, together with the regularity of ϕ, and the Cauchy‐Schwarz inequality,It remains to analyze the expressions G2(m) can be estimated in a similar way.The term F30(m) − F3(m)| → 0. The is completely analogous to the previous arguments, sinceFinally, we need to show that |Fi0(m) − Fi(m)| → 0 for i = 1, 2, 3, and since F1(m) + DF2(m) − DF3(m) = 0, the convergence u solves Summarizing, we have proved that |61/2−) inside the channel region. These ions contribute to the permanent charge density f = −uox/2 in the Poisson equation, but also to the total sum of the concentrations. We consider three further types of ions: calcium , sodium , and chloride . While the concentrations of these ion species satisfy the evolution equations NA ≈ 6.022 × 1023 mol−1 is the Avogadro constant and utyp = 3.7037 × 1025 L−1 the typical concentration  and centers inside the channel region. On these discs, the oxygen concentration is set to Ek with respect to the stationary solution, whereui,K∞Φ∞ is the steady state determined from the boundary data. Figure L1 norms of the concentrations and electric potential decay with exponential rate. Interestingly, after some initial phase, the convergence is rather slow and increases after this intermediate phase. This phase can be explained by the degeneracy at u0 = 0, which causes a small entropy production slowing down diffusion. Indeed, as shown in Gerstenmayer and Jüngel The simulations suggest that the solution tends towards a steady state as λ in the Poisson equation has the value λ2 = 4.68 × 10−4. Therefore, the drift term is moderately convective (in the sense that the modulus of the electric field |∇Φ| is moderately large). Figure L1 error for a smaller value of λ = 2 × 10−4. It turns out that the scheme is still entropy dissipative, but the entropy decay is much slower and the L1 error decreases with smaller rate.The scaled permittivity constant h𝒯≈0.01. This mesh is obtained from the coarse mesh by a regular refinement, dividing the triangles into four triangles of the same shape. The reference solution is compared to approximate solutions on coarser nested meshes. In Figure L1 norm between the reference solution and the solutions on the coarser meshes at two fixed time steps k = 50 and k = 1,400 are plotted. We clearly observe the expected first‐order convergence in space.Since Assumptions (A1)–A3) are not satisfied in our test case, the convergence result of theorem  are not