Ed in relation to the polarization is studied. In particular, the value of permittivity for Ed (εd) in prototypical situations of ferroelectrics, including Mehta formula, is examined by ab initio calculations. By using spontaneous polarization PS corresponding to accurate experiment ones, we show εd = 1, which suggests that the results of εd ≫ 1 indicate hidden mechanisms; εd = 1 suggests that the effect of Ed is significant to induce intriguing important phenomena overlooked by εd ≫ 1. A bridge between εd = 1 and εd ≫ 1, i.e. the consistency of εd = 1 with conventional results is presented. The exact electrostatic equality of head-to-head–tail-to-tail domains to free-standing ferroelectrics is deduced. Hence, most stoichiometric clean freestanding monodomain ferroelectrics and head-to-head–tail-to-tail domains are shown unstable regardless of size, unless partially metallic. This verifies the previous results in a transparent manner. This conclusion is shown consistent with a recent hyperferroelectric LiBeSb and “freestanding” monolayer ferroelectrics, of which origin is suggested to be adsorbates. In addition, this restriction is suggested to break in externally strained ultrathin ferroelectrics. The macroscopic formulas of Ed are found valid down to a several unit-cells, when electronic and atomic-scale surface effects are unimportant and accurate PS is used.Electrostatics of depolarization field  PS that is useful in various applications, for which high insulativity is desired. Therefore, ideal insulativity of FE is assumed in most studies. In such high insulativity FEs, the depolarization field Ed exists universally even in the absence of external electric field Eext, owing to the charge − ∇·PS originating from inhomogeneity or the existence of surface; For a homogeneous PS, Ed disappears for infinite FE without surface or FE with no surface effect and ideal metal electrodes.Ferroelectrics (FEs) have reversible spontaneous polarization Ed are unnecessary in ab initio calculations, they are indispensable for non ab initio examinations of PS configurations, stability of domains, and critical thicknesses of FEs17. These formulas use the permittivity for Ed (εd), of which difference affects critically the results. No controversy exists for εd = εf used in combination with an initial PS that is an ideal bulk PS for no macroscopic field in FE as in Kittel models2. Here, εf is static linear permittivity εr of FE. Otherwise, the choice of εd is controversial.Although analytical formulas of P under Eext is P = PS + (εf − 1)ε0Eext (ε0: vacuum permittivity) and the permittivity of FE underEd is still εf ≫ 1 ] thin films by using Ed = − PS/εfε0 with PS from remnant polarization measurements (7 μC/cm2) and εd = εf = 10. In the electrostatic study of Tian et al.7, Ed in BiFeO3 thin films was estimated by using εf = 60 (in Eq. (1) of Ref.7), where this P is initially a total polarization of a single domain state. Kim et al.8 estimated Ed in BaTiO3 (BTO) ultrathin films through the formula by Mehta4 with the experimental remnant P obtained from pulse train methods and εd = εf = 80 from εf − Eext curves. In the Ginzburg–Landau–Devonshire (GLD) theory of BTO ultrathin films by Jo et al.9, Ed was given through the Mehta formula4 with the same P as P in the GLD equation and εd = εf = 80 of Ref.8, where P in GLD theory is the total polarization. Schroeder et al.10 estimated Ed in HfO2 and PZT ultrathin films through the Mehta formula4 with experimental PS and εd = εf = 20–300. Similar analyses with εd = εf ≫ 1 are frequently employed17.Examples of , a primitive considerations show εd = 1 for PS, i.e. a total PS (≡ P(Ef(Eext = 0)))13, where Ef is the total macroscopic electric filed in FE. For freestanding FEs, for example, Ef(Eext) = Eext − P(Ef(Eext))/ε0 or Ef(Eext) = Eext/εf − P(Ef(Eext = 0))/ε0 )/ε0 “”. This iεd is due to explanations based on macroscopic quantities. Because macroscopic explanations are abstract, they are unsuitable to bridge the gap between two conflicting views of εd. On the other hand, ab initio estimation of εd is considered as the clearest method for this problem but is not reported to our knowledge. Hence, we clarify εd in the formulas of Ed, by ab initio simulations in which ab initio PS is exactly P(Ef(Eext = 0)), which is considered as PS obtained by accurate experiments of ideal samples. Here, the standard theoretical assumptions: pure, stoichiometric, clean FEs are used.We think that the existence of this controversy on Ed is related to fundamental issues such as stability of monodomains, critical thicknesses of FE, and the emergence of ferroelectricity in superlattices. Some of these subjects require the consideration of other effects such as strain-induced FE and electronic effect at electrodes11. To avoid the complexity, we concentrate on free-standing insulating FE and its electrostatic identicals, i.e. head-to-head–tail-to-tail (HH–TT) domains. Thus, we estimate the value of εd in the formula of Ed in a clear manner.11, it may be argued that the formula of Ed based on electrostatics is not possible for nm-FEs. We resolve this by focusing on the formulas of Ed and using ab initio PS in the formulas. Therefore, PS in these formulas contains the effects of the interactions in slabs or superlattices, whereas the absence of these effects in conventional studies has limited the applicability. The use of nonlinear εf is often better than linear εf but can be approximated by an average linear εf12. Therefore, the conclusions below are applicable also to the nonlinear εf (≫ 1).As expected from the electronic interaction at the electrodeEext = 0, where FEs with thickness lf are in periodic slabs  superlattice, respectively, while the latter mimics an inhomogeneous FE. Iadj stands for both vacuum and insulator, which is dielectric or FE having different PS. The polarization, thickness, and permittivity of Iadj are PI, lI, and εI, respectively, and the thickness of slab is lSC = lf + lI (lV). The angles of the polarization P of FE and Iadj to the slab direction are θ and θI, respectively.For simplicity, we discuss 1-dimensional (1D) cases with ect Fig.  but to tϕ) of these models are represented by Fig. Ed (Edab initio) was obtained from the envelope of the peak tops of atomic electrostatic potential, of which example is Fig. 3 (BTO) capacitors are examined, where metal layers are standard electrode materials for FEs: SrRuO3 or Pt and ~ 20 Å  BTO in vacuum is metallic, when FE is enforced18. To achieve insulativity, we used tetragonal (P4mm) SrTiO3 (STO) of which a-axis lattice constant increased by 0.5% and decreased by 0.01% from that of the theoretical cubic phase. For these a-axis lattice constants, bulk STO was FE19. We refer to them as STO1.005 and STO.9999, respectively, of which bulk PS’s were 3.56 μC/cm2 and 6.15 μC/cm2, respectively, by VASP28.Accurate estimation of Ed are obtained in a following manner. The normal component of P of FE (P⊥) under Ed is P⊥ = PS cos θ + (εd − 1) ε0Ed in standard approaches17. The equation of continuity of electric flux is PScos θ + εdε0Ed = PI cos θI + ε0EI, where EI is the macroscopic electric field in Iadj. The validity of this continuity in the presence of peaks at the surfaces /ε0(εd + lf/lI). In the present study, PS and PI in Eq. ε0EI, Eq. .For ≠ 0, Eq.  is Ed = Ed in FE capacitors, because equations of continuity of electric flux similar to the above hold; A short-circuited FE capacitor is modelled as a perfect-metal/insulator(lI/2)/FE/insulator(lI/2)/perfect-metal29, where the perfect metal refers to a metal with zero screening length and the thickness of each screening layer λ is lI/2. Assuming PI = εIε0EI in screening layer, Eq.  is the distance between an outermost electron density and a center of ion position   lf is thd layer “”; The efεd was examined through the comparison of ab initio Ed with Ed of Eq. (εd. For FE/vacuum (PI = 0), PS in Eq.  t tεd was S in Eq.  was the I in Eq.  and (2) Ed’s by Eq. , which, with εd = 1, quantitatively agrees with bandgap Eg that decreases with lf and lV in Fig. In Fig. Ed’s by Eq. . Equatulation “ with εd Ed by Eq. ) and Ed ≡ Ef(Eext = 0). When the potential difference between the electrodes is V, Eq. (Ef(V) = V/(lf + lI/εI) − P(Ef(Eext))/ε0(1 + εIlf/lI)13. EI, the field in the screening layer of the electrode, is EI(V) = V/εI(lf + lI/εI) + P(Ef(Eext))lf/ε0lI(1 + εIlf/lI).Using the total electric field in FE s V, Eq.  changes Ef(V) can be written as Ef(Eext) = Eext + Ed(Ef(Eext)), where Eext ≡ V/(lf + lI/εI) (= V/lf for lf ≫ lI/εI) and Ed(Ef(Eext)) ≡ − P(Ef(Eext))/ε0(1 + εIlf/lI) similar to Eq. (r to Eq. .εf is defined by εf − 1 = {P(Ef(Eext)) − PS}/ε0(Ef(Eext) − Ed), where Ed = Ef(Eext = 0) = − PS/ε0 for lf ≫ lI/εI, and εf (≫ 1) is linear for |Eext| ≪ |PS|/ε0. Hence, P(Ef(Eext)) = PS + (εf − 1)(ε0Ef(Eext) + PS) for lf ≫ lI/εI.P(Ef(Eext)) in the expression of Ef(Eext) yields Ef(Eext) = Eext/εf − PS/ε0 for lI/εI ≫ lf (freestanding), suggesting that the measured permittivity is εf.The substitution of this Q = εIε0(EI(V) − EI(V = 0)) = V/(lf/εfε0 + lI/εIε0)13 from the above expression of EI(V). This is equal to ΔQ = CV, where C = (Cf−1 + CI−1)−1  is a series capacitance per area. In particular, for lf ≫ εflI/εI, ΔQ = εfε0V/lf = Cf V. Therefore, the permittivity of FE under Ed = Ef(Eext = 0) = − PS/ε0 is εf.We show an example: Δεd = 1 shows D = PS − PS/(1 + εIlf/lI), while D = Q. Because lI/εI ~ 0.1 Å, the difference between the real PS and the measured PS is detectable only for lf < 10 Å. As for potential difference, Eq. (Ed = − PSlI/(εIε0lf) for lf > 10 Å, because lI/εI is short ~ 0.1 Å. Therefore, the potential difference across the capacitor is independent of the FE thickness lf, when the quality of FE is independent of lf and FE is ideally stoichiometric.Additionally, Eq.  with εd nce, Eq.  is well Here, we discuss only monodomain FE Fig. c.P in standard GLD theories34 are formulated with a single total polarization, εd = 1 should be used in standard GLD theories.Because the polarization P in response to Eext is εf (≫ 1) even for FE under Ed. By the same logic, the change of P by built-in internal field Ebi is also expressed by εf (≫ 1), where Ebi is not due to P or a dipole that is not expressed by P. Ebi exists in FEs by various mechanisms such as the diffusion potentials at pn and Schottky junctions and chemical orders, e.g. LaAlO3 in the polar catastrophe model.The preceding results have shown that the permittivity that expresses the change of PS = 0 and Ed = 0 in a bulk cubic BTO. However, Ebi ≠ 0, when the surfaces of a cubic BTO slab are asymmetrically terminated to form a dipole, e.g., BaO/TiO2/BaO/…/TiO2/BaO/TiO2. Hence, to achieve Ebi = 0, the present study used chemically symmetric slabs , for which Eq.  are bulk Eg normalized by 2 eV and PS of bulk FE at TC normalized by 10 μC/cm2, respectively, and the unit of lf and lV is Å. PS of bulk FE at TC approximates the critical PS of FE that is about to become paraelectric by Ed12. Equations , Eq. (lf < 1.8 Å(Eg/2 eV)/(PS(TC)/10 μC/cm2). This suggests that freestanding FEs with normal bulk properties are FEs with metallic layer or insulating paraelectrics15 as explained by the following GLD analysis. This conclusion is valid also for HH–TT domains with θ = 0.The giant permittivity and large piezoelectric coefficients of FE are regarded as an  0), Eq.  yields lP as the order parameter. We approximate the polarization possibly missed in such GLD theory29 by an extra permittivity εNG − 136, while εNG is speculatively close to electronic permittivity29. The GLD energy F of an insulating FE is F = (T − T0)P2/2Cε0 + βP4/4 + γP6/4 − PEd/2, where T0, C, β, γ, and θ are Curie–Weiss temperature, Curie constant, and GLD coefficients, respectively. The effect of strain can be incorporated in T0 and β37. Curie temperature TC is T0 + ΔT, where ΔT = 3β2/16γCε0. For 2nd order transition, γ = 0, β > 0, and TC = T0. The effect of Ed = − PS/ε0εNG is the change of T0 to T0 − C/εNG in F, where Eq. /Cε0 + 3βP2 + 5γP4}ε0 at T < TC. Stable state satisfies ∂F/∂P = (T − T0)/Cε0 + βP2 + γP4 = 0. Therefore, χGL = 1/(4T0/Cε0 − 2βP2)ε0 < C/4T0 at T = 0 (We assume T0 > 0), and TC = T0 + ΔT = T0 + 3β2/16γCε0. Because ΔT ≪ T0 in almost all FEs34, we may assume TC < 2T0. Under this assumption, χGL < C/2TC at T = 0, and C < εNGTC means χGL < εNG/2 at T = 0. εf = χGL + εNG < 3εNG/2 at T = 0, which is < 7.5 for εNG = 5.Similarly, for FE undergoing 1st order FE transition, it is known that εf’s of FEs undergoing 2nd and 1st transitions appear too small for experimentally observed bulk metal oxide FEs. Therefore, freestanding insulating FEs satisfying C < εNGTC are unlikely to exist, unless εNG is far larger than 5; That is, for freestanding FE materials, there exists virtually one choice between a partial loss of insulativity and a loss of FE.These θ = 0), C < εNGTC was shown, while Δϕ ~ PS(Ed)lf/ε0. Therefore, FE materials having a very large εNG (≫ 5) can retain FE and remain insulating, when ultrathin. Such FE materials are unlikely to exist. Alternatively, we may consider electrically freestanding FE or FE with clean surface that is not mechanically freestanding. In this case, T0 (~ TC) of common FEs increases to T0eff by inplane strain, while ab initio calculations shows that T0eff is much larger than those of standard GLD theories37. Therefore, heavily strained FE materials may retain FE and remain insulating, when ultrathin (Formula estimating an effective T0 from ab initio PS is Ref. [88] of Ref.37). The above indicates that εf of such FE is extremely low for Ed = 0 but can be large for Ed ≠ 0, because the coefficient of the first term GLD energy F is (T − T0eff + C/εNG)/2Cε0 (freestanding).For freestanding insulating FE  in vacuum  that retains FE down to monolayer. This appears to contradict both the reports of metallicity at HH–TT domains of BFO and the present results, esp. the single choice between insulating paraelectric and partially metallic FE.Mechanically freestanding FE is customarily referred to as freestanding; Ji et al.εd = εf = 100, the potential difference Δϕ across freestanding insulating BFO of 1–4 unit-cell thickness with a moderate PS ~ 20 μC/cm2 is 0.09–0.36 V by Eq.  and may move ions43, those by Ji et al.42 are not that of freestanding FE. Consequently, all the measurements of Ji et al.42 do not contradict the conclusion of the present paper.Hence, we shall look at the measurements of Ji et al.Ed-limited domain and size effect, which was later attributed to adsorbates44. This agrees with recent ab initio study45. Further, photoemission spectroscopy in UHV showed that SrTiO3 surface was covered by adsorbates even in ultrahigh vacuum (UHV)46. Actually, the free surface with PS ⊥ surface is insulator-like in air and metallic in UHV when cleaned14. Because the insulating freestanding FE42 was exposed to air and water, we suggest adsorbates as its hidden mechanism.More importantly, the interdisciplinarity of nano FE hides true mechanisms. In the present case, “freestanding” is defined by surface science and electrostatics. For example, Fong et al. found monodomain FE of 3 unit-cell thickness as opposed to Ed, especially, the value of permittivity εd in the formula of Ed by ab initio simulations, where ab initio PS corresponded accurately to experimental PS. For this, the standard theoretical assumptions: pure, insulating, stoichiometric, and clean FEs were used. To validate the analyses of Ed based on electrostatics, we concentrated on the formulas of Ed for accurate ab initio total P(Ef(Eext = 0)) that contained various atomic effects and corresponded to experimental PS. Further, we focused on the simplest cases of Ed: freestanding 1D-FE, HH–TT domains, and superlattices that mimicked inhomogeneous FE and FE/dielectric.We studied the electrostatics of εd = 1 ± 0.06–1 ± 0.2. That is, εd = 1 should be applied to experimental and standard-GLD PS’s. A contradiction between εd = 1 and εd = εf was resolved by a bridge; Even under Ed, the permittivity for Eext and built-in field Ebi was εf. Therefore, if a study requires εd ≫ 117, the value of PS is incorrect, the values of the parameters are inappropriate, or, most likely, hidden screening mechanisms exist48.The present ab initio simulations showed lI = ∞), Eq. (Ed = − PS/ε0 (or Ed = − PScos θ/ε0), while, for HH–TT insulating domains, Eq. , Eq.  yields Eins, Eq.  with PI Ed and the FE free energy of insulating freestanding and HH–TT FEs scale linearly with lf. This implies that the stability of 1D-freestanding and HH–TT insulating FEs is independent of size15, when the energy increase by surface effect and domain walls energy is ignored. A strain effect to overcome this restriction was suggested.Consequently, both the electrostatic energy of lfPS/ε0 < Eg/e by εd = 1, the insulativity required an extremely small bulk PS ≪ 1 μC/cm2 or paraelectricity  required a partially metallic FE. This conclusion verified the previous results41 in a material-independent manner and was confirmed also for hyper-FE LiBeSb that was reported to be insulating in FE/paraelectric38. This conclusion appeared inconsistent with “freestanding” monolayer BFO42. But, the examinations of experimental procedures42 suggested adsorbates as a hidden mechanism46.Because ity Fig. . AlternaEd 28, which was used also for BTO/Pt. In the slab calculations, graphic processing units acceleration50 was used. The supercells were produced by VESTA51.The ab initio calculations with VASPεd, accurate estimations of a total polarization PS under Ed are essential. Because we compare Eqs.   εd, accuPSfor given ion positions is accurate but only possible for insulators. For example, the present Berry phase calculations yields PS of bulk BaTiO3 that agree with experimental PS within 4%, when experimental ion positions and lattice constants of at 303 K are used37.Berry phase calculation of PS, the dipole moment of a whole slab was calculated with Berry phase; We treated these slabs as unit-cells to apply Berry phase calculations directly, unlike conventional approaches. PS was obtained by dividing the dipole moment by the volume of FE part of the slab. These PS’s were referred to “rigorously calculated PS’s of the slab” and obtained for all the FE/vacuum and BTO/STO slabs. Here, STO1.005 and STO.9999 slabs are insulating, allowing accurate Berry phase calculations.Therefore, to obtain accurate PS of the unit-cell that possessed exactly the same ion positions as those in the slab was calculated with Berry phase and, then corrected with atomic polarization by Ed by the procedures in Ref.29. These PS’s agree perfectly with “rigorously calculated PS’s of the slab”, which further confirmed the accuracy of the present PS’s of FE/vacuum and BTO/STO. These corrected PS’s29 were used for capacitors. Therefore, in the present study, PS’s are accurate total PS’s and self-consistent with Ed. Hence, PS’s in Figs. Additionally, Equations  and 2)  2) are aσ+R = σ+L and σ−R = σ−L as expected from their origin, where σ+R, σ−R, σ+L, and σ−L are positive and negative charge densities that yield the right and left peak, respectively. Because of the charge neutrality of FE, σ+R + σ+L + σ−R + σ−L = 0, i.e. σ−R + σ+R = 0. Therefore, the continuity of the electric fluxes DFE in FE and DI in Iadj .Because surface buckling in vacuum is electrostatically dipole due to ion displacements, the arguments exactly the same as the above hold. Therefore, the electric flux of the inside lf (lfeff) was estimated from the planer averaged electron density ρ profiles29. Below, z = 0 corresponds to the position of bottom ion. Because ρ at z = − 0.8 Å was same as the minimum ρ of inner part in all the ρ–z curves, the region of z = 0 ~ − 0.8 Å was considered as a part of FE (λsmear = 0.8 Å), and lfeff was lfeff = lf + 2λsmear. In addition, λsmear = c/2  was also tested, and lV = lSC − lfeff. For BTO/STO, lf was defined as the distance between the top and bottom Ti ions of BTO , for which the quantum mechanical smearing29 may be responsible. The estimations with lT-B − 1.5ucBTO were also tested. The effective thicknesses of the screening layer, i.e. the effective passive layer lI/2εI of BTO/SrRuO3 and BTO/Pt were estimated as 0.1 Å and approximately 0.05 Å, respectively29.For FE capacitors,


Scientific Reports 10.1038/s41598-020-60750-8, published online 12 March 2020Correction to:The Article contains errors.In the Results section under the subheading “Identifying differentially expressed language features prior to a hospital visit” “Dictionary-based: Prior to ED visits, patients less more likely to post about leisure (d =  − 0.225), associated words such as ‘family’, ‘fun’, ‘play’, ‘nap’, internet slang (netspeak) (d =  − 0.374) words such as ‘:)’, ‘fb’, ‘ya’, ‘ur’, and informal language (d =  − 0.345) with words such as ‘u’, ‘lol’, ‘smh’, ‘da’. Patients also use personal pronouns less (d =  − 0.345) prior to ED visits compared to random time windows.”should read:“Dictionary-based: Prior to ED visits, patients were less likely to post about leisure (d =  − 0.225) with words such as ‘fun’, ‘play’, ‘nap’, internet slang (netspeak) (d =  − 0.374) such as ‘u’, ‘da’, ‘smh’, and informal language (d =  − 0.345) with words such as ‘lol’, ‘:)’, ‘b’.”In Table 2 under “Categories that decrease in usage before emergency visit”, the row “1st person singular” is a duplication of row “informal speech”.In Table 2 under “Change in linguistic topics”, the row “even, still, tho, though, yet, blah” is a duplication of the row “kids, child, their, children, mother, father”.The correct Table 2 appears below as Table 1In Table 3 under “Categories that decrease in usage before inpatient visit”, the row “1st person singular ’, ‘b’)” is a duplication of the row “Informal Speech ’, ‘b’)”.In Table 3 under “Change in Linguistic Topics”, the row “even, still, tho, yet, blah, mad” is a duplication of the row “kids, child, their, children, mother, father”.In Table 3 under “Topics that decrease in usage before inpatient visit”, the row “better, feeling, little, hope, bit, type” is a duplication of the row “lol, ha, ctfu, lmao, funny, haha” and the row “no, what, matter, how, always, end” is a duplication of the row “:), show, crew, awesome, fashion, guys”.The correct Table 3 appears below as Table 2These corrections do not affect the conclusions of the Article.


We confirm Kreps' conjecture if the consumer's utility function U has asymptotic elasticity strictly less than one, and we provide a counterexample to the conjecture for a utility function U with asymptotic elasticity equal to 1, for ζ such that E[ζ3]>0.We examine Kreps' conjecture that optimal expected utility in the classic Black–Scholes–Merton (BSM) economy is the limit of optimal expected utility for a sequence of discrete‐time economies that “approach” the BSM economy in a natural sense: The  The price process for the stock is generated as follows: For an i.i.d. sequence {ζj;j=1,2,…}, where each ζk has the distribution of ζ, the law for the price of the stock at time k/n isξ(0)≡0 and S(0)≡1.) At time 1, the bond pays a consumption dividend of 1, and the stock pays a consumption dividend of S(1) defined as above.Fix a random variable ζ with mean 0, variance 1, and bounded support. For Ω=C0. Let ω denote a generic element of Ω, with ω(t) the value of ω at time t. Endow Ω with the sup‐norm topology; let F denote the Borel σ‐field, and let {Ft;t∈} denote the standard filtration on Ω. For each n, let Pn be the probability measure on Ω such that the joint distribution of (ω(0),ω(1/n),…,ω(1)) matches the distribution of (ξ(0),ξ(1/n),…,ξ(1)), and such that ω(t) for k/n<t<(k+1)/n is the linear interpolate of ω(k/n) and ω((k+1)/n). And let S:Ω→R+ be defined by S=eω(t).Embed this model into the standard state space Pn⇒P, where P is Wiener measure on C0; that is, ω under P is a standard Brownian motion, starting at ω(0)=0, and S(ω) under P is geometric Brownian motion, so that P, together with the riskless bond, prescribes the simple continuous‐time economy of Black and Scholes  the information she possesses at time k/n, on which basis she trades, is [only] the history of the stock price up to and including time k/n, and (b) any purchase of stock after time 0 is financed by the sale of bonds, and the proceeds of any sale of stock are invested in bonds), seeking to maximize the expectation of a utility function U:→R applied to the final dividend generated by the portfolio she holds at time 1.We imagine an expected‐utility‐maximizing consumer who is endowed with initial wealth nth discrete‐time economy (where the stock and bond trade (only) at times 0,1/n,2/n,…,(n−1)/n), does the optimal expected utility she can attain approach, as n→∞, what she can optimally attain in the continuous‐time BSM economy?The question that forms the basis for this paper is: If we place this consumer in the un(x) be the supremal expected utility she can attain in the nth discrete‐time economy if her initial wealth is x, and let u(x) be her supremal expected utility in the BSM economy. Kreps (liminfnun(x)≥u(x). And he proves limnun(x)=u(x) in the very special cases of U having either constant absolute or relative risk aversion. But he only conjectures that the second “half,” or limsupnun(x)≤u(x) is true for general (concave and differentiable) U.Let y. Kreps  obtains asymptotic elasticity of utility from Kramkov and Schachermayer (limnun(x)=u(x) if the utility function U has asymptotic elasticity less than 1. However, we show by example that if the asymptotic elasticity of U is 1, it is possible that u(x) is finite while limnun(x)=∞, both for all x>0.Employing the notion of hermayer   like the continuous‐time limit, at least for the BSM continuous‐time limit. But what if ζ has support of, say, size three, but there are only the two securities? Markets are incomplete for any finite n; does this incompleteness mean very different economic outcomes? Or, if the probability laws Pn that govern the discrete‐time security‐price processes converge weakly to P, is it then true that limnun(x)=u(x)?Our interest, motivated by the discussions in Kreps , is in cU has constant relative risk aversion with risk‐aversion parameter less than 1/2, the optimal strategy in the BSM economy is to short‐sell bonds, leveraging to achieve a (fixed) fraction greater than 100% of current wealth in the risky asset. Suppose that, in our special discrete‐time setting, where the security‐price process is driven by scaled copies of a single random variable ζ, ζ has support that is unbounded below. Trying to achieve such a leverage strategy in any of the finite‐time economies would give a positive probability of bankruptcy, which is incompatible with these utility functions. The best an investor can do for large enough n in these circumstances is to hold 100% of her wealth in the risky asset, which results in limnun(x)<u(x).It is already known that weak convergence is insufficient. Merton  observesPn to P alone does not preclude the possibility of asymptotic arbitrage =∞, even when u(x) is finite valued =u(x) for all x>0 if U has asymptotic elasticity less than 1. But if U has asymptotic elasticity of 1, even if markets are complete for each n, convergence can fail, and fail in spectacular fashion.By assuming in our setting that ζ has bounded support, we avoid the first problem. And, in our setting, asymptotic arbitrage is precluded n=1∞ are general semi‐martingales and the limiting market S is a continuous semi‐martingale. Also, the utility function U in Bayraktar et al. ((Sn) is induced by a single  random variable ζ and S is geometric Brownian motion. In this more general setting, they make assumptions sufficient to show that limnun(x)=u(x). The key assumptions in Bayraktar et al. =∞ and limx→∞U′(x)=0. Moreover, without loss of generality and for notational convenience later, we assume unless otherwise specified that U is normalized so that limx→∞U(x)>0, without precluding the possibility that limx→∞U(x)=∞.  can be either finite or −∞.)The utility function We always assume the followingV denote the conjugate function to U:I:→ be the inverse of U′; that is, I(y)=(U′)−1(y). Then for every y∈, V(y)=U(I(y))−yI(y).Let y→V(y) is strictly convex, continuously differentiable, and strictly decreasing.The function V(∞)=U(0) and V(0)=U(∞), where the values of U and V at 0 and ∞ are interpreted as the limits as x and y approach 0 and ∞, respectively.V′(y)=−I(y), so limy→0V′(y)=−∞ and limy→∞V′(y)=0.U(x)=infy>0[V(y)+xy],forx>0.We let asymptotic elasticity of U, defined in Kramkov and Schachermayer , is defined byU(x)=xα/α for α∈, then AE(U)=α.The notion of hermayer , plays aU implies that AE(U)≤1 in all cases; if U is bounded above and if U(∞)>0, then AE(U)=0. But if U(∞)=∞, AE(U) can equal 1; an example is where U(x)=x/ln(x) for sufficiently large x.The concavity of (U)<1, which derives from a comparison of the average and marginal utilities provided by U as the argument of U approaches ∞: AE(U)<1 is equivalent to:x0. As(U)<1 is equivalently:x0.Many of our results depend on the assumption that AElimx→∞−xU′′(x)/U′(x), exists and is strictly positive, then limx→∞xU′(x)/U(x) exists and is less than 1; that is, U has asymptotic elasticity less than 1. As it is believed to be “common” for economic agents to have nonincreasing relative risk aversion, this belief implies that agents with this common property have asymptotic elasticity less than one.As noted in Schachermayer , the con4P∗; that is, a probability measure on Ω that is probabilistically equivalent to P and such that {S;t∈} is a martingale . This measure P∗ has Radon–Nikodym derivative with respect to P given byX that she can afford, where what she can afford is given by the single budget constraint EP∗[X]≤x, where EP∗[·] denotes expectation with respect to P∗. Hence, with wealth x, the consumer's problem is tou(x) is the supremum of expected‐utility level that the consumer can achieve in the BSM economy, starting with wealth x.As is well known, the continuous‐time BSM economy admits a unique equivalent martingale measure (emm) denoted by Z:C0→ byZ is the unique continuous (in ω) version of the random variable dP∗/dP. Of course, EP∗[X]=EP[X·Z] for any random variable X such that (at least) one of the expectations makes sense. And, in this notation u(x)=sup{EP[U(X)]:EP[X·Z]≤x}.It is convenient for later purposes to define the density function 4.1.u(x)≥U(x) .4.2.x→u(x) is continuously differentiable, strictly increasing, and concave.4.3.u(x0)<∞ for any x0>0, then u(x)<∞ for all x>0.From 4.1 and 4.2, if 4.4.u(x)<∞, and if the consumer's problem has a solution , then there exists y(x)>0 such that the solution has the form X(ω)=I(y(x)·Z(ω)), where y(x)=u′(x) and (as noted earlier) I=(U′)−1.If We have the following from Cox and Huang , Karatzax>0, then it has a solution for all wealth levels x′>0 such that x′<x.If the consumer's problem has a solution at wealth level x, u(x)<∞ and yet the supremum that defines u(x) is not attained by any contingent claim X.  this is true for some finite x, let x¯ be the infimum of all x for which there is no solution (but u(x)<∞); there is a solution at x¯, and so the range of x for which there is a solution is the interval  is continuously differentiable and the “Lagrange multiplier function” x→y(x)=u′(x) is continuous and strictly decreasing on .4.5.v be the conjugate function to u. That is,y→v(y) is convex and nonincreasing, and it is strictly decreasing and continuously differentiable where it is finite.Let The function u(x)≡∞, in which case v(y)≡∞. But suppose u(x)<∞ for some, and therefore for all, x>0. While u(x) is necessarily concave, differentiable, and strictly increasing, it is not in general true that limx→∞u′(x)=0. That is, the marginal  utility of wealth need not approach zero as the wealth level goes to ∞. Roughly speaking, this can happen when a consumer can purchase ever larger amounts of consumption on events of ever smaller probability, but where the ratio of the amount purchased to the probability of the event approaches infinity at a rapid enough rate. This idea was exploited by Kramkov and Schachermayer (U that satisfies AE(U)=1, by choosing, based on U, specific measures that are different from but play an analogous role to P∗ and P. Here, P∗ and P are fixed—they come from BSM—so we show this sort of possibility through the selection of specific utility functions U.Of course, it may be that hermayer  for any v and, in particular, for v around the value y0, where y0=limx→∞u′(x). To the left of y0 (y<y0), we have v(y)=∞. To the right, v(y) is finite. But what is the limiting behavior of v as y approaches y0 from the right? In theory, we could have limy↘y0v(y)=∞. Or limy↘y0v(y)<∞, and limy↘y0v′(y)=∞. Or limy↘y0v(y)<∞, and limy↘y0v′(y)<∞. All of these are possible. In fact, giving the full catalog, we have Proposition Proposition 4.1U satisfies the conditions (3.1). It is possible that u(x)=∞ . But if u(x)<∞ for some x, hence for all x, it must be that x→u(x) is strictly increasing. Moreover, we have the following possibilities.(a)U, limx→∞u′(x)=0, in which case v(y) is finite for all y>0. (As limy→0v(y)=limx→∞u(x) and limy→∞v(y)=limx→0u(x), the function v can have limit ∞ or a finite limit as y approaches 0; and v can have limit −∞ or a finite limit as y→∞.)For some utility functions (b)U, limx→∞u′(x)>0. If we denote limx→∞u′(x) by y0, then v(y)=∞ for y<y0, while v(y)<∞ for y>y0. As for the behavior of v as y↘y0, we have the following possibilities:For other utility functions (i)limy↘y0v(y)=∞;(ii)limy↘y0v(y)<∞ and limy↘y0v′(y)=∞;(iii)limy↘y0v(y)<∞ and limy↘y0v′(y)<∞. y0>0.Moreover, all are possible for any value of Finally, AE(u)≤ AE(U); hence, AE(U)<1 implies limx→∞u′(x)=0. That is, asymptotic elasticity less than 1 removes the cases given by part b.Assume that Admitting this is possible (we show that it is), consider the implications for a is simple to show: Take utility functions with constant relative risk aversion, for which solutions are well known and fit case a. And the final assertion needs no proof; it derives from ,For the sequence of discrete‐time economies as described in Section u(x)=∞, this proposition still applies, implying that limn→∞un(x)=∞.If X satisfies EP[U(X)]=z and EP∗[X]=x (so that u(x)≥z), then for every ε>0, there exists N such that, for all n>N, the consumer in the nth discrete‐time economy can synthesize a claim Xn for an initial investment of x such that EPn[U(Xn)]≥z−ε.Kreps (u(x)<∞ and, for the given x, a solution to the consumer's problem exists  is a max). We then know, as the solution is of the form X=I(yZ) for some multiplier y>0 (see (4.4) above), that the solution X:Ω→ is a continuous function of ω. By truncating the solution X, we get approximately u(x), with what is a bounded and continuous claim. Hence, we conclude thatSuppose we know that u(x)<∞ but no solution exists and where u(x)=∞ are a bit more delicate, because we don't know, a priori, that we approach the upper bound  with bounded and continuous contingent claims. But we can show this is so. Suppose for some level z, there is a measurable contingent claim X such that EP[U(X)]=z and EP∗[X]=x. In this context, of course X≥0.The cases where ε>0. We first replace X with a bounded claim X′, bounded away from ∞ above and away from 0 below, in two steps. First, for α<1 but close to 1, let Xα:=αX+(1−α)x. Of course, EP∗[Xα]=x. And by a double application of monotone convergence (split EP[U(Xα)] into EP[U(Xα)1{Xα≥x}]+EP[U(Xα)1{Xα<x}]), we have limα→1EP[U(Xα)]=EP[U(X)]=z. So, choose αo close enough to 1 so that EP[U(Xαo)]≥z−ε/4. Of course, Xαo is bounded below by (1−αo)x. As for the upper bound, cap Xαo at some large β. That is, let Xαo,β be Xαo∧β. For large enough βo, this is bounded above and will satisfy EP>z−3ε/4. It may be that EP∗[X′]>x, but the last ε/4 is used to replace X′ with X′−(EP∗[X′]−x), giving a bounded and continuous contingent claim that costs x (or less) and provides expected utility z−ε, at which point Proposition 5.2 of Kreps  where E[ζ3] is the third moment of ζ, and that limnbn=1/8. (The notation E[·] is used to denote expectations over ζ.) Of course, Pn⇒P  and, for this specific emm, Pn∗⇒P∗.We fix one particular emm for each EPn[·] denotes expectation with respect to Pn and EPn∗[·] denotes expectation with respect to the specific emm Pn∗. In words, we allow the consumer any consumption claim X she wishes to purchase, subject only to the constraint that she can afford X at the “prices” given by dPn∗/dPn.So, suppose we pose the following problem for the consumer:Zn be the function on C0 given by Zn(ω)=exp[−anω(1)−bn]. Hence, Zn is a specific version of the random variable dPn∗/dPn and the constraint EPn∗[X]=x can be rewritten as EPn[ZnX]=x. That is, we can think simply of a consumer facing complete markets with Zn the “pricing kernel” for contingent claims. Using this interpretation, we denote the supremal utility the consumer can obtain in the problem (4.1) asLet nth discrete‐time economy; in problem (limnunZn(x)=u(x), we will know that limsupnun(x)≤u(x). This, together with Proposition limnun(x)=u(x). So, this is what we set out to do.The point of this is that the problem  relaxes  problem  she face7Z is the function Z(ω)=exp(−ω(1)/2−1/8), which is a version (the unique continuous version) of dP∗/dP. DefineunZ(x) is the supremal expected utility that the consumer can attain if she faces complete markets and “prices” Z in the nth discrete‐time economy. That is, moving from the consumer's problem in the BSM model to the problem described by Equation =su. DefineunZ(x)=su8Theorem 8.1U satisfies conditions (3.1) and that AE(U)<1. Then, for all x>0, the value function x→u(x) is finite‐valued andSuppose that the utility function  The proof of Theorem Lemma 8.2limn→∞EPn[exp(γω(1)]=EP[exp(γω(1))].For any constant γ, In this section, we prove the following result.Pn, ω(1)=∑k=1nζk/n for {ζk} an i.i.d. sequence of random variables with the law of ζ. Hence,exp(γζ/n) is 1+γζ/(n)+γ2ζ2/(2n)+o(1/n), where the o(1/n) term is uniform in the value of ζ because ζ has bounded support. Therefore,eγ2/2, which is (of course) EP[exp(γω(1))].□Under Step 1.(U)<1, u(x)<∞ for all x>0.Because AENow we turn to the proof of Theorem not guarantee that the optimal expected utility is finite. The result here strongly depends on the price processes being given by the BSM model.This step rates a remark: For “general” price processes as investigated, for instance, in Kramkov and Schachermayer , having (U)<1, but only for 0<y≤y0, for some y0>0. If we have that V(∞)=U(0)<0, we get the bound for all y>0. And, for purposes of this theorem, it is without loss of generality to shift U by a constant. So, if U(0)≥0, simply replace U with U(x)−U(0)−c, for a suitable constant c>0. Then we have, and henceforth assume, .The second key to Step 1 is the following bound.hermayer  establisThereexisy→v(y) as well as the primal value function x→u(x) have finite values; and we deduce as well from part c of Proposition (u)≤ AE(U)<1.Step 2.u∞Z(x) for x>0 byu∞Z(x)<∞ for all x>0.Using the notation , define  Define for each n the conjugate function vnZ, which isHence, we may estimatev(y)=EPVyvnZ(y) is uniformly bounded in n for fixed y, which, by standard arguments concerning conjugate functions, proves that u∞Z(x) is finite for each x.Step 3.limnunZ(x) exists and equals u(x) for all x>0.Indeed, We show that limnvnZ(y)=v(y) for all y>0, which proves Step 3, again using standard arguments concerning conjugate functions.Lemma vnZ(y) and v(y):V were a bounded function , the conclusion would follow immediately from Pn⇒P. But V is typically not bounded, and so we must show that the contributions to the expectations from the “tails” can be uniformly controlled. We do this by showing the following two uniform bounds.Compare U(0) is finite, it follows that V(y)≥V(∞)=U(0) for all y, so taking M=−U(0) immediately works. The (slightly) harder case is where U(0)=−∞. In this case, recall that, by the Inada conditions, limy→∞V′(y)=−limy→∞(U′)−1(y)=0. As V is convex, this implies that for large enough M, |V(y)|≤εy, provided V(y)≤−M. But thenlimn→∞EPn[Z]=EP[Z]=1, which shows Assertion (Begin with Assertion . If U(0)ssertion .L that give the bound EPn→EP and(2)Pn({ω(1)≥B})→P({ω(1)≥B}), for all B>0.And to show the (uniform) Inequality : For thehe bound  and for V(y)≤Ly−α for all y>0, and son.And lety and ε both >0, and find M such that (VM(yZ):=max{−M,min{M,V(yZ)}}; that is, VM(yZ) is V(yZ) “truncated” at ±M. This truncated function is bounded and continuous, so Pn⇒P implies EPn[VM]→EP[VM]. And the differences EPn[V(yZ)−VM(yZ)] are uniformly bounded by 2ε. Hence, vn(y)=EPn[V(yZ)]→EP[V(yZ)]=v(y) for all y, which implies that unZ(x)→u(x) for all x>0.Step 4.y>0 and ε>0, there exists M>0 such thatFor every  The parallel to the uniform inequality (vnZ(y)=EPn[V(yZ)]; here we are uniformly controlling the right‐hand tail of the integral vnZn(y)=EPn[V(yZn)]. And the same proof works; indeed, in this case we even have EPn[Zn]=1 (as opposed to EPn[Z]→1 before).Step 5.C>1 such that, for all n,There is a constant  (The reason for this step is to prove a uniform bound for EPn[V(yZn)] analogous to the uniform bound (Having shown the two uniform bounds  and 12)12), whatuch that  and (12)equality  is obvioequality  uniformlan=1/2+d/n+o(1/n), for d=E[ζ3]/24, and bn=1/8+o(1). Hence,K such that |ζ|≤K with probability 1, and so |ω(1)|≤Kn, Pn‐a.s. The ability to find a constant C>1 that gives the inequalities in /C≤Zn(ω), on the support of Pn. As V is a decreasing function, this implies that, for all y>0,Pn. Therefore, for any M>0,M such that the inequality in /ality in  is satisV(0)=U(∞) and/or V(∞)=U(0) are finite valued. So we first give the argument in the case where V(0)=U(∞)=∞ and V(∞)=U(0)=−∞, and then sketch how to handle the easier cases where one or the other is finite.The argument for this step changes a bit when ε>0 and y>0, and pick M>0 large enough, so that )=M′ and V(yZ(w2))=−M′, where by Z(w), we temporarily mean exp(−w/2−1/8). This implies that if ω is such that ω(1)∈, then V(yZ(ω))∈. Moreover, by the continuity and monotonicity of V, Z, and Zn (the latter two viewed as functions of ω(1)), and the fact that for fixed w, Zn(w)→Z(w), there exists n0 such that for all n>n0, ω(1)∈ implies that V(yZn(ω))∈.Fix  so that , 12), , so there is a Lipschitz constant ℓ such thatn>n0 and w∈. And, on the event ω(1)∈, there is n1 such that for all n>n1, |Zn(ω)−Z(ω)|<ε/ℓ.The functions D={ω(1)∈}, and for n>max{n0,n1}, we have |V(yZ(ω))−V(yZn(ω))|≤ε, and EPn[|V(yZ(ω))−V(yZn(ω))|·1D]<ε. By construction, the complement of D is a subset of the union of the four events on which we have uniformly controlled the integrals of V(yZ(ω)) and V(yZn(ω)), so for n>max{n0,n1}, EPn[|V(yZ(ω))−V(yZn(ω))]<5ε, uniformly in n, which proves Step 7.Hence, on the event V(0) and/or V(∞) are finite, the argument needs a bit of modification. Suppose V(0)<∞. This is relevant when yZ and yZn are both close to zero, which is for paths ω where ω(1) is large. And for those paths, Zn(ω) can be quite far from Z(ω). However, even if these terms are far apart, V(yZ(ω)) and V(yZn(ω)) will be close together, as each is close to the finite V(0). A similar argument works for cases where V(∞) is finite.Step 8.limn→∞unZn=u(z) for all z>0.Combine Steps 7 and 3 to conclude that Step 3 shows that vnZ(y)→v(y) for all y>0 (which is how we concluded that unZ(x)→u(x)). Step 7 then implies that vnZn(y)→v(y) for all y>0. This, in turn, implies limn→∞unZn(x)→u(x) by standard arguments on conjugate functions.Step 9.□Combine Steps 8 and Proposition When Pn faces complete markets and prices given by Z: Comparing this with the BSM model, the conjugates vn and v to optimal expected utility functions un and u are the expectations of a fixed function for different probability measures. So, after controlling the tails of the integrals that define these conjugate functions, we have a more or less standard consequence‐of‐weak‐convergence result in Step 3. In Step 7, both the probability assessments and the prices (for one of the two problems being compared) change with n. Although the pairs of problems being compared differ only in the prices, because both the integrand and the integrating measure Pn change with n, a level of finicky care is required.This proof clarifies why we introduced the analogous problem, where a consumer with probability assessment 9U that satisfy the conditions (3.1) and have asymptotic elasticity less than one, everything works out nicely within the context of the BSM model and the discrete‐time approximations to BSM that we have posited.Theorem 1 guarantees that for utility functions U for which AE(U)=1. In such cases, it may be that things work out in the sense of Theorem 1. But it is also possible that limsupn→∞un(x)>u(x). That is, when AE(U)=1, Kreps' conjecture can fail. In this section, we provide an example to illustrate this failure in stark fashion: In this example, u(x)<∞ while limsupn→∞un(x)=∞, both for all x>0.It is natural to ask, then, what can be said if we maintain (3.1) and these specific models of the financial markets, but we look at utility functions V taking the formαk,βk>0 and the sequences {αk} and {βk} are chosen so that the sum defining V(y) is finite for all y>0.In this example =βy−α.Lemma 9.1α>0 and β>0, denote by Vα,β(y) the function Vα,β(y):=βy−α for y>0. For the utility function Uα,β that is conjugate to Vα,β, if x0=−Vα,β′(y0)=βαy0−α−1 for given y0, thenUα,β isα>0, α/(1+α)∈, and AE=α/(1+α)<1.For We begin with some standard facts about conjugate pairs Vα,β.Lemma 9.2Uα,β is given by Equation ‐distributed, so that Y−1/8,1/4=−Y/2−1/8 has the law of ln(dP∗/dP)=ln(Z). Hence, the random variable β[yexp(−Y/2−1/8)]−α has the law of Vα,β(yZ), and so□Let e(α+α2)/8 recurs occasionally, so to save on keystrokes, let ϕ(α):=e(α+α2)/8.The factor Lζ(λ) the Laplace transform of the law of ζ; that is,Y a standard  Normal variate and writeYn be the scaled sum of n independent copies of ζ,LYn(λ) converges to LY(λ). On the other hand, if E[ζ3]>0, by considering—similarly as in the proof of Lemma exp(λY) around λ=0, it follows that, for small enough λ0>0,Vα,β(y)=βy−α as above, its conjugate Uα,β, the corresponding value functions for the BSM economy uα,β and its conjugate vα,β, as in Lemma Zn. In this section, we do not require value functions for discrete‐time economies in which the consumer faces prices Z, so to simplify notation, we write vα,βn for the conjugate‐to‐the‐value‐function for the nth discrete‐time economy—that is,uα,βn to denote that primal value function .Denote by k, the ratio vα,βn(k)/vα,β(1/k). From Equations , letLemma 9.3k, there exists n large enough so that, for α:=2λ0n1/2, M≥22k.For each integer Consider, for integer quations  and 23)k, the rak. Without loss of generality, set β=1. For given n and α=2λ0n1/2, calculate the denominator and numerator in the M separately.Fix Y is a standard  Normal variate and E denotes the expectation with respect to Y.For the denominator, we havey and β before specializing to y=k and β=1:ζj's are i.i.d. copies of ζ, and E denotes expectation with respect to these random variables.And for the numerator, which we calculate for general We therefore have thatbn→1/8), so as n→∞, this term, raised to the power n1/2 is bounded above by Gn1/2 for some constant G. And the term within the second set of square brackets converges to Lζ(λ0)/LY(λ0), which, per Equation ,nk≥k and nk/k is increasing, and we do so. As αk=2λ0nk1/2, this implies that limk→∞αk=∞.For each LetEquation ,(29)vαky>0, and the function has all the properties required to be conjugate to a utility function U that satisfies Condition (3.1).U the conjugate (utility) function to V, and for u the  utility‐of‐wealth function corresponding to this U within the BSM economy, u(x)<∞ for all x.Clearly, the sum is well defined for all Vαk,βk(y)≤V(y). This implies that, for each k,Concerning the discrete‐time economies, begin by noting that all the terms in the sum  are posiyk>0, letNow we enlist Lemma yk=k1/2. As vαk,βknk(k)>2k and vαk,βknk(y) is decreasing in y, we know that vαk,βknk(k1/2)>2k. As αk=2λ0nk1/2 and nk/k is, by construction, nondecreasing, we know that αk/k1/2 is nondecreasing. Putting these two observations together, we know thatChoose yk=k1/2, yk=k1/2, uαk,βknk is concave and has value 0 at x=0. So, Equation >k1/2x. Hence, from Inequality (unk(x). But as k increases toward ∞, the intervals  over which this is true expand to all of  asymptotically generate infinite expected utility, although she can only generate finite expected utility in the BSM economy. The limit in Equation  is precisely what she can attain in the nkth discrete‐time economy, even with the synthesizability constraint imposed.But this final step is easy. The properties of ζ that are used to get to Equation  >LY(λ0) for some λ0>0, fails.For such ζ, the above reasoning does not apply. To the contrary, in the specific case of the symmetric random walk, we havelimsupnun(x)>u(x), for some utility function U =1). Or it may be that equality holds true, in the case of the symmetric binomial . We leave this question open.It may still be true for the symmetric binomial that 10Z=dP∗/dP=e−ω(1)/2−1/8, the law of Z is that of exp, where Y−1/8,1/4 is Normal variate with mean −1/8 and variance 1/4. By standard calculations, then, the law of Z has density functionBecause Consider the functionV0 shows that it is decreasing in y as long as ln(y)<3/8, and a similar computation shows that V0 is also convex on  for small enough z0>0. Hence, we may define a function V:R++→R++ that coincides with V0 on the interval  and is extended to all of  and that is convex, decreasing, differentiable, and satisfies V′(∞)=0. The conjugate function to this V, denoted U, therefore satisfies the conditions of Assumption Differentiating y for which the value functionZ=dP∗/dP. Write the expectation  asw=yz.We want to determine the range of values of strictly positive functionv(y)=EPVyK depending on y,4ln(y)≤−1; that is, if y≤e−1/4.By straightforward calculation, we find that, for some constant u and v, this demonstrates, for y0=e−1/4, the possibility that v(y)=∞ for y≤y0 and is finite for y>y0; moreover, as y↘e−1/4=y0,v(y)↗∞. This is possibility b(i) in Proposition y0>0, it suffices to pass from the function V0(y) toBy the duality between ϕ(α):=e(α2+α)/8. Begin by defining, for y>0,αk and βk are given byy>0 and, in fact, does so faster than geometrically past some k0 (that depends on y).For possibility b(iii) in Proposition for y>0,V(y):=∑k=V is strictly positive, convex, and twice (and more) continuously differentiable. And from Lemma αk and βk, Equations (v(y)=∞ for y<1 and is finite for y≥1. And v′(y) is finite for y≥1. This, then, is the possibility b(iii) in Proposition y0=1.Moreover, it is evident that om Lemma v(y)=EPVy(αk)y−αkandv′(y)=b(ii) and examples where the pole is y0≠1.We leave to the reader the construction of an example of possibility


Scientific Reports 10.1038/s41598-020-62327-x, published online 26 March 2020Correction to: This Article contains errors in the Results section.Under the subheading ‘Systematic Bias’,H = 120° and θF = 90°, θH = 90° and θF = 60°, and θH = 60° and θF = 30°, i.e., postures under which the difference between the finger and head angles was −30° (θH − θF = −30°).”“Interestingly, the systematic bias was approximately zero for postures of θshould read:H = 120° and θF = 90°, θH = 90° and θF = 60°, and θH = 60° and θF = 30°, i.e., postures under which the difference between the finger and head angles was −30° (θF − θH = −30°).”“Interestingly, the systematic bias was approximately zero for postures of θUnder the subheading ‘Nonsystematic Bias’,2 = , t = , p = , df = 22, data from participant #1.”“However, no correlation was found between any of the three postures and the amplitude of the nonsystematic bias; indicating that the amplitude of the nonsystematic bias is not posture-related .”should read:“In particular, the slope is negative for the systematic bias and positive for the phase of the nonsystematic bias given changes in the finger posture and finger-head posture. Similarly, the slope is positive for the systematic bias and nonsignificant for the phase of the nonsystematic bias given changes in the head posture .”


Bacillus amyloliquefaciens has been used to synthesize the immunomodulatory γ‐D‐glutamyl‐L‐tryptophan (γ‐D‐Glu‐L‐Trp) and the kokumi‐active γ‐D‐glutamyl peptides. The optimum yield of γ‐D‐Glu‐L‐Trp was 55.76 mM in corresponding to a minimum yield of by‐product (γ‐D‐Glu‐γ‐D‐Glu‐L‐Trp) in the presence of 75 mM D‐Gln and 100 mM L‐Trp. The glutaminase has a low Km values for the donors (D‐Gln and L‐Gln:5.53 and 0.98 mM), but high ones for the acceptors ‐L‐Val/L‐Phe/L‐Met, ranging from 32.51 to 193.05 mM). The highest Km value appearing when n = 2 ‐L‐Val/L‐Phe/L‐Met) suggested the rising difficulty for synthesis when the number of donor increases in the reaction mixtures. The γ‐[D‐Glu]‐L‐Val/L‐Phe/L‐Met at 5 mM can impart the blank chicken broth an enhancing monthfulness, thickness, and umaminess taste.Glutaminase of  Glutaminase of Bacillus amyloliquefaciens has been used to synthesize the immunomodulatory γ‐D‐glutamyl‐L‐tryptophan (γ‐D‐Glu‐L‐Trp) and the kokumi‐active γ‐D‐glutamyl peptides. The optimum yield of γ‐D‐Glu‐L‐Trp was 55.76 mM in corresponding to a minimum yield of byproduct (γ‐D‐Glu‐γ‐D‐Glu‐L‐Trp) in the presence of 75 mM D‐Gln and 100 mM L‐Trp. The glutaminase has a low Km values for the donors (D‐Gln and L‐Gln:5.53 and 0.98 mM), but high ones for the acceptors ‐L‐Val/Phe/Met, ranging from 32.51 to 193.05 mM). The highest Km value appearing when n=2 ‐L‐Val/Phe/Met) suggested the rising difficulty for synthesis when the number of donor increases in the reaction mixtures. The γ‐[D‐Glu]‐L‐Val/Phe/Met at 5 mM can impart the blank chicken broth an enhancing monthfulness, thickness, and umaminess taste. D‐Gln, L‐Gln, L‐Val, L‐Trp, L‐Met, L‐Phe, Gly‐Gly, and γ‐glutamyl‐p‐nitroanilide (γ‐GpNA) were purchased from Sigma‐Aldrich . L‐Glutaminase of Bacillus amyloliquefaciens was acquired from Amano Enzyme China Ltd. . The HPLC grade acetonitrile and formic acid were purchased from CapitalBio Corporation .Commercial synthetic 2.2γ‐D‐Glu‐L‐Trp: a reaction mixture that contained D‐Gln , L‐Trp , and glutaminase (0.05 U/ml) was placed in a water bath shaking table for incubating for 8 hr at 37 ºC, and then the mixture was inactivate enzyme in a boiling water bath at 90 ºC for 15 min. The optimum ratio of substrate was also analyzed including the various D‐Gln  and a fixed L‐Trp (100 mM). The isolation of γ‐D‐Glu‐Dipeptides was carried in accordance with the method of Suzuki with some modification , and then purified γ‐D‐Glu‐L‐Trp was dissolved in D2O and then analyzed with a Bruker 600 MHz Superconducting Fourier Transform NMR Spectrometry . HPLC  equipped with XSelect HSS T3 column  was used to determine the content of peptides and the L‐Trp. Acetonitrile and water with 0.1% formic acid were as the mobile phase. The other parameter contained: 1 ml/min flow rate, 220 nm detection wavelength (λ), 40°C column temperature, and 20 μL injection volume. The objective products in the HPLC chromatogram were preliminarily identified on the basis of the retention time of external standard compounds.: Synthesis of γ‐D‐Glutamyl peptides: The reaction mixture contained one of the substrate mixtures of D‐Gln:L‐Met = 75:100, D‐Gln:L‐Val = 75:100 or D‐Gln:L‐Phe = 75:100, 0.05 U/ml enzyme, incubated in a water bath shaking table at 37°C for 12 hr, then the mixtures were heated to the point of inactivating enzyme in a boiling water bath at 90 ºC for 15 min. These γ‐D‐Glu‐peptides in the mixtures were analyzed by UPLC‐Q‐TOF MS/MS system  with an Agilent ZORBAX RRHD SB‐C18  110 column, referring to Yang's methods  of glutaminase for the synthesis of γ‐D‐glutamyl peptides.Enzymatic kinetic parameters  and a fixed L‐Trp (25 mM).The apparent Kγ‐GpNA, γ‐glutamyl acceptors and enzyme in the borate‐NaOH  buffer solution, then adding 0.1 M HCl to terminate the reaction after incubation at 37 ºC for 30 min. The content of p‐nitroaniline was determined by a UV spectrophotometer  at 410 nm. The release of 1 μmol of p‐nitroaniline per min from γ‐GpNA was defined as the amount of one unit (U) of enzyme through the transpeptidation reaction.The substrate specificity of the enzyme with different acceptors were measured as described previously with some modification  were added into a blank chicken broth to evaluate the flavor characteristic. Taste profiles mainly included mouthfulness, thickness and umaminess. Thereinto, mouthfulness describes an overall perception associated with food texture, structure and morphological complexity, and thickness which was measured as the increased taste intensity evaluated 10 s after food tasting describes rich complexity. The panels were consisted of 10 males and 7 females (age 30–40) from a professional company. 10 ml of solutions was given to these panelist individually and then put into the mouth for 15 s before swallowing, finally recorded the taste experienced in 25 s. Panelists were asked to rate the intensity of a given taste on a scale from 0 (not detectable) to 5 (strongly detectable).5 mM of these synthetic 2.3p < .05 between samples using SPSS 16.0 statistical software.Statistical analysis was performed in triplicate and analyzed by Microsoft Excel 2000. Duncan test was used to determine significant differences at 33.1γ‐D‐Glu‐L‐Trp.γ‐D‐Glu‐L‐Trp (12.245 min) indicating that γ‐D‐Glu‐L‐Trp could be enzymatically synthesized in the action of a commercial glutaminase of Bacillus amyloliquefaciens : δ 7.61−7.55 , 7.41 , 7.19−7.13 , 7.08 , 4.71−4.64 , 3.81−3.72 , 3.74−3.69 , 3.67 , 3.59 , 3.31 , 3.16 , 2.35−2.27 , 1.95−1.87 .Three main intense chromatographic peaks were observed, the most intense peak corresponded to lu‐L‐Trp 2.245 minγ‐D‐Glu‐γ‐D‐Glu‐L‐Trp (12.389 min), suggesting that there was a significant coexisting product (γ‐D‐Glu‐γ‐D‐Glu‐L‐Trp) in the postenzymatic reaction mixture. The production of γ‐D‐Glu‐L‐Trp and γ‐D‐Glu‐γ‐D‐Glu‐L‐Trp, as well as the consumption of L‐Trp with time‐course are showed in Figure γ‐D‐Glu‐L‐Trp would reach a maximum yield of 60.90 mM after 3 hr incubation (its yield increasing during the first 3 hours but without significant differences) and then leveled off, the result was closed to that of GGT‐catalyzing , finally maintained stability (p > .05). The observation illustrates there was a time difference between obtaining the maximum of γ‐D‐Glu‐L‐Trp (3 hr of incubation) and γ‐D‐Glu‐γ‐D‐Glu‐L‐Trp (5 hr), so the concentration of by‐product can be minimized through controlling the incubation time. Correspondingly, the consumption of L‐Trp can be summarized in roughly three stages during the synthesis of γ‐D‐Glu‐L‐Trp: firstly, L‐Trp sharply dropped to 43.12 mM after 1 hr reaction, then dropped at a lower rate from the 1st to 5th h, and finally reached a plateau (L‐Trp at about 28.36 mM) after the 5th h .The other two intense peaks were the L‐Trp (7.762 min) and γ‐D‐Glu‐L‐Trp and γ‐D‐Glu‐γ‐D‐Glu‐L‐Trp (a 2‐fold difference of glutamic acid residue), it can be speculated the concentration of D‐Gln would influence the production of by‐product more than that of objective product. Therefore, the effect of D‐Gln  with a fixed L‐Trp (100 mM) was analyzed for reducing the yield of γ‐D‐Glu‐γ‐D‐Glu‐L‐Trp . The yield of γ‐D‐Glu‐L‐Trp was 55.85 mM, with a below detectable limit of γ‐D‐Glu‐γ‐D‐Glu‐L‐Trp.When comparing the sequence characteristic between p Figure . With a γ‐D‐Glutamyl Phenylalanine, Methionine, and Valine.γ‐D‐Glu‐L‐Phe, γ‐D‐Glu‐L‐Val, and γ‐D‐Glu‐L‐Met were with the significant signals at m/z 295.1295, 247.1299, and 279.1016, respectively in positive ESI mode , [AA(Acceptor)‐COOH]+ (y‐CO2 type ion), [AA(Acceptor)‐NH3]+ (y‐CO2 type ion), [AA(Acceptor)+H]+ (y type ion) in each of the mass spectrum. Their yield was 30.23 ± 4.18, 23.12 ± 3.97, 17.19 ± 2.04 mM, respectively ‐L‐AA as an acceptor to participate in the γ‐glutamyl transfer reaction to form the by‐products (γ‐D‐[Glu]n‐L‐AA) ‐L‐Trp/Phe/Met/Val) during γ‐glutamyl transpeptidation by glutaminase‐catalysis were getting bigger with the increased “n,” that is, the lowest Km with L‐Trp/L‐Phe/L‐Met/L‐Val , while the highest  with γ‐[D‐Glu](n=1)‐L‐Trp, γ‐[D‐Glu](n=2)‐L‐Phe, γ‐[D‐Glu](n=2)‐L‐Met and γ‐[D‐Glu](n=2)‐L‐Val. Thus, it was more and more difficult for the synthesis of γ‐[D‐Glu]n‐L‐Trp/L‐Phe/L‐Met/L‐Val became more difficult with the increase of the number of γ‐D‐glutamyl residues in the acceptor, which were consistent with γ‐[L‐Glu]n‐L‐Phe/L‐Met/L‐Val synthesis. The Km values of γ‐[D‐Glu]‐L‐Trp/L‐Phe/L‐Met/L‐Val during the hydrolysis reaction by glutaminase‐catalysis were in the range of 36.73 ~ 122.73 mM, which were lower than those for the synthesis of γ‐[D‐Glu]n‐L‐Trp/L‐Phe/L‐Met/L‐Val. Moreover, the affinities of glutaminase for the same acceptors with the L‐Gln as the donor were obviously lower than that of D‐Gln. The affinities of glutaminase for the acceptors illustrate L‐Trp is the most natural acceptor, indicating the glutaminase is very suitable for the synthesis of γ‐D‐Glu‐L‐Trp.The results of the Km values of glutaminase show high Km values for acceptors, but low ones for s, Table . The Km γ‐glutamyl acceptors with the D‐Gln as the γ‐glutamyl donor  an enhancing mouthfulness, thickness and umami taste with the taste‐enhancing score of 0.60 ~ 1.07  and γ‐[D‐Glu]n‐L‐Phe/L‐Met/L‐Val have been synthesized in the presence of D‐Gln catalyzed by the glutaminase of B. Amyloliquefaciens. Differ from the GGT enzyme, this glutaminase can catalyze the binding of D‐Gln to the synthetic γ‐D‐glutamyl di‐/tri‐peptides. When comparing the D‐Gln and L‐Gln as the donors, the number and content of the by‐products with the D‐Gln as the donor were significantly less than that of L‐Gln in the γ‐glutamyle peptides synthesis. The Km values of glutaminase of B. Amyloliquefaciens for D‐Gln and L‐Gln (the donors) were 5.53 and 0.98 mM, and for the acceptors were range from 32.51 (L‐Trp) to 193.05 mM . The optimal yield of γ‐D‐Glu‐L‐Trp was 55.85 mM with a below detectable limit of γ‐D‐Glu‐γ‐D‐Glu‐L‐Trp under the reaction condition: 75 mM D‐Gln and 100 mM L‐Trp, incubating for 2 hr at 37°C (pH 10.0). The addition of the γ‐[D‐Glu]n‐L‐Phe/Met/Val can impart the blank chicken broth an enhancing kokumi taste.The authors declare that they do not have any conflict of interest.This study does not involve any human or animal testing.


During the early Byzantine period, the therapeutic herb “Colchicum autumnale”, or “ermodaktylon” was introduced in the treatment of podagra (gout). Podagra presented throughout the Byzantine period a disease with high incidence, since 14 out of the total 86 Emperors seem to have suffered from it. The lead pipes of the city of Constantinople’s sewer system, utensils, but also the production of the sweetening grape syrup sapa contributed to its appearance. Although Alexander of Tralles considered to be the physician who discovered the properties of the plant, Severus Iatrosophista, Theodosius the Philosopher and Jacobus Psychrestos, were the healers who introduced ermodaktylon as the pioneering treatment of podagra in the early Byzantine period. Colchicum autumnale (Greek: κολχικό του φθινοπώρου), or bitter hermodactyl , or cochlicon (Greek: κοχλικόν), or articulorum (the soul of the joints), later named as saffron, or autumn crocus, was most probably known since the early rhizotomi (Greek: ριζο-τόμοι), the pharmacobotanists practicing herbal medicine since the Prehippocratic era . Although it was Pedanius Dioscorides (ca 1st century AD) who firstly mentioned Colchicum as a therapeutic plant with strong poisonous action , it was Alexander of Tralles (Greek: Αλέξανδρος ο Τραλλειανός) (525–605 AD) who was wrongfully credited with having introduced it intothe treatment of gout.7He himself, mentioned Severus Iatrosophista, Theophilus the Philosopher, and Jacobus Psychrestos who were the first healers to use the plant Colchicum. Although Alexander had proposed the use of cantharides as a blister to treat podagra, he had adopted previous knowledge to emphasise on a new promising herbal treatment. Alexander gave us six recipes containing hermodactyl; the final two of which he had attributed to other physicians: one to Jacobus Psychrestus, and another to Theodosius the Philosopher.8The plant’s name comes from Colchis, a region in Asia Minor, were Colchicum was endemic at the era .9Podagra  disease, a complex form of arthritis currently known as gout, was apparently accumulated among Byzantine Emperors, and common citizens of Constantinople. It has been considered that gout may be a disease caused by lead poisoning: a contributing cause for this accumulation may have been exposure to high levels of lead, originating from Constantinople’s water pipes, wine containers and cooking pots used for producing the sweetening grape syrup sapa (Greek: έψημα). Hippocrates (ca 460-370 BC), 10 eons earlier, suggested that podagra (Greek: ποδάγρα) was the result of the excess-accumulation of the body humours (4 humours theory) inside the articular capsule of the joints. The disease was called “podagra”, meaning in Greek severe pain grabbing the leg. As podagra seems to have been a significant, widespread and invalidating disease at the era, a disease that was prevalent in Byzantium, but also a well-known entity in ancient Greece.10Our historical note presents the first Byzantine physicians who introduced Colchicum autumnale as a treatment for podagra. Although Colchicum autumnale is no longer recommended due to its significant toxicity, its active components, colchicine in particular, are still widely used. The plant contains colchicine , and smaller amounts of secondary alkaloids, such as demecolcine. German botanist Hyeronimus Bock (Latinized: Tragus) (1498–1554 AD) mentioned in his treatise “Kreuterbuch” during 1550, that Arab and Byzantine physicians employed the plant in the early Byzantine period.11Theodosius the Philosopher, an eminent scholar of the same era, had also proposed the same plant as a botanic cure against podagra.3Severus Iatrosophista (Greek: Σευήρος ο Ιατροσοφιστής), was a 5th century medicophilosopher, known for his masterpiece “De Clysteribus” (Greek: Περί Ενετήρων); suggesting clysters for various diseases such as skin rash. He was the first Byzantine physician to describe the allergic shock from food and drugs, while he had noted women’s hysteria proposing acute intervention. He had also presented a thorough report upon the surgical instruments availliable in that period. He is considered by Alexander of Tralles the first to introduce hermodactyl (Colchicum).Figure 4).12Jacobus studied podagra, « τας ζεούσας φλεγμονάς των ποδών » , to be put upon the inflamed joint.14Today it is known that the plants of the genus Colchicum contain colchicine, a toxic natural product and secondary metabolite, which is still in use for the treatment of gout.2Jacobus was a scholar with a broad spectrum of knowledge concerning human physiology. Thus, he was also considered as an exceptional dietician, proposing a moistens temperate diet (Greek: υγραίνουσα εύκρατον δίαιτα) for the organism’s homeostasis to be balanced.15Although Jacobus was an eminent physician in the imperial court of the Byzantine empire, his work was lost, and only fragments survived inside the treatises of other important medical figures such as Damascius the philosopher (458–550 AD), Alexander of Tralles, and Aetius of Amida (ca mid 5th-mid 6th century AD). His pioneering thought to use colchicine to treat podagra should grant him a place among the significant figures of the history of rheumatology.The Byzantine physician Jacobus Psychrestos , was the offspring of Hesychios from Damascus, a known physician of the era. He had exercised both medicine and philosophy in Constantinople during the 5th century AD. He was considered an authority (Greek: άριστος), capable to treat a plethora of diseases. He had soon become the personal head physician (Greek: αρχίατρος) of the Byzantine emperor Leo the 1st, known as the Thracian (Greek: Λέων Α’ ὁ Θρᾷξ) (401–474 AD) , thus, perhaps, suggesting the speed of cure that the herbal drug provided for the patients. Although by many researchers Alexander of Tralles is considered to be the physician who introduced this drug in the treatment of podagra, it was the personal physician of Emperor Leon the 1st, Jacob Psychrestus, who epitomized Severus’ and Theodosius’ proposals and systematized its use; quickly gaining substantial fame.It appears that a great number of Sovereigns (14 of the 86 Byzantine Emperors) of the Byzantine Empire and officials of the State and leaders of the Church suffered from what in most cases seems to have been the podagra disease. The texts of the most Byzantine writers referred to the main medicine for arthritis during that era, hermodactylus, a constituent of the herb Colchicum autumnale. Hermodactylus means, in the Hellenic language, the “finger of Hermes”; the Olympian messenger god of the ancient Greeks (In the history of medicine, a plethora of forgotten physicians, such as our three Byzantine healers, still seek their rightful place in medical history. Rheumatology owes them an honourable citation.


The contributions of γδ T cells to immune (patho)physiology in many pre‐clinical mouse models have been associated with their rapid and abundant provision of two critical cytokines, interferon‐γ (IFN‐γ) and interleukin‐17A (IL‐17). These are typically produced by distinct effector γδ T cell subsets that can be segregated on the basis of surface expression levels of receptors such as CD27, CD44 or CD45RB, among others. Unlike conventional T cells that egress the thymus as naïve lymphocytes awaiting further differentiation upon activation, a large fraction of murine γδ T cells commits to either IFN‐γ or IL‐17 expression during thymic development. However, extrathymic signals can both regulate pre‐programmed γδ T cells; and induce peripheral differentiation of naïve γδ T cells into effectors. Here we review the key cellular events of “developmental pre‐programming” in the mouse thymus; and the molecular basis for effector function maintenance vs plasticity in the periphery. We highlight some of our contributions towards elucidating the role of T cell receptor, co‐receptors (like CD27 and CD28) and cytokine signals (such as IL‐1β and IL‐23) in these processes, and the various levels of gene regulation involved, from the chromatin landscape to microRNA‐based post‐transcriptional control of γδ T cell functional plasticity. This ordered rearrangement results in timed production of defined γδ T cell populations, which leave the thymus and populate different epithelial‐rich tissues in the adult animal  γ chains. Early work on the genetics of TCR rearrangement during fetal, neonatal and adult thymocyte development revealed an organized and sequential rearrangement of specific + γδ T cells are generated.+ γδ T cells develop exclusively in the fetal thymus and have no junctional diversity due to absence of terminal deoxynucleotidyl transferase expression. DETC are found in the adult skin, while Vγ6+ γδ T cells localize to diverse tissues such as, eg, uterine epithelia, tongue and meninges.+ γδ T cells is restricted to the confined window of fetal development and cannot be induced in adult animals.+ and Vγ4+ γδ T cells develop from late fetal life onwards, throughout adulthood. These show higher junctional diversity and localize to diverse sites including peripheral lymphoid tissues, where they represent the majority of γδ T cells in the adult mouse. The distinctive developmental phases, orchestration of TCR rearrangements and specialized tissue localization suggest a diversity of physiological roles for γδ T cells encompassing features of both innate and adaptive – or “adaptate” – immune surveillance.The first T cells produced in the fetal thymus are dendritic epidermal T cells (DETC), a specialized subset of γδ T cells expressing an invariant Vγ5Vδ1 TCR. Subsequently, Vγ6Listeria monocytogenes.E coli infection.Staphylococcus aureus and Candida albicans, among other infections, or detrimental roles in inflammatory diseases and in cancer, where they can promote angiogenesis, immune suppression and tumor cell growth  and the majority of Vδ6.1/Vδ6.3+ γδNKT cells (liver and spleen); and post‐natally generated cells expressing more polyclonal TCRγδ (mostly Vγ1+ or Vγ4+) with junctional diversity .IFN‐γ‐producing (γδIFN) T cells, including a fetal/ perinatal‐derived subgroup expressing invariant and semi‐invariant TCRs with no junctional diversity, which is made up by Vγ5+ or Vγ4+ TCRs, although minor populations expressing Vγ1 and Vγ2/3 chains have been described in the thymus and most notably in the liver.+ γδ17 T cells are dominated by one invariant Vγ6Vδ1 clone lacking additional N‐nucleotide insertions and few semi‐invariant Vγ6Vδ1 clones,+ γδ17 T cells present a more oligoclonal population encompassing multiple (semi)invariant TCRs.γδ17 T cells usually expressing Vγ6In this review, we will focus on the biology of effector γδ T cells making IFNγ or IL‐17, from their early steps of differentiation in the thymus to their functional properties in the periphery, and the underlying molecular mechanisms.2highCD4−CD8−CD45RBlowCD44highCD62Llow, implicating a memory‐like phenotype.Over the last decade or so, the field has put much effort into establishing markers to identify and analyze the two main effector γδ T cell subsets. The initial description of γδ17 T cells characterized them as CD3− in the steady state. We found that CD27− γδ T cells were CD44highCD62Llow and contained all IL‐17 producers but very few γδIFN T cells. In contrast, CD27+ γδ T encompassed most CD122+ γδ T cells, had lower expression of CD44 and produced IFNγ but essentially no IL‐17.Plasmodium berghei). Most interestingly, we found that the effector phenotypes were already established during thymic development, and since embryonic life. Our detailed analysis of γδ T cell development revealed that manipulating CD70‐CD27 signals on developing γδ T cells in fetal thymic organ cultures (FTOC) impacted effector phenotype acquisition. In particular, CD27 signals were required for expression of the lymphotoxin‐β receptor (LTβR) and downstream genes previously associated with IFNγ production in γδ T cells.+ while γδ17 cells are exclusively Ly‐6C−.+ αβ T cells, the authors suggested γδ T cell naïve‐like and memory‐like subsets sharing characteristics with adaptive αβ T cells.CD27 had been previously used to characterize functional subsets of αβ T cells and natural killer (NK) cells.+ γδ T cells in the adult thymus and peripheral tissues was especially prominent and resulted in establishment of CCR6 as a γδ17 T cell marker. Of note, in the neonatal thymus only Vγ6+ γδ17 thymocytes express CCR6, whereas the vast majority of Vγ4+ γδ17 thymocytes do not; it was suggested that CCR6 expression might be induced extrathymically on dermal Vγ4+ γδ T cells.−) γδ T thymocytes.− Vγ4+ γδ T cells are mostly absent from the thymus. Therefore, it remains possible that CCR6 induction occurs intrathymically on Vγ4+ cells that promptly leave to peripheral sites. Interestingly, the expression of the two scavenger receptors SCART1 and SCART2 was shown to be mutual exclusive on Vγ6+ and Vγ4+ dermal γδ17 T cells, respectively, possibly due to different ontogenic origins.−CD44highCD45RB+) and IL‐17 (CD24−CD44highCD45RB‐) committed subsets, as well as their developmental trajectories from immature (CD25+CD24+) γδ thymocytes.Besides lacking CD27 expression, γδ17 T cells were found to express SCART1 and SCART2 and CCR6 in the adult mouse.+ γδ T cells within the adult thymus.+ γδ T cells in adult mice; and confirmed that CD24− γδ T cells in the adult thymus mostly represent long‐lived resident effector cells generated during fetal/ perinatal life.Different thymic trajectories for the generation of γδIFN cells have been recently described by introducing transient surface markers, namely CD117, CD200 and CD371, as useful tools to segregate CD2433.1Among the surface markers that segregate the two γδ T cell effector subsets ‐based subsetting described above. The synthesis of T10/T22‐tetramers  for flow cytometry analysis enabled the first study of a minor but antigen‐specific γδ T cell subset within wildtype (ie non‐transgenic) mice.+ γδ17 T cells were decreased in early life but Vγ4+ γδ17 T cells developed normally, resulting in normal γδ17 T cell abundance in the periphery of adult mice. In contrast, γδIFN T cells were diminished throughout life in the thymus and in peripheral lymphoid organs of CD3DH mice. Most importantly, CD122+ NK1.1+ γδIFN T cells (mostly Vγ1+) were virtually absent, but their thymic development could be rescued upon injection of an agonist CD3 antibody. The reduction in peripheral γδIFN T cells in CD3DH mice associated with less susceptibility to P berghei ANKA‐driven experimental cerebral malaria, an inflammatory syndrome dependent on IFNγ and γδ T cells.A subsequent study from our laboratory has provided additional data supporting a role of TCR signaling in the development of specific γδ T cell effector subsets. The research was based on a new mouse model, haploinsufficient for both CD3γ and CD3δ , where we found both TCRγδ surface expression and TCRγδ signal strength to be substantially reduced.The CD3DH phenotype is surprising considering that single haplodeficient mice did not show a similar impairment, and because several studies on TCR composition reported that CD3δ was not incorporated into the murine TCRγδ at least on mature γδ T cells.+NK1.1+ γδIFN T cells appear to be especially dependent on TCRγδ signaling during development. It may be that these CD122+NK1.1+ γδ T cells represent a unique subset that has yet to be fully explored; we next summarize additional findings that support this possibility. In the thymus of adult wildtype mice, CD122+NK1.1+ γδ T cells are found among the CD44highCD24low mature γδ T cells .Bcl11b knockout mouse, two hepatic γδ T cell subsets with different developmental requirements were described: a NK1.1+CD5− subset generated early in newborn mice producing exclusively IFNγ rapidly upon infection; and a NK1.1−CD5+ subset that comprised both IFNγ and IL‐17 producers.Bcl11b deficiency resulted in a complete loss of the NK1.1−CD5+ γδ T cell subset , whereas the NK1.1+CD5− proved to be Bcl11b‐independent and retained CD122, high PLZF expression and IFNγ production upon stimulation. These observations suggest that the NK1.1+ γδIFN T cell subset has specific developmental properties and represents a unique innate‐like population.It is also intriguing that CD122Another important clue on the role of TCR signaling in effector γδ T cell differentiation came from the analysis of Skint‐1‐deficient mice. Vγ5Vδ1 TCR‐expressing DETC were shown to rely on encountering Skint‐1 during development in the embryonic thymus to embark on the IFNγ effector program. Absence of Skint‐1 expression in a naturally occurring mutant strain resulted in Vγ5Vδ1 T cells that atypically produced IL‐17 instead of IFNγ.A cell‐intrinsic program indeed appears to drive the development of γδ17 T cells in the fetal/ perinatal thymus. Thus, IL‐17 expression in early precursors prior to TCR rearrangement has been documented in the fetal thymus; and γδ17 T cells appear to require only weak or even no TCR signaling for their development.+ γδ17 T cell development.As a final point on TCR signaling, several studies have reported variable expression and dependency on associated proteins by different γδ T cell subsets. The main Src family kinase Lck was surprisingly poorly expressed in mature γδ17 T cells in the thymus3.2The identification of pre‐programmed γδ T cell effector subsets and their key contributions to peripheral immune responses highlighted, once more, the importance of understanding thymic development. How γδ T cell effector subsets acquire their effector functions during thymic development remains an area of active research and controversies, particularly whether γδ17 and γδIFN T cells arise from common or distinct thymic progenitors.−CD8−) stage 2 (DN2), in which they simultaneously rearranged and transcribed Trb, Trg and Trd,In the adult mouse, bipotential thymocytes were found up to the DN (double negative/CD4prior to TCR expression. One notable example is the expression of the interleukin 7 receptor (IL‐7R), since intrathymic injection and reconstitution of fetal thymic lobes with purified DN2 populations from adult mice demonstrated that the progeny of IL‐7Rhigh DN2 cells presented a higher γδ T cell to αβ T ratio compared to that of IL‐7Rlow DN2 cells.TCRγ locus,+ γδ17 T cells, as these were absent in a spontaneous Sox13 mutant mouse strain, while other γδ T cell subsets appeared to develop normally.In support of pre‐commitment, cell heterogeneity within the DN2 subset has been described and linked to different lineage biases loxP site flanked version of the gene of interested and the expression of Cre recombinase from a specific promoter whose activity is restricted to a certain cell type or tissue. The heterogeneity within the γδ T cell lineage and our incomplete understanding of its developmental trajectories impose limitations to the use of several conditional KO mice for the analysis of γδ T cells. Indeed, conditional deletion using the proximal promoter of Lck‐driven Cre (pLckCre) was shown to occur in DN2 cells and to efficiently target the αβ T cell lineage,On the other hand, TCR signal “strength” emerged as a key determinant based on studies using transgenic TCRγδ expression, in which the manipulation of downstream signaling mediators had major effects on γδ vs αβ T cell commitment. In such settings, γδ T cells required stronger TCR signals to develop than their αβ T cell counterparts.+ and IL‐17+ γδ T cells, c‐kitlow DN3 cells failed to give rise to γδ17 T cells.An interesting possibility is the existence of various points of divergence from the αβ T cell lineage path for discrete γδ T cell subsets. Support for such a “multiple branching‐off model” first came from studies on E17 fetal thymi. While DN2 cells expressing high levels of c‐kit developed into both IFNγ+ γδ17 T cells derive from CD44+CD25−CD24+c‐Kit− (DN1d) but not DN2 precursors.+ γδ17 T cell development.+ γδ17 T cells, which was inferred to be TCR‐independent from the analysis of mutant mice.More recently, a progenitor fate analysis and single‐cell RNA sequencing of discrete DN1 subpopulations, as previously defined by Petrie and colleagues,high DN2 cells can develop into γδ17 T cells.+ cells, whereas Spidale et al employed DN1 and DN2 precursors from 10‐day‐old thymi, which were introduced into FTOC, and focused on Vγ4+ γδ17 T cell development. Since γδ17 T cell thymocyte development is supposed to be terminated in 10‐day‐old mice,+ γδ17 T cell development excluding a DN2 stage, while directly assessing the differences to earlier fetal Vγ6+ γδ17 T cell development.In contrast with the conclusions of Spidale et al,Tcrg locus, may develop into IL‐17 producers by default, with no or low TCR signaling involved – unless the rearranged TCRs encounter their ligands, as it is the case for Vγ5Vδ1 DETC precursors, and potentially for Vγ1Vδ6 γδNKT cells. Ligand encounter does not seem to positively select the TCR repertoire, but instead divert the effector fate towards IFNγ production. Later on, a second wave of T cell progenitors enters the thymus; these differentiate towards γδIFN cells upon TCR signaling but are no longer prone to differentiate towards a default γδ17 fate, instead generating naïve, uncommitted γδ T cells that can display adaptive‐like behavior in the periphery.In sum, there may be two separate waves of γδ T cell development that differ substantially with regard to thymic precursors and dependence on TCR signaling. The first wave in the fetus is made up by a set of progenitor cells whose origin is not yet fully elucidated, but some reports suggest they may arise in the yolk sac.Next, we will focus on the current view of the molecular mechanisms that underlie the acquisition and maintenance  of γδ T cell effector functions.4+ T helper cells.The acquisition of the capacity to secrete IFNγ and IL‐17 during thymic development distinguishes γδ T cells from their αβ T helper cell counterparts, whose effector cell differentiation  occurs in peripheral lymphoid organs upon activation in specific inflammatory milieus.Some of the key questions addressed over the past decade have been:How peripheral γδ T cell subsets expressing IFNγ or IL‐17 compare at the molecular level to their Th1 and Th17 counterparts, respectively;Whether the molecular determinants identified in peripheral γδ T cell subsets are imprinted during thymic development;What are the relative contributions of epigenetic, transcriptional and post‐transcriptional (including microRNA‐based) mechanisms to the regulation of IFNγ and IL‐17 expression in effector γδ T cell subsets.In the next sections, we review the currently available data that provide some answers to these broad and still outstanding questions.4.1+ T helper cell subsets, where genes encoding transcription factors exhibit a large spectrum of epigenetic marks and allow for functional plasticity.Epigenetic mechanisms operating at the chromatin level control the maintenance of transcriptional networks ensuring autonomous maintenance of lineage phenotypes in differentiated cells, even through mitotic divisions.+ and IL‐17+ γδ T cells, we isolated CD27+ (γδ27+) and CD27− (γδ27−) γδ T cells, respectively, from peripheral organs of C57BL/6 mice and subjected them to Chip‐seq (chromatin immunoprecipitation followed by deep sequencing) analysis of activating H3K4me2 and repressive H3K27me3 marks.+ vs γδ27− T cell subsets were not segregating between Th1 and Th17 cells, suggesting that lineage‐specific mechanisms operate in γδ T cell differentiation.+T cells,− T cells but not in γδ27+ T cells nor in either CD4+ helper T subset.−/−γδ27− T cells were enriched for IL‐17 producers when compared with wildtype controls, although the underlying mechanism is yet to be clarified.Recurring to CD27 levels to segregate IFNγBcl11b, Id3 or Etv5) and survival  displayed, as expected, very similar histone marking in γδ27+ and γδ27− T cells.Ifnγ and its transcriptional regulators Tbx21, Eomes and Hlx, were all “primed” for expression in both γδ T cell subsets  between the two subsets, whereas Ifng was only mildly (<10‐fold) higher in γδ27+ cells compared to γδ27− cells. Importantly, the key epigenetic and transcriptional signatures observed in peripheral γδ T cell subsets were also present in their thymic counterparts,Our detailed analysis of chromatin marks in genes involved in global γδ T cell biology revealed that those implicated in γδ T cell development  in highly inflammatory settings, ie, upon stimulation with high amounts of IL‐1β and IL‐23, or in an ovarian cancer microenvironment.+ T cells, IL‐17‐producing γδ27− cells are endowed with functional plasticity, which can be deployed (to co‐express IFNγ) under quite specific conditions. Subsequent work from our group showed that the polyfunctional γδ27− (γδ17) T cell population can also secrete IL‐17F, IL‐22 and GM‐CSF upon IL‐1β and IL‐23 stimulation,+ T cells, Th17 cells are substantially more prone to plasticity than their Th1 or Th2 counterparts. Thus, Th17 cells are able to transdifferentiate into Th1, regulatory T (Treg) and follicular helper T (Tfh) cells,Tbx21 or Gata3, also underlying their functional plasticity.Building on these molecular analyses, we challenged the functionalities of γδ T cell subsets in vitro and in vivo. We found that while γδ274.2Following the genome‐wide analysis of γδ T cell subsets,+ T cells both in vitro and in vivo upon infection with murid herpes viruses, while also impairing IFNγ production by γδ27−CCR6+ T cells stimulated with IL‐1β or IL‐23 in vitro or during Listeria infection in vivo.−/− γδ27+ cells compared to controls,T‐bet deficiency was found to significantly reduce IFNγ expression by peripheral γδ27Rorc) deficiency completely abolished the production of IL‐17 by γδ27− T cells.+ T cells, IL‐17 production had been shown to be co‐regulated by the auxiliary transcription factors RORα and BATF,On the other hand, RORγt (+ and Vγ4+), the IFN‐γ producers  and DETCs (Vγ5+), with Rorc, Maf, Sox13, and Sox4 associated with the IL‐17A producers; and Tcf7 (TCF‐1), Lef1, Tbx21 (T‐bet) and Eomes with the IFNγ producers. These results were confirmed in additional studies that showed, for example, that Sox13, Sox4, and the Ets family member ETV5, are key regulators of the development of γδ17 T cells.Of relevance, in γδ T cells, an additional and critical layer of TF‐mediated regulation occurs during thymic development,Sox13, Maf and Rorc knockout mice, Sagar and colleagues have shown a sequential activation of these factors during both fetal and adult γδ17 cell differentiation.+ Rorc+ Il17a+ Il17f+ γδ T cells in the fetal thymus, and displayed reduced levels of Maf, Blk and Roc in γδ T cells from adult thymus, whereas Maf‐deficient fetal thymi lacked Rorc+ Il17a+ γδ T cells, and Rorc‐deleted γδ T cells did not show reduced Sox13 or Maf expression. Thus, during thymic γδ17 cell development, Sox13 acts upstream of c‐MAF which is essential for RORγt function in orchestrating the γδ17 program.A recent study has added an important temporal dimension to the role of some of these transcription factors regulating thymic γδ17 cell differentiation. Upon performing single‐cell analyses of + T helper cell differentiation. This is the case for BATF and IRF4, which promote opening of chromatin at Th17 cell‐specific loci, allowing access to RORγt.+ γδ T cells has not been addressed, STAT3 was shown to be dispensable for the generation of IL‐17‐producing γδ T cells.A more detailed discussion on the transcriptional networks operating in effector γδ T cell subsets, especially during thymic pre‐programming, is provided by Anderson and colleagues elsewhere in this issue. Of note, an important regulatory function of TFs is to act as chromatin remodelers, mostly as promoters of open chromatin, thus leaving specific T cell loci accessible to other TFs/regulatory factors. Although, so far, no studies have specifically addressed this issue in γδ T cells, several TFs have been implicated in chromatin remodelling during CD4Other issues that require further elucidation are (i) which extracellular cues, including potential TCR ligands, may feed in and regulate the transcriptional programs required for effector γδ T cell differentiation; and (ii) how the transcriptional networks described separately for thymic and peripheral γδ T cells are integrated, and potentially cross‐talk, within the cell. Future studies, based on single‐cell analysis, will likely contribute to improving our understanding of these phenomena.4.3Our group has been recently addressing the role of non‐coding RNAs, with special attention to microRNAs, in effector γδ T cell differentiation.+ T cells show reduced proliferation and survival after in vitro stimulation, but increased frequencies of IFNγ producers, implicating miRNAs in T helper cell differentiation.MicroRNAs (miRNAs) constitute a fundamental layer of post‐transcriptional regulation, acting as negative regulators of expression for most mammalian genes by promoting the degradation of mRNAs or preventing their translation.Interestingly, the development of γδ T cells is not impaired by miRNA ablation; on the contrary, there is a substantial increase of γδ T cells in the double negative thymic compartment of mice conditionally lacking Dicer in early thymocytes.− cells  and in vivo, upon Listeria monocytogenes infection.More recently, our group identified miR‐146a as functionally relevant for γδ T cell differentiation, and an important determinant of the limited functional plasticity of γδ27−Vγ4−/Vγ6+ γδ T cells) compared to γδ27+T cells, in accordance with the expected inverse correlation between the levels of a given miRNA and its target mRNA. In fact, the high expression of Nod1 in γδ27+ T cells is consistent with Nod1‐mediated promotion of IFNγ production in CD4+ and CD8+ T cells, given that Nod1‐deficient αβ T cells have impaired IFNγ responses in vivo.−/− γδ27− cells were unable to differentiate into IL‐17+ IFNγ+ double producers, in opposition to the phenotype (accumulation of IL‐17+ IFNγ+ cells) of miR‐146a−/− γδ27− cells .−/− with Nod1−/− mice  revealed that heterozygous Nod1 reduction (in miR‐146−/− Nod1+/− mice) prevented the accumulation of double producers among γδ27− T cells observed in miR‐146−/− Nod1+/+ mice. These results collectively indicate that Nod1 is the key miR‐146a target implicated in the regulation of γδ27− cell plasticity. Of note, until now, only one other miRNA has been implicated in T helper cell plasticity, miR‐10a, which restricts regulatory CD4+ T cells from acquiring Th17 and follicular helper T cell characteristics.A differential Argonaute‐2 (Ago2) RIP‐seq strategy allowed us to identify Nod1 as a novel target of miR‐146a in γδ T cells,In sum, we have shown that miR‐146a limits γδ T cell plasticity by targeting Nod1, an intracellular pattern recognition receptor that is an important mediator of endoplasmic reticulum (ER) stress‐induced production of pro‐inflammatory cytokines,4.4Peripheral effector γδ T cell responses rely either on the activation and expansion of thymically  pre‐programmed cells; or on activation and de novo differentiation of effectors from naïve γδ T cells exported from the adult thymus.−CD27 pathway, besides critical during γδ thymocyte development,+ γδ T cell responses via expansion of γδ27+ cells.+ cells, thus controlling their responses to viral and parasitic infections in vivo.Plasmodium berghei infection, which contrasted with the γδIFN‐specific effect of CD27.Our group has shown that the CD70+ Vγ1+ T cells, which protected mice from Listeria infection in an IFNγ‐dependent manner.+ Vγ6+ T cells in mucosal tissues in the steady‐state and upon Listeria infection, which associated with reduced bacterial clearance, and could be rescued upon administration of an agonist anti‐CD30 antibody.Other “costimulatory” (or inhibitory) receptors reported to differentially impact on γδIFN and γδ17 T cells are CD137, CD30, BTLA and PD‐1. Agonist anti‐CD137 (4‐1BB) antibodies promoted the expansion of IFNγ− cells in vitro, even in the absence of TCR stimulation.− T cells,Another important molecular layer of γδ T cell activation and differentiation are cytokines, including usual suspects like IL‐2Finally, IL‐1β and especially IL‐23 can also drive de novo differentiation of γδ17 cells from naïve peripheral γδ T cells. While the acquisition of the IL‐17‐producing capacity of “natural” or thymic γδ17 T cells occurs exclusively during fetal/ perinatal development,5This review focused on several layers of regulation of mouse effector γδ T cell differentiation, while highlighting our group's main contributions. Despite the significant progress made over the past decade, various issues remain incompletely understood.The signals involved in dictating effector cell commitment continue to be an area of intensive research, now benefiting from full transcriptomic comparisons and pseudotime alignments of single thymocytes through the use of single‐cell RNA sequencing approaches. These may allow the dissection of novel TCR‐dependent vs independent thymic γδ T cell developmental pathways, as well as resolve the contradiction of having low or no TCR signalling in γδ17 T cell development within a γδ T cell lineage promoted (at the αβ/γδ bifurcation) by strong TCR signals. Moreover, such single‐cell approaches will permit further validation of the proposed model suggesting that γδ17 and γδIFN T cells arise from distinct thymic progenitors.bona fide ligands remains a priority,Besides the TCR, for which the identification of il17a‐GFP Ifng‐YFP), which we are using to define the full mRNA and miRNA transcriptomes of “pure” effector γδ T cell subsets. By combining with single‐cell technologies it will be possible to enquire the potential heterogeneity even within IL‐17‐ or IFNγ‐expressing populations.Following the seminal research performed on the basis of surface markers or TCR Vγ chain usage, we believe future studies should employ cytokine reporter mice to isolate pure populations of IL‐17‐ or IFNγ‐expressing cells, so that cellular and molecular properties can be directly associated to effector functions (within heterogeneous γδ T cell subsets). We have generated mice with reporter gene markers for both cytokines .Tcrd‐Cre showing limitations to delete genes in most γδ T cell populations (except DETC).Another critical aspect will be how to best assess the function of particular genes within the γδ T cell lineage. Indeed, a major limitation of our (and other group's) studies so far is the analysis of γδ T cell phenotypes in full KO animals, rather than having specific gene ablation in γδ T cells. Although some promoters, such as Upcoming research should further elucidate the extracellular signals that drive peripheral effector γδ T cell responses. A current view that requires more experimental support is that γδ17 cells respond to innate signals, whereas γδIFN cells may be involved in adaptive‐like responses, including antigen specificity . This is an outstanding topic also in human γδ T cell biology.Finally, while this review was focused on IL‐17 and IFNγ production as hallmark effector functions of murine γδ T cells, one should highlight their versatility in mice and humans: besides being highly cytotoxic, as acknowledged for decades and currently being explored for cancer immunotherapy,The authors declared no conflicts of interest.