We consider call option prices close to expiry in diffusion models, in an asymptotic regime (“moderately out of the money”) that interpolates between the well‐studied cases of at‐the‐money and out‐of‐the‐money regimes. First and higher order small‐time moderate deviation estimates of call prices and implied volatilities are obtained. The expansions involve only simple expressions of the model parameters, and we show how to calculate them for generic local and stochastic volatility models. Some numerical computations for the Heston model illustrate the accuracy of our results. Today's price of the underlying, the spot value S0, is known and fixed. Discrete option data are available from the market, typically quoted in (Black–Scholes) implied volatilities; see Figure Consider a European call option struck at C as t↓0 exhibits very different behavior in the respective cases K>S0 (“out‐of‐the‐money”) and K=S0 (“at‐the‐money”). We argue that there is a significant asymptotic regime in between, namely,More specifically, small‐maturity approximations of option prices have been studied extensively in recent years. Starting with Carr and Wu , it was c for the normalized call price as a function of log‐moneyness k=log(K/S0)c depends tacitly on S0, the (fixed) spot value.k=0. In the Black–Scholes model, writing c=c BS  with volatility parameter σ>0, we have the following ATM call price behaviorσ0=v0>0),v0>0)σ imp  for the Black–Scholes implied volatility with log‐moneyness k and maturity t.) Higher order terms in t will be model dependent. For instance, in the Heston case, with variance dynamics dV=−κ(V−v¯)dt+ηVdW, implied variance has the ATM expansiona(0) has no easy interpretation in terms of the model parameters.To put our results into perspective, we recall some well‐known facts on option price approximations close to expiry. We write and Nutz , the sam and Lee , and we k=0 to kt=o(t) amounts to what we dub “almost ATM” (AATM) regime.kt∼tβ is in the AATM regime if and only if β>1/2.) Again for generic semimartingale models with diffusive component and spot volatility σ0>0, it is easy to see from Caravenna and Corbetta (kt ceases to be o(t). Indeed, for kt=θt with constant factor θ>0, we have, from Caravenna and Corbetta  stands for a Gaussian random variable with mean −θ and variance σ02. This, too, holds true in the stated semimartingale generality. In any case, the proof is based on the Lévy case with nonzero diffusity v0, and the result follows from comparison results, which imply that the difference is negligible to first order. For a thorough discussion of the regime k=O(t) in the  diffusion case, see Pagliarani and Pascucci Pascucci .log(1/t),t‐behavior of call prices described above and in factℓ(t), see Mijatović and Tankov =12k2/σ2 in the Black–Scholes model. Similar results appear in the literature, with different levels of mathematical rigor, for other and/or generic diffusion models; see Berestycki, Busca, and Florent , which we call moderately out‐of‐the‐money (MOTM). We have a threefold interest in this MOTM regime,Throughout the paper, we reserve the term OTM for (i)reality of quoted (short‐dated) option prices, where strikes of option price data with acceptable bid–ask spreads tend to accumulate “around the money,” as illustrated in Figure k=O(tβ) for some β>0. There is no reason why quoted strikes should always be almost ATM (β>1/2), which effectively means an extreme concentration around the money; we are thus led to study the regime First, it is related to the (ii)mathematical convenience. In contrast to the genuine OTM regime (large deviation regime) in which the rate function Λ(k) is notoriously difficult to analyze—often related to geodesic distance problems—MOTM naturally comes with a quadratic rate function and, most remarkably, higher order expansions are always explicitly computable in terms of the model parameters. The terminology moderately OTM (MOTM) is in fact in reference to moderate deviations theory, which effectively interpolates between the central limit and large deviations regimes.The second reason is (iii)Finally, our third point is that MOTM expansions naturally involve quantities very familiar to practitioners, notably, spot (implied) volatility, implied volatility skew, and so on.fractional stochastic volatility models, see Forde and Zhang (first‐order moderate deviation principle (MDP) for diffusions under classical regularity assumptions from SDE theory. The main difference to the bulk of our results is that we develop higher order expansions, until Section In the Black–Scholes model, it is easy to check that we have the MOTM asymptoticsTheorems  below asTheorems ; then, cnd Zhang  and Guennd Zhang . Guillinnd Zhang , who connd Zhang , the dynnd Zhang  contains(Xn)n≥1 with finite exponential moments. Then the empirical means X^n:=n−1∑k=1nXk converge to zero , and this is quantified by an LDP according to Cramér's classical theorem: P[X^n>x]=exp(−I(x)n+o(n)) decreases exponentially as n→∞ for fixed x>0, governed by a rate function I(x)=supy∈R(yx−logE[eyX1]). On the other hand, by the CLT, nX^n=n−1/2∑k=1nXk has a Gaussian limit law. Moderate deviations cover intermediate scalings, i.e., nanX^n with an→0 and nan→∞. It turns out  This is sometimes called an MDP. Formally, an MDP is thus just a certain LDP with appropriate scaling and speed function. Still, the terminology is often useful because of the trichotomyTo round off the introduction, we briefly recall some background on moderate deviations. Consider the classical setting of a centered i.i.d. sequence Zeitouni . SeveralZeitouni  and refeS0=1.The rest of the paper is organized as follows. Section 2general stochastic volatility model, i.e., a positive martingale (St)t≥0 with dynamicsS0=1. We assume that the stochastic volatility process (σt)t≥0 itself is an Itô‐diffusion, started at some deterministic value σ0, called spot volatility. Recall that in any such stochastic volatility model, the local (or effective) volatility is defined byS∼t=St (in law) for all fixed times. See Brunick and Shreve  match in both models. Recall also Dupire's formula in this contextAssumption 2.1t>0, St has a continuous pdf K↦q, which behaves asymptotically as follows for small time:K=ek in some neighborhood of S0=1. The energy function Λ is smooth in some neighborhood of zero, with Λ(0)=Λ′(0)=0. Moreover, limk→0γ(k)=γ(0)>0.For all We consider a d Shreve  for precAssumption 2.2t↓0 and K→S0=1, the local volatility function of (St)t≥0 converges to spot volatilityFor 0  here) and the Hessian of the energy function Λ=Λ(k),Theorem 2.3ℓ>0 varies slowly at zero and β∈.(i)The call price satisfies the moderate deviation estimate(ii), then the following moderate second‐order expansion holds trueIf we restrict β to Under Assumptions σ02, equal to σ imp 2, and implied variance skew S=∂∂k|k=0σ imp 2.with spot‐variance The latter assumption is fairly harmless  expansionθ↦θ2/2σ02, typical of moderate deviation problems.In particular, for c=exp(−Λ(k)/t(1+o(1))) yields a correct result, upon Taylor expanding Λ. Mind, however, that this needs a proof using the specifics of our situation, in light of the fact that validity of a large deviation principle does not automatically imply an MDP.In a nutshell, Λ′′(0),Λ′′′(0),⋯ appearing above are always computable from the initial values and the diffusion coefficients of the stochastic volatility model. This is in stark contrast to the OTM regime, where one needs the function Λ(·), which is in general not available in closed form . We quote the following result on N‐factor models from Osajima  is Markov, started at  with σ¯0∈RN−1 and σ¯01>0, with stochastic volatility σ≡σ1, where the generator has (nondegenerate) principal part ∑aij∂ij in the sense that a−1 defines a Riemannian metric. ThenAssume that The quantities  Osajima  and refeT=1).□See Osajima .1/β is not an integer, then ktm/t tends to infinity for m=⌊1/β⌋, of order tβ⌊1/β⌋−1 (up to a slowly varying factor). If, on the other hand, 1/β is an integer, then the last summand of the sum ∑m=2⌊1/β⌋ in ℓ(t), which means that the following term log(1/t) may be asymptotically larger. The upper summation limit ⌊1/β⌋ thus ensures that no irrelevant ) terms are contained in the sum. Note that ⌊1/β⌋=2 for β∈, and ⌊1/β⌋≥3 for β∈, and so If Theorem 2.6Λ′′(0)=σ0−2=v0−1, and that the Berestycki–Busca–Florent formula σ imp 2=k2/2Λ(k) holds. Then the small‐time ATM implied variance skew and curvature, respectively, relate to Λ viaSuppose that Λ is a function with the properties required in Assumption The passage from the derivatives of the energy function to ATM derivatives of the implied volatility in the short time limit is best conducted via the BBF formula that was proved in Berestycki et al. . That s. In this□By the BBF formula and our smoothness assumptions on Λ,Proposition 3Proof of Theorem 2.3St satisfies q=∂KKC, we have, by Dupire's formula Kt=ekt with kt↓0 as stated, we apply Assumption t↓0,Λ(0)=Λ′(0)=0, we havet, the integrand in x=1, and by the Laplace method  if β∈, and that kt3/t dominates log(kt2t−3/2) if β∈. For m∈{2,3}, we calculatelogℓ(t)=o(logt), and som=2 and β∈, and for m=3 and β∈, as desired.□As the density of  Teugels , we knowlogc further.Inspecting the preceding proof, it is easy to see that we can expand Proof of Theorem 2.5ktm/t=o(1) for m≥⌊1/β⌋+1.□Taking logs in 4Corollary 4.1kt=tβℓ(t) with β∈ and ℓ>0 slowly varying. Then the implied volatility has the MOTM expansionUnder the assumptions of Theorem L, V of Gao and Lee  resp. V=t1/2σ imp , the dimensionless implied volatility. Then corollary 7.2 of Gao and Lee  in Bingham et al., □We use our main result Theorem  in conju and Lee  means L= and Lee  implies β∈, under very mild assumptions , one can compute the implied volatility expansion using the results of Gao and Lee  if β is close to zero. Suppose, on the other hand, that β is very close to 12: Then the summands m>⌊1/β⌋=2 in o(1)‐term of We have no doubt that Corollary  and Lee . Howevert≪kt, and so the k‐term dominates the O(t)‐term, which in turn identifies the implied variance skew as−13σ04Λ′′′(0). We have thus arrived at an alternative proof of the skew representation Corollary 55.1C2 on . An expansion of the pdf q of St has been worked out in Gatheral, Hsu, Laurence, Ouyang, and Wang  This shows that Assumption kt=θtβ and θ>0S the  implied variance skew, and so the implied volatility skew is given by S/2σ0, which equals S/2σ(1)=S/2σ(S0) in model local skew σ′(1)=σ′(S0) equals twice the implied volatility skew,Clearly, Assumption and Wang . They asand Wang . Proposiand Wang  states tabordère . Generic)2,and5.u0=σ5.22Λ(k) is the squared geodesic distance from  to {:σ>0} with K=ek. Theorem 2.2 in Berestycki, Busca, and Florent ≡1) and the 3/2‐model (ν(v)=v; see Lewis, L of the stochastic process , neglecting first‐order terms, can be written asD2f denotes the Hessian matrix of f, and the coefficient matrix g=(gij) is given byb1=g11|v=v0=v0 and b2=34∑i=12g1i∂ig11|v=v0=34ρηv0ν(v0). If we assume that the coefficients in k→0. Therefore, the quantities Λ′′(0)=v0−1=σ0−2 and Λ′′′(0)=−32ρην(v0)/v02 can easily be computed, as well as the small‐time ATM implied variance skewNow we describe how the expressions appearing in the expansions from Theorem  Theorem  have ver5.3v¯,κ,η>0, and d⟨W,Z⟩t=ρdt with ρ∈. According to Forde and Jacquier (He is the (not explicitly available) Legendre transform ofρ¯2=1−ρ2.) This expansion impliesThis section contains an application of the results of Sections Jacquier , the firSt, we get the density approximationk, where φt is the characteristic function of Xt=logSt, and p∗ and U are defined on p. 693 of Forde et al.  From U(p∗(0))=U(0)=1, we see that the factor γ(k) from k→0.The locally uniform density asymptotics e et al. .  in the numerator of St, the analysis of which we have just described. Virtually the same saddle point approach can be applied to the numerator ∂tC, yielding convergence of the quotient to σ02.To verify Assumption He is given by Λ He (k)=supx{kx−Γ(x)} with maximizer x*=x∗(k). From general facts on Legendre transforms,x∗(0)=0, which implieskt=θtβ and θ>0, we then obtain the MOTM call price estimateβ∈, Theorem t↓0:We now calculate our MOTM asymptotic expansions for the Heston model. The Legendre transform ΛΛ′′′(0), we get the explicit expression S He =ηρ/2 for the skew. This agrees with Gatheral  However, Gao and Wang Assumption 6.1β∈, the rescaled mgf satisfiesFor all In this section, we discuss a different approach at small‐time moderate deviations. While yielding only first‐order results, its conditions are usually easy to check for models with explicit characteristic function. Assumptions We expect that this assumption holds for diffusion models in considerable generality. It is easy to check that q by a Freidlin–Wentzell LD argument for x sufficiently large. As for x≈0, because Λ(x) increases with |x|. Finally, Heuristically, Assumption St is defined byp+(t) must grow faster than tβ−1 as t↓0. In the Heston model, e.g., the critical moment is of order p+(t)∼(const)/t≫tβ−1 for small t, as follows from inverting p+(t)≡p+ does not depend on t, and is finite for most models used in practice. Therefore, p+=∞, but it is easy to check that it does not satisfy The critical moment of Xt  in Theorem call prices in Theorem Theorem 6.2kt=θtβ with β∈ and θ>0, we have a first‐order MD estimate for the cdf of Xt:Under Assumption After this discussion of Assumption ΓZ is finite and differentiable on R, the Gärtner–Ellis theorem (theorem 2.3.6 in Dembo & Zeitouni (Zt)t≥0 satisfies an LDP as t↓0, with rate at and good rate function ΛZ, the Legendre transform of ΓZ. Trivially, ΛZ is qquadratic, too:θ>0. Applying the lower estimate of the LDP to  yields is finite for all t∈. Let v0=σ02>0. Then the following are equivalent(i)kt=ℓ(t)tβ, with β∈ and ℓ>0 slowly varying at zero, it holds thatFor (ii)Under the assumptions of (i), we haveLet As in the LD/OTM regime, first‐order cdf asymptotics translate readily into call price asymptotics. The proof of the following result is similar to Pham  by a milε>0 and define k∼t=(1+ε)kt. Thenk∼t.ε↓0 to get the desired lower bound for c.First assume (i). Let p>1 and note that, by definition of p↦t∗(p), we have E[Stp+1]<∞ for all t∈. Define S¯t=sup0≤u≤tSu for t≥0. By Doob's inequality  has a finite pth moment,S, we thus conclude1/p+1/q=1 and apply Hölder's inequality.p↑∞, i.e., q↓1. The same argument yields the lower bound of the implication (ii) ⟹ (i). The remaining upper bound of (ii) ⟹ (i) is shown very similarly to the lower bound of the implication (i) ⟹ (ii).□As for the upper bound, we let kt=ℓ(t)tβ is replaced by t≪kt≪1.In the light of the general MDP result by Gao and Wang  quoted a


We show that an analogue of the ball-box theorem holds true for a class of corank 1, non-differentiable tangent subbundles that satisfy a geometric condition. We also we give examples of such bundles and give an application to dynamical systems. C1 contact bundles. Another place where sub-Riemannian geometry pops up in dynamical systems is in (p) = TpM, then such a bundle is called step 2, completely non-integrable at p.ℓ(⋅) denotes length with respect to the given metric g on Δ. dΔ is called the sub-Riemannian distance. We let BΔ denote the ball of radius 𝜖 around p with respect to the distance dΔ. Given p ∈ M and a coordinate neighbourhood U, coordinates z =  are called adapted at p for Δ if, p = 0, q is transverse to q ∈ U. Given such coordinates and α > 0, we define the coordinate weighted box as follows: ball-box theorem ))K1, K2, rms (see  to applyC1 bundles). Motivated by the discussion above, our proof is very geometric and uses Stokes’ theorem as the main tool rather than classical ODE analysis. The techniques that we use are inspired from a proof of the Frobenius theorem given in . We denote this integral curve by t → etX(q).We call a continuous vector-field uniquely integrable if it is integrable and if X defined on U1 has Cr family of solutions if there exists some U2 ⊂ U1 and an 𝜖0 such that for all p ∈ U2, there exists an integral curve passing through p with 𝜖p ≥ 𝜖0 so that this choice of solutions seen as maps from  × U2 → U1 are Cr.We say that an integrable vector field Next proposition is almost a direct corollary of the Theorem 6.1  be continuous. Then, the ODEhasunique andC1solutionsy(t) = η,y(t0) = y0for all  ∈ U, if for anyp ∈ U, there exists a neighbourhoodUp, a non-singularn × nmatrixA and a continuousn × mmatrixC such that the 1 −formshavecontinuous exterior derivative.z1,…,zm). This generality is not needed for our purposes; therefore, we assume that m = 0 . For thC1 solutions above is equivalent to unique integrability of the extended vector field C1 family of integral curves which would be given by t → ) ⊂ U for t small enough. Let t, y) space. Then, this bundle is the intersection of the kernel of the 1-forms ηi = dyi − fidt, i.e ηi is a basis of sections for A simply asks that there must exist a basis of sections for C1 solutions. To prove Proposition 2.1, we will show that the conditions given there imply that each vector field Xi is equipped with a continuous exterior derivative. We have that αj = dxj for j≠i and αi = η which is the 1-form given in the proposition. Note that η(Xi) = 0 since η annihilates Δ which contains Xi. Moreover, by our choice of adapted coordinates, η≠dxi on some neighbourhood around p0. Now by assumption, η has continuous exterior derivative and dxj has continuous exterior derivative 0, so Xi is uniquely integrable. □We start by translating the terminology of Theorem 2 to something more closer to ours. Note that the unique integrability of the non-autonomous ODE U is given, we can find 𝜖0 be such that for all k = 1,...n, t ≤ 𝜖0, C1. We will impose some more conditions which will fix U once and for all for the rest of the paper. The conditions are as follows:k = {1,…,n}For all, i, j ∈{1,…,n}p ∈ U.For some, i, j ∈{1,…,n}p ∈ U and for some constant B(n) > 0.For some, With this proposition, once a suitable neighbourhood Xk(0) = ∂k and η is non-involutive at 0.These are possible since 𝜖0 starting with choosing it small enough so that K2 is as in Theorem 1 and We will also do some modifications on Xk|∞≤ 2 is that uniqueness and differentiablity of solutions is extended to n + 4-tuple compositions. That is for all ik ∈{1,…,n} and k = 1,…,n + 4, one has that the following is well defined:C1 map with respect to q and 𝜖 ≤ 𝜖0 where K2 is as in Theorem 1.One consequence of this and the fact that |E(n),G(n) > 0 we fill shrink 𝜖0 so that 𝜖0 later on.In the remaining part of the proof, there will be places where for some constants γ :  → U with 𝜖 ≤ 𝜖0 that are either unit parameterized or satisfy t. Therefore, the reader should keep in mind that for such curves x be the projection to x coordinates.From now on, we will always be working with admissible curves Letγℓ :  → Uforℓ = 1, 2 be two Δ −admissiblecurves such thatγ1(0) = γ2(0) = q, Πxγ1(ε1) = Πxγ2(ε2), Letε = max{ε1, ε2} andβbe the segment in the∂ydirection that connectsγ1(ε1) toγ2(ε2). Letαℓbe the 1-chains which are projections ofγℓalong∂yto Δqand assume thatαℓ ⊂ U. Then, for any 2-chainP ⊂Δq ∩ Uwhose boundary iswe have thatandγ1(ε1) = q1, γ2(ε2) = q2. Assume w.l.o.g that ∂y direction. Since γℓ are admissible curves, we have that t a.e. for some piecewise C0 functions αℓ are integral curves of Zi starting at t = 0 and q which are inside Δq ∩ U. In particular, U bounded by these 1-chains as follows: εℓ] to U. Since αℓ(s) and γℓ(s) are piecewise C1 in the s variable, the domain of this map can be partitioned into smaller rectangles on which vℓ are differentiable and therefore whose images are two cells. Then for each ℓ the images of vℓ become 2-chains which we denote as Cℓ. Note that v1 = v2 = q for all t. And also let the image of v2 be a curve τ. Since v1 is β ⋅ τ. It is also clear that vℓ = αℓ(s) and vℓ = γℓ(s). Then we orient these curves and Ci such that modulo some 1-chains whose image is q: ∂C1 = γ1 − β − τ − α1, ∂C2 = τ − γ2 + α2.Denote U bounded by concatenating γ1, γ2 and β  in the right orientation so that ∂Γ = β − γ1 + γ2. Finally, also orient P so that ∂P = α1 − α2. Then, Γ,C1, C2 and P form a closed 2-chain CC ≠ 0 for all p ∈ U and that Since Now using Eqs. P satisfies some properties, we can get a more precise estimateIn the case Assume thatP, γ1, γ2, qare as in Proposition 2.4, P ⊂ B𝜖0), area of P is less thanD(n)ε2andε ≤ F(n)𝜖0for someD(n),G(n),F(n) > 0. Then for𝜖0small enough, there exists a constantC(n) such thatMoreover, ifPis everywhere tangent toXi(q), Xj(q) (q fixed),andP is everywhere tangent to Δq, q has modulus of continuity ω, we have that E(n) > 0. But |p − q|≤ G(n)𝜖0. Choose 𝜖0 small enough so that 𝜖0 small enough so that P is everywhere tangent to Xi(q),Xj(q) for q is fixed. So in this case, we use the fact that (letting p ∈ P) 𝜖0 small enough so that dηp(Xi(p),Xj(p))≠ 0 and never changes sign then dηp(Xi(q),Xj(q)) never changes sign. Then choosing a parametrization P such that k = i, j, one gets𝜖0 small enough so that c and Eq. η(∂y) always has constant sign which wlog we can assume positive. So signSince 𝜖, we will construct a certain n-dimensional manifold Xi, which is admissible and transverse to ∂y direction. This is carried out in Section The part of the Theorem 1 about smooth adapted coordinates is divided into two separate parts. First given 𝜖|≤ 2n𝜖0, due to condition in Eq. C1 function T𝜖 : n → VXi are uniquely integrable and since due to their form, an integral curve of Xi can intersect an integral curve of Xj for i≠j only once. Therefore, the image of T, which we denote as C1 manifold. Moreover, every point on it is obviously accessible. Finally, it can be given as a graph over , in fact due to the form of the vector fields T𝜖 = ) for some C1 function a. In particular, note that if  = T𝜖, then |x| = |t|. Also Given any |Letbeas above and |𝜖|≤ 2n𝜖0. Then,For anyp =  ∈ Uwith |xi|≤ 𝜖, there exists a uniquewiththe samex coordinates as p.For anyonehas that |y|≤|x|C(n)ωx|2|.ti = xi one has T𝜖 = ). For the second, letting q is the end point of composition of flows tangent to Xi which have modulus of continuity ω, total time of flow less than |x| and length less than 2|x|. Since the flow starts at the point 0, where Xk(0) = ∂k, its rise in y coordinates is bounded by |x|C(n)ωx|2|. □For the first item, taking p =  with |xi|≤ 𝜖, p to the point q on x coordinate as p. For |𝜖|≤ 𝜖0, we define the boxes K1𝜖2 (which are inside U by conditions on 𝜖0). Next, Lemma finalizes the proof of existence of smooth adapted coordinates in Theorem 2.Define for There exists an𝜖0 > 0 andK1, K2 > 0 such thatfor all𝜖 ≤ 𝜖0onehas,𝜖0 that has been fixed before Eq. 𝜖0) for all 𝜖 ≤ 𝜖0, one has V𝜖, we can travel with admissible curves of length less than or equal to ∂y up to some uniform amount 𝜖 and reach anywhere in τ :  → U with We start with the Now, we build the curves that allows us to travel horizontally from any κ4 ∘ κ3 ∘ κ2 ∘ κ1, so that s with ℓ(s) = i, j. Also U and always lies on the ∂y axis passing above q1. Moreover, by Eq. n + 4 integral curves of some ± Xℓ with integration times less than 𝜖0. Denoting the image of this curve as 𝜖 ≤ 𝜖0 such that setting There exists some Xi or Xj, we can go in the opposite direction along the ∂y.By reverting either We let ∂y) while the first item guarantees that we can travel more than |p − q1|. Then, by mean value theorem by reducing 𝜖, we can reach any point on the ∂y axis passing through q1 whose distance to q1 is less than Note first that these are admissible curves which have length less than γ2 = κ3 ∘ κ4 (defined up to γ1(0) = γ2(0) = q and Πx(γ1(ε1)) = Πx(q1) = Πx(γ2(ε2)) = Πx(q2). Projection of γi along ∂y to Δq gives us αi the boundary of a parallelogram P with sides’ lengths less than αi contain q and |q|≤ 𝜖, they are inside B ⊂ U  + ℓ(γ1) ≤ 2𝜖. Therefore, P ⊂ B𝜖). Also area of P is less than D(n)𝜖2. So since γ1(𝜖1) = q1, using Eq. Now, it remains to show that The claim about opposite directions is an automatic consequence of Eq. p =  ∈ U such that dΔ ≤ 𝜖, then |x|≤ 2n𝜖 and K2.To prove this inclusion, we need to prove that if dΔ ≤ 𝜖 means that there exists an admissible curve γ1 :  → 0 that connects 0 to p such that ℓ(γ1) = 2𝜖. Since for all i, |xi|≤ ℓ(γ1) = 2𝜖 we get |x|≤ 2n𝜖. So it remains to estimate The condition dη|∞ appearing there is replaced by |dη|Δ|∞ which will be thanks to Proposition 2.4. We first state a lemma ) ⊂ U. Then, there exists a 2-chain cc such that∂(cc) = c, a(cc) ≤ C(n)ℓ(c)2andcc ⊂ B) wherea(⋅) denotes Euclidean area.x|≤ 2n𝜖, there exists a point q on q − p|≤ K24n2𝜖2 for some K2. We can connect q to 0 by along γ2 and apply Proposition 2.4 to the curves γ1, γ2 again  = γ2(0) = 0 and γ1(𝜖1) = p, γ2(𝜖2) = q which have the same x coordinates. Then, Now since |γ1 and γ2 along ∂y to Δ0 forms a closed piecewise C1 curve 𝜖 + 4n𝜖 ≤ 6n𝜖. Since they also contain 0, these curves are in B which is in U by condition in Eq. P that satisfies the conditions in Proposition 2.4 and Corollary 2.6. By Eq. 2.4B) ⊂ B ⊂ U. So by Lemma 2.9, α bounds a 2-cycle P ⊂ B ⊂ U ∩Δ0 such that |P|≤ C(n)𝜖2. Since γ1(𝜖1) = p and γ2(𝜖2) = q we have by Eq. The projection of Now, it is easy to prove the rest using Lemmas 2.7 and 2.8. Note again that for x|≤ 2n𝜖 and z − y|≤ 4K2n2𝜖2. But we know that First, we prove that Then, we prove C1 adapted coordinates. Note that for each 𝜖, C1 surfaces given as the image of the map T𝜖 with |ti|≤ 𝜖 and for 𝜖1 < 𝜖2, ϕ : V → U on some appropriately sized domain V ⊂ U such thatC1 diffeomorphism onto its image and takes each 𝜖 ≤ 2n𝜖0) to the x plane. It also maps Xℓ(0) to ∂ℓ. This is again an adapted coordinate system. In particular in this adapted coordinate system x|≤ 𝜖 and |y|≤ K𝜖2. So we get Finally, we prove the statement about the existence of d2 = 0 and usual homotopy theory, but it is out of the scope of this paper. These will be consider in a separate paper but the interested reader can have a look at the preprint [In this section, we give some of the very basic properties of continuous exterior derivative preprint .To construct examples, we give an alternative characterization Lemma 3.1 which is proven in  -form inC0topology, which becomes the continuous exterior derivative ofη.Although this construction is local, one can show that the usual glueing techniques can be used to produce global examples. Since we are only interested in local examples here, interested read should refer to . A classLetsuchthat eachaiis continuous and has continuous partial derivatives with respect to allxjforj≠i. Then,ηhas continuous exterior derivative.C1 functions (by mollification) C0 topology to ai, such that i≠j. Then, define ηk converges in C0 topology to η and dηk converges to dη is the continuous exterior derivative of η by Lemma 3.1. □By assumptions on regularity of the functions, we can find Now, we create an example from η = ady + bdx + cdz. Then since a = 1. To create a non-integrable and non-differentiable example, we have to satisfy (cx − bz + byc − cyb)(p) > 0 at some point where η is non-differentiable. Consider α, β > 0, which gives η (or any of its product with a differentiable function) is non-differentiable at x = 0,z = 0,y = 0 but η ∧ dη(0) = dx ∧ dz ∧ dy. Therefore, there exists a neighbourhood of 0 on which η is defined but non-differentiable and yet satisfies the sub-Riemannian properties mentioned in this paper. To create a non-Hölder example out of this, we can replace for instance x = 0,z = 0,y = 0 and also non-integrable.We first consider the class of examples given by Lemma 3.2. We restrict our-self to It is possible to given an example in which non of the components are differentiable in any variable but it requires to prove some analytic properties of continuous exterior derivative. We state the example without proof but the interested reader can see Example 2.17 and Lemma 2.15 in  for the Consider the following 1 −form In this subsection, we give an application to dynamical systems.M be a compact Riemannian manifold of dimension n + 1 and f : M → M is a C3 diffeomorphism. Assume moreover that there exists a continuous splitting Dfx. This splitting is called partially hyperbolic if there exists constants K, λσ, μσ > 0 for σ = s, c, u such that μs ≤ 1, μs < λc, μc < λu, λu > 1 and for all σ = s, c, u, x ∈ M and vσ ∈ TxM with |vσ| = 1 one has, f, the Dfx expands Eu and Es are uniquely integrable into what are called as unstable and stable manifolds. Given p ∈ U, we denote the connected component of the stable and unstable manifold in U that contains p as Ec, Ecs = Ec ⊕ Es or Ecu = Ec ⊕ Eu maybe be non-integrable both in the case the bundles are continuous (see [Let ous (see ), a wellous (see ) and or ous (see ). The inous (see ). It is Ec and Es are C1 and center bunched, partially hyperbolic then Ecs is uniquely integrable. As far as we are aware, there is no general result on integrability of such continuous bundles that do not make any assumptions on the differentiable and topological properties of the manifold M  inside partially hyperbolic systems.A dynamical assumption that we will make is center bunching. A system is called center bunched if instance  where ceinstance  where a instance , the autinstance  where thtorus or  where thOur aim is to make one small step towards an integrability condition for continuous bundles that relies only on the constants. The only place where differentiability is required in the proof of the theorem in  is whereAssumef : M → Mis a diffeomorphism of a compact manifoldwhich admits a center bunched partially hyperbolicsplittingwhereAssume moreover thatadmitsa continuous exterior derivative. Then,Ecsis uniquely integrable with aC1foliation.η ∧ dη(p) > 0 where Ecs = ker(η) and dη is the continuous exterior derivative. Then by Theorem 1, there exists a C1 adapted coordinate system and a neighbourhood U of p on which every point q on the local unstable manifold κ(t) such that κ(0) = p, κ(ℓ(κ)) = q and d ≥ c|κ|2. To show that the last condition can be achieved, we choose first a smooth adapted coordinate system at p for Esc so that the unstable bundle is very close to the ∂y direction (since both are transverse to Esc this is possible). By choosing it close enough, we can make sure that when we pass to the C1 adapted coordinates using the transformation given in Eq. ∂y direction are still close enough so that in a small enough neighbourhood U and for any xi|≤ δ|y| where C1 adapted coordinate system, for 𝜖 small enough, we pick y| = K1𝜖2. But the ball-box theorem tells us that there exists a length-parameterized admissible curve κ such that ℓ(κ) ≤ 𝜖, κ(0) = p and κ(ℓ(κ)) = q. Since p = 0). Therefore, for some constant c, η ∧ dη = 0 everywhere. Then, by the integrability theorem of Hartman in [Ecs integrates to a unique C1 foliation. □As in , one stam 4.1 of  are fullrtman in , this me


Anatidae); herons and egrets (Ardeidae); terns (Sternidae); and gulls (Laridae). This article describes the level of lead in the most commonly studied tissue types: feathers, bones and the liver. The study also presents data concerning the concentration of lead in the eggs of water birds. The highest levels of lead pollution can be observed in China and Korea, related to their high level of industrialization. In Iran too, environmental lead pollution is high, likely due to the developed petrochemical industry. Lead pollution in Japan, as well as in Western European countries , seems to be much lower than in China, India or Iran. Nevertheless, the level of pollution in Europe is higher than satisfactory, despite the introduction of a number of bans related to, for example, the use of leaded petrol or lead-containing paints. Finally, the USA and Canada appear to be the areas with the lowest lead pollution, possibly due to their low population densities.Due to the ability of birds to travel long distances in the air, the potential feeding area of each individual is much larger than that of typical terrestrial animals. This makes birds a convenient indicator of environmental lead (Pb) pollution over large areas, in particular areas of inland and coastal waters. The aim of this study was to assess the concentrations of Pb in various organs of water birds from a variety of locations. The focus was on ducks, geese and swans ( The ubiquity and toxicity of lead (Pb) have it ranked as the second most dangerous environmental poison in the world ATSDR .Currently, Pb is still used in the production of batteries, fishing sinkers and bullets for firearms . Nevertheless, all these birds play essential roles in the functioning of the ecosystem and, due to their position in the trophic chain, they are susceptible to bioaccumulation of pollutants, including heavy metals. All individuals differ in body weight, metabolic rate, habitat, range and diet, and therefore, it can be expected that birds living in the same area will accumulate Pb in different amounts depending on these factors. Many of these birds can also live directly in or on the outskirts of cities . Moreover, many of them are hunted species, which makes them available for ecotoxicological studies (e.g. ducks).The aim of this study was to make a comparative analysis of Pb content in the liver, bones, eggs and feathers of birds living in the Northern Hemisphere on the basis of published data from the last 20 years (1998–2018). The available literature provided data on members of the families With the wealth of available information using wild bird species as biomonitors, it should be possible to assess the state of the entire aquatic ecosystem of the Northern Hemisphere in terms of Pb pollution over of the last 20 years.https://www.ncbi.nlm.nih.gov/pubmed/) was used to search for all articles on a given topic  and ”Sternidae and Laridae: “(Pb or lead) and (gull or larus or tern or sterna)”Ardeidae: “(Pb or lead) and (heron or egret)”The PubMed search engine  chicks from Hongdo Island, Gyeongsangnam-do (Korea), feathers showed a lead concentration of 3.24 ± 1.75 μg/g dry weight (dw) (Kim and Oh Larus saundersi) in Dongtai, Jiangsu province (China), the concentration of lead in feathers was 2.05 ± 0.47 μg/g dw  population in the Hara Biosphere Reserve of Southern Iran, a country with a very active oil industry, the concentration of lead in feathers was 7.04 μg/g dw  collected in Midway, the measured concentration of lead was 0.289 ± 0.063 μg/g dw (Burger and Gochfeld Sterna fuscata), the Pb level was 0.519 ± 0.048 μg/g dw. In the same study, the concentrations of lead in grey-backed terns (Onychoprion lunatus) and white terns  were 0.942 ± 0.312 μg/g dw and 1.380 ± 0.693 μg/g dw, respectively. The levels of lead concentration among the brown noddy (Anous stolidus) and grey-backed tern (Onychoprion lunatus) populations nesting in Midway were 0.289 ± 0.063 μg/g dw and 0.942 ± 0.312 μg/g dw, respectively  from Long Island, NY (USA) was 4.10 ± 0.26 μg/g dw (Burger Leucophaeus pipixcan) in Agassiz National Wildlife Refuge, MN (USA), the concentration of lead was 2.86 ± 0.67 μg/g dw (Burger and Gochfeld Larus glaucescens) from the clean areas of the Aleutian Islands in Alaska (USA), the concentration of this heavy metal was 0.855 ± 0.133 μg/g dw , the concentration of lead was 0.707 ± 0.131 μg/g dw  in the National Park of the Galician Atlantic Islands on the north-western coast of Spain measured 1 year later was 1.2 μg/g dw and in 2007 as low as 0.2 μg/g dw  chick feathers, the concentrations of lead were 0.298 ± 0.065 μg/g dw and 1.365 ± 0.518 μg/g dw, respectively  feathers ranged from 0.08 ± 0.01 μg/g dw in 2011 to 0.13 ± 0.01 μg/g dw in 2012  and the USA show lower levels of lead pollution. However, more research on European and American populations of Sternidae and Laridae is required to accurately assess the state of the environment there.The aforementioned numbers demonstrate that certain Asian countries  are the most highly polluted with lead, overall. The average feather lead levels in Anatidae indicate high levels of contamination in Korea (Table Anas poecilorhyncha) and white-fronted goose  from Gimpo, Gyeonggi-do (Korea), measured lead concentrations in feathers were 1.69 ± 1.54 μg/g dw and 1.96 ± 1.04 μg/g dw, respectively (Kim and Oh Cygnus cygnus) contained 3.64 ± 1.13 μg/g dw of lead  feathers from Baroghil valley (sparsely populated areas) and Soan valley (densely populated areas), the concentration of lead was 0.91 ± 0.03 μg/g dw and 1.97 ± 0.57 μg/g dw, respectively  showed higher lead levels of 1.19 ± 0.74 μg/g dw  and 2.34 ± 0.31 μg/g dw  , pintail ducks (Anas acuta) and greylag geese (Anser anser) of 2.02 ± 3.13 μg/g dw, 3.05 ± 3.51 μg/g dw and 0.81 ± 1.16 μg/g dw, respectively  was 0.530 ± 0.066 μg/g dw  was also low, at 0.488 ± 0.075 μg/g dw , the concentration of lead in the feathers of the common eider (Cygnus olor) caught at Keszthely Bay, Lake Balaton, Hungary, the level of lead in feathers was 1.11 ± 1.23 μg/g dw  feathers were 0.45 μg/g dw and 0.18 μg/g dw, respectively  and great egret  in a recent study were 4.55 ± 0.96 μg/g dw and 5.15 ± 4.62 μg/g dw, respectively (Table Nycticorax nycticorax), the lead concentration was 5.28 ± 2.22 μg/g dw  of 4.80 ± 0.67 μg/g dw  and little egrets (Egretta garzetta), the levels were 4.6 ± 0.4 μg/g dw and 4.4 ± 0.6 μg/g dw, respectively (Burger and Gochfeld Nycticorax nycticorax) nestlings was 9.1 ± 2.2 μg/g dw (Burger and Gochfeld Egretta garzetta), lead concentrations ranged from 0.8 to 4.4 μg/g dw, depending on where the feathers were collected  and the black-crowned night heron (Nycticorax nycticorax) were 4.2 ± 1.0 μg/g dw and 5.6 ± 0.7 μg/g dw, respectively  and grey heron (Ardea cinerea) nestlings from the city of Pyeongtaek in Gyeonggi-do (Korea), the concentrations of lead in feathers were 2.65 ± 0.76 μg/g dw and 2.05 ± 1.27 μg/g dw, respectively (Kim and Oh Nycticorax nycticorax) was 2.57 ± 1.49 μg/g dw (Kim and Oh Bubulcus ibis) in the areas of Lahore and Sialkot indicate a contamination level of 297 ± 11 μg/g dw and 286 ± 18 μg/g dw, respectively  were 32.5 ± 10.3 μg/g dw (Shorkot) and 43.1 ± 13.4 μg/g dw (Mailsi)  near the Chenab River and Ravi River were as high as 37.5 ± 10.7 μg/g dw and 76.5 ± 8.6 μg/g dw, respectively  near Rawal Lake reservoir site, not far from Islamabad City, the level of lead was 60.2 ± 20.7 μg/g dw  was 4.22 μg/g dw  and the black-crowned night heron (Nycticorax nycticorax) were 1.110 ± 0.309 μg/g dw and 0.671 ± 0.105 μg/g dw  in 1989 was 1.460 ± 0.765 μg/g dw and has systematically become lower since that time  and tricolored heron (Egretta tricolor), the concentrations of lead in feathers were 0.247 μg/g dw and 0.296 μg/g dw, respectively  and black-crowned night heron (Nycticorax nycticorax) populations, the concentrations of lead in feathers were 4.52 μg/g dw and 3.36 μg/g dw, respectively  and Enmedio Island (south coast of Spain), the levels of lead were 0.087 ± 0.097 μg/g dw and 0.462 ± 0.824 μg/g dw, respectively  have since the 1990s until now shown high levels of pollution (lead concentration above 2 μg/g dw). Opposite results are shown by studies conducted in North American and European countries. In the populations of Ardeidae in these countries, the concentration of lead in feathers has consistently been below 1 μg/g dw. However, further environmental studies are still needed to show the current and accurate state of the environment in these countries.Studies on Due to their high level of calcium in the form of phosphates, bones accumulate xenobiotics, including heavy metals. However, the level of heavy metals, including lead, in bones does not indicate a temporary state of the animal, but the average exposure throughout its whole lifetime. Therefore, toxicological studies of bone materials from animals allow for the analysis of the average state of the natural environment over several years prior to the collection of the studied material (Hać and Krechniak Anas platyrhynchos) and spot-billed ducks (Anas poecilorhyncha) were measured at 10.6 ± 11.1 μg/g dw and 10.30 ± 6.94 μg/g dw, respectively , lead concentration was 0.93 ± 1.22 μg/g dw  and the black-crowned night heron (Nycticorax nycticorax) from Pyeongtaek, the levels of this heavy metal were 2.60 ± 1.11 μg/g dw and 4.71 ± 3.29 μg/g dw, respectively (Kim and Oh Egretta intermedia) and the little egret (Egretta garzetta), the lead concentrations were 1.17 ± 0.90 μg/g dw and 1.26 ± 1.36 μg/g dw, respectively , the concentrations of lead in the bones of mallards (Melanitta perspicillata) and wood ducks/carolina ducks (Aix sponsa) were 5.06 μg/g dw and 5.86 μg/g dw, respectively (Bagley and Locke Anas platyrhynchos) and the Canada goose (Branta canadensis), the concentrations were 13.3 μg/g dw and 2.66 μg/g dw, respectively (Bagley and Locke Bubulcus ibis) and laughing gulls (Larus atricilla) were 10.57 ± 5.12 μg/g dw and 9.24 ± 1.23 μg/g dw  and sandwich terns , the concentrations of lead in bones were 3.28 ± 1.69 μg/g dw and 1.49 ± 3.31 μg/g dw, respectively , in the 1960s, at the time when leaded petrol and paint were used, the concentrations of lead in the populations of surf scoters (Marmaronetta angustirostris) and white-headed ducks , the concentrations of lead in bones were 5.19 μg/g dw and 91.75 μg/g dw  in the north-western part of Poland around the city of Szczecin and the Słońsk Waterfowl Reserve, the concentration of lead was 5.908 ± 6.70 μg/g dw and 1.574 ± 1.863 μg/g dw, respectively  from West Galveston Bay, TX (USA), the concentration of lead in livers was 18.0 ± 13.0 μg/g dw among males and 12.0 ± 8.0 μg/g dw among females  in the same area, the level of lead in the liver was 17.70 ± 2.47 μg/g dw , the concentration of lead in the liver was 1.433 ± 0.333 μg/g dw , the level of lead in the liver was 1.767 ± 0.667 μg/g dw  were 0.800 ± 0.166 μg/g dw and 0.833 ± 0.167 μg/g dw  and the thick-billed murre (Uria lomvia) in Baffin Bay (Canada), the concentrations of lead in the liver were 0.110 ± 0.043 μg/g dw and 0.303 ± 0.440 μg/g dw, respectively  was 0.057 ± 0.030 μg/g dw  livers from 1983 and 1992 show that lead levels in Baffin Bay (Canada) have been below 0.09 μg/g dw for a while , in the population of black-tailed gulls (Larus crassirostris), the level of lead in the liver was 0.022 ± 0.009 μg/g dw  was 5.1 ± 0.8 μg/g dw  from Hongdo Island and Rando Island (Korea), the concentrations of lead in the liver were 4.82 ± 1.80 μg/g dw and 3.71 ± 2.17 μg/g dw  and little egrets (Egretta garzetta) from the provinces of Gyeonggi-do, Chungcheongnam-do and Seoul city in South Korea, the concentrations of lead in the livers were 5.32 ± 2.01 μg/g dw and 4.19 ± 1.57 μg/g dw, respectively (Kim and Oh Ixobrychus eurhythmus), the concentration was 7.97 ± 4.38 μg/g dw (Kim and Oh Nycticorax nycticorax) and grey herons (Ardea cinerea), the concentrations of lead in livers were 4.43 ± 2.42 μg/g dw and 3.56 ± 1.93 μg/g dw, respectively (Kim and Oh Egretta intermedia) and little egrets (Egretta garzetta), the levels of lead were 2.98 ± 1.19 μg/g dw and 3.36 ± 1.29 μg/g dw, respectively  and the little egret (Egretta garzetta) from Nilgiris district, Tamil Nadu (India), the concentrations of lead in the livers were 13.26 ± 1.23 μg/g dw and 3.23 ± 1.60 μg/g dw  was 2.10 ± 0.43 μg/g dw  of 0.728 ± 0.368 μg/g dw  and spot-billed ducks (Anas poecilorhyncha), the concentrations of lead in the liver were 4.74 ± 2.92 μg/g dw and 4.61 ± 2.51 μg/g dw, respectively , lead contamination was 1.32 ± 2.61 μg/g dw , the median lead level was 2 μg/g dw (Nam and Lee Anas strepera) and common teals (Anas crecca) wintering on Miankaleh and Gomishan International Wetlands, located in the southern part of Caspian Sea (northern Iran), the concentrations of lead in the liver were 10.73 and 3.60 μg/g dw , the concentrations of lead were 3.87 ± 1.37 μg/g dw and 7.87 ± 3.33 μg/g dw  showed that the concentration of lead was 1.641 μg/g dw (Ayas and Kolankaya The use of the ia Table . AccordiAnser anser) the concentration of lead in livers was as high as 151.4 ± 95.8 μg/g dw  (5.8 μg/g dw), the common teal (Anas crecca) (2.2 μg/g dw), the eurasian wigeon (Anas penelope) (1.7 μg/g dw), the gadwall (Anas strepera) (1.6 μg/g dw), the mallard (Anas platyrhynchos) (17 μg/g dw), the northern pintail (Anas acuta) (18 μg/g dw), the northern shoveler (Anas clypeata) (3.3 μg/g dw), the red-crested pochard (Netta rufina) (1.6 μg/g dw) and the tufted duck (Aythya fuligula) (6.1 μg/g dw) (Mateo and Guitart Marmaronetta angustirostris) and white-headed ducks , lead concentrations were 1.5 μg/g dw and 54 μg/g dw, respectively , the levels of lead were 2.39 μg/g dw and 2.43 μg/g dw, respectively, while in northern shovelers (Anas clypeata) and common teals (Anas crecca), levels were 0.986 μg/g dw and 0.465 μg/g dw, respectively  was 0.77 ± 0.17 μg/g dw (Levengood Spatula discors) from the area of the Gulf of Mexico, TX (USA), the concentration of lead was 0.40 ± 0.67 μg/g dw  was 0.830 ± 0.150 μg/g dw  1.7 μg/g dw, the brent goose (Branta bernicla) 4.3 μg/g dw, the canada goose (Branta canadensis) 1.7 μg/g dw, the mallard (Anas platyrhynchos) 3.0 μg/g dw, the snow goose (Chen caerulescens) 4.0 μg/g dw, the surf scoter (Melanitta perspicillata) 3.0 μg/g dw, the velvet scoter (Melanitta fusca) 2.7 μg/g dw and the wood duck/carolina duck (Aix sponsa) 5.0 μg/g dw  collected more recently in Chesapeake Bay, USA, the level of lead was 0.41 ± 0.29 μg/g dw, despite the collection of research material from military areas (Beyer and Day Somateria spectabilis) and spectacled eiders (Somateria fischeri) caught near Barrow in the Beaufort Sea in northern Alaska, the concentrations of lead in the liver were 0.13 ± 0.09 μg/g dw and 0.19 ± 0.27 μg/g dw, respectively  from Table Bay  and from Tern Island in Foxe Basin in the Arctic Circle, the level of lead in the livers of these birds was 0.088 μg/g dw and 0.181 μg/g dw, respectively , in the eggs of Canada geese (Branta canadensis), the concentration of lead measured in recent studies was 0.483 ± 0.108 μg/g dw and in eggs of mallards (Anas platyrhynchos), it was 0.186 ± 0.036 μg/g dw  from South Brother Island (New York Harbor) and Mill Rock (New York Harbor) contained 0.313 ± 0.072 μg/g dw and 0.837 ± 0.587 μg/g dw, respectively (Burger and Elbin Somateria sp.), in the Yukon-Kuskokwim Delta, Alaska (USA), the concentration of lead was significantly higher at 3.21 ± 0.15 μg/g dw  was only 0.306 ± 0.099 μg/g dw  from the areas around the industrialized cities of Pakistan—Lahore and Sialkot—showed a very high concentration of lead, 47 ± 13 μg/g dw and 41 ± 8 μg/g dw, respectively (Table Bubulcus ibis) eggshells was lower, at 5.40 ± 3.01 μg/g dw  egg contents were 0.89 ± 0.25 μg/g dw and 0.84 ± 0.54 μg/g dw  who spend their breeding period in the same uncontaminated areas were 0.37 ± 0.28 μg/g dw and 0.44 ± 0.33 μg/g dw , the degree of contamination was 0.13 ± 0.30 μg/g dw for the Islam Headworks and 0.58 ± 0.88 μg/g dw in the Trimmu Headworks  eggshells, lead concentration in the Trimmu Headworks was 1.9 ± 1.3 μg/g dw and in the Islam Headworks was 1.09 ± 0.83 μg/g dw  were 1.34 ± 1.25 μg/g dw and 1.44 ± 1.13 μg/g dw, respectively —0.172 ± 0.090 μg/g dw and 0.191 ± 0.094 μg/g dw, respectively  in a study from the early 1990s was 0.341 μg/g dw (Ayas and Kolankaya Ardea cinerea), the lead content in eggs and eggshell was 13.6 ± 3.2 μg/g dw and 3.6 ± 1.4 μg/g dw, respectively  was very low , the great egret  and the black-crowned night heron (Nycticorax nycticorax) were 88.5 ± 14.6 μg/g dw, 82.1 ± 6.9 μg/g dw and 84.9 ± 6.9 μg/g dw, respectively  and the little egret (Egretta garzetta) were very low, at the level of 0.007 ± 0.007 μg/g dw and 0.014 ± 0.015 μg/g dw  was 0.039 ± 0.011 μg/g dw (Burger and Gochfeld Nycticorax nycticorax) eggs on South Brother Island (New York Harbor) and Mill Rock (New York Harbor) were 0.370 ± 0.141 μg/g dw and 0.054 ± 0.034 μg/g dw, respectively , the concentration of lead in eggs of the black-crowned night heron  from Long Island, NY (USA) (Burger and Gochfeld Sterna hirundo) ranged from 0.022 to 0.528 μg/g dw, depending on the place where the eggs were collected and the breeding season. Lead concentrations in common tern eggs collected from Mike’s Island in New Jersey (USA) in the years 2000 and 2002 were 0.100 ± 0.021 μg/g dw and 0.022 ± 0.004 μg/g dw, respectively (Burger and Gochfeld Larus argentatus) and the great black-backed gull (Larus marinus) were 0.273 ± 0.069 μg/g dw and 0.227 ± 0.075 μg/g dw (Burger Sterna hirundo) and the Forster’s tern (Sterna forsteri), the concentrations were 0.164 ± 0.025 μg/g dw and 0.056 ± 0.007 μg/g dw, respectively (Burger Leucophaeus pipixcan) in Agassiz National Wildlife Refuge, MN (USA), the concentration of lead was 0.129 ± 0.016 μg/g dw (Burger and Gochfeld Larus argentatus) population study from Mill Rock in the New York/New Jersey Harbor Estuaries, where lead concentration in eggs in 2011 and 2012 was 0.040 ± 0.024 μg/g dw and 0.451 ± 0.095 μg/g dw, respectively (Burger and Elbin Larus marinus), the concentration of lead in eggs collected at this location was 0.138 ± 0.044 μg/g dw  during the first decade of the twenty-first century was 0.107 ± 0.029 μg/g dw . The lead content of the eggs of the glaucous-winged gull  from Hongdo Island (Korea) were 3.10 ± 1.36 μg/g dw and 0.92 ± 0.24 μg/g dw, respectively (Kim and Oh Larus saundersi) near the city of Dongtai, in the province of Jiangsu (China), the concentration of lead is as high as 79 ± 11 μg/g dw  in Hong Kong, the concentration of lead in egg contents in the end the twentieth century was 0.010 ± 0.021 μg/g dw  were 0.023 ± 0.015 μg/g dw and 0.061 ± 0.042 μg/g dw, respectively  egg contents and eggshells, lead concentrations were 0.286 ± 0.064 μg/g dw and 0.260 ± 0.049 μg/g dw  found near the mouth of the river, Pb levels were 3.70 ± 0.80 μg/g dw and 30.0 ± 8.8 μg/g dw  and the Mediterranean gull  revealed concentrations of 0.204 μg/g dw and 0.060 μg/g dw, respectively  in Axios Delta (Greece), lead concentration was 0.249 μg/g dw  in northern Poland, the concentration of lead ranged from 0.4 ± 0.1 to 0.7 ± 0.8 μg/g dw, depending on the egg collection site  on the eggs of the yellow-legged gull  are also heavily polluted with this heavy metal. Western Europe  is less polluted with lead in comparison. Nevertheless, the level of pollution in these European countries is greater than satisfactory, despite the introduction of a number of bans restricting the use of lead, for example, the use of leaded petrol or lead-containing paints. Lead does not decompose and so it can persist in the environment for many years after its use has ceased. Finally, the USA and Canada appear to be the areas with the lowest level of lead pollution. This may be due to the low population density in these countries and a high concern for the environment, encouraging the use of modern technologies that do not require the use of this element. Abandoning the use of lead in petrol has also been crucial in reducing environmental lead contamination. As a result, the current level of lead pollution in the USA is much lower than in the 1970s and 1980s. However, due to the fragmented nature of the data, it is difficult to determine any such a trend across Europe.


Coreference resolution is a challenging part of natural language processing (NLP) with applications in machine translation, semantic search and other information retrieval, and decision support systems. Coreference resolution requires linguistic preprocessing and rich language resources for automatically identifying and resolving such expressions. Many rarer and under-resourced languages (such as Lithuanian) lack the required language resources and tools. We present a method for coreference resolution in Lithuanian language and its application for processing e-health records from a hospital reception. Our novelty is the ability to process coreferences with minimal linguistic resources, which is important in linguistic applications for rare and endangered languages. The experimental results show that coreference resolution is applicable to the development of NLP-powered online healthcare services in Lithuania. Digital means of medical informatics, especially when applying natural language processing (NLP), are indispensable in the application of e-health and digitalization of medical records and processes . The useWith the development of Semantic Web technology, web information retrieval (IR) is changing towards meaning-based IR. The quality of retrieved documents relevant to the user also highly depends on the information extraction (IE) methods applied. In general, IE focuses on automatic extraction of structured information from the unstructured source. Standard document text preprocessing steps used in IE are lexical analysis, morphological analysis, and named entity recognition (NER), which can be complemented by coreference resolution and semantic annotation. The main issue here is the ambiguity and complexity of the natural language, thus making the progress in IE dependent on the evolution of the NLP techniques. While for widely used languages (such as English), the IE-related NLP research has already reached the levels of maturity and practical application on a massive scale  , but theNER when applied to biomedical texts is a critical step for developing decision support tools for smart healthcare. Examples for it are as follows: drug name recognition (DNR), which recognizes pharmacological substances from biomedical texts and classifies them for discovering drug-drug interactions , 13; bioOur previous work included the development of the semantic search framework  for answThe rest of the paper is structured as follows. In Machine-learning and rule-based approaches are efficient methods in semantic processing, especially when enhanced with external knowledge and coreference clues derived from the structured document, while often still performing better (in comparison with classic implementations) in coreference resolution when provided with ground truth mentions , while fIn general, the coreference resolution methods can be classified into knowledge rich and knowledge poor. Both methods require large resources such as semantic information, syntactic annotations, or preannotated corpora of hospital transcripts from a hospital reception. Under resourced, rarer languages, like Lithuanian, usually do not have such resources available.While up-to-date, no research has been performed to solve coreferences in Lithuanian, but many solutions have been proposed for other languages, mostly for English . Note thFor Latvian, the only solution is LVCoref . It is aFor Polish, rule-based Ruler  for scorFor Russian, RU-EVAL-2014  was an eFor Czech, coreferences are annotated in the tectogrammatical layer of Prague Dependency Treebank (PDT) and their first coreference resolution approach was rule based . At firsConsidering languages that are more-or-less grammatically similar to Lithuanian , we summarize the related work on Latvian  and Slavic languages such as Polish, Russian, and Czech in In summary, the rule-based solutions have the advantages of easier adaptability and provide comparable results when good training data are not available as is the case for Lithuania. Many of more advanced solutions cannot be fully adapted for rarer and under-resourced languages due to the lack of available linguistic resources, as is the case with Lithuanian language. For example, BART at the time supported 64 feature extractors, but due to lack of language-specific resources for the Polish language, only 13 could be utilized. The solutions that are not heavy on linguistic resources can be very useful for resource-poor languages in general.Coreference resolution (or anaphora) is an expression, the interpretation of which depends on another word or phrase presented earlier in the text (antecedent). For example, “Tom has a backache. He was injured.” Here the words “Tom” and “He” refer to the same entity. Without resolving the relationship between these two structures, it would not be possible to determine why Tom has the backache, nor who was injured. In such cases, semantic information would be lost.Anaphoric objects are expressed with pronouns and cannot be independently interpreted without going back to its antecedent. In this work, such expressions are called coreferences, unless it is required to make a distinction. Usage of such expressions can vary depending on the type and the style of the text. Here we focus on texts from medical-related domains.The role of coreference resolution in the semantic search framework is to provide additional semantic information after named entity recognition before semantic annotation .What features of text, sentence, and word help us recognize the existence of coreference (they are specified in the package Concepts of Input Flow)What kind of text preprocessing is requiredWhat additional resources are required for resolution of certain type coreferences (they are specified in the package Database of Public Persons and Classification of Professions)In this chapter, a conceptualization of coreference resolution is presented. A given model, which is expressed as UML class diagram , specifiA text segmented into sentences and lexemesMorphological features of lexemes identifiedNamed entities recognizedFor example, from the model provided, it is clear that, before coreference resolution starts, it is important to preprocess text and obtain the following:Text preprocessing itself is not a task of coreference resolution, so it is out of the scope of this paper.It is worthy to mention that the model is quite abstract, language independent, and technology independent. Therefore, it is applicable not only for Lithuanian but for grammatically similar languages as well. Concepts of this model are used for the formalization of coreference resolution rules in the next section. The concepts are explained in more detail below.Text, Lexical_Unit, and Named_Entity. The concept Text assumes a textual document whose content should be analysed. Each test has an associated publication date, which is important for solving coreferences. Each text consists of at least one Lexical_Unit, which includes paragraphs, sentences, words, and punctuations, classified into the Sentence and Lexeme categories. Lexeme assumes lexical units such as words, punctuations, and numbers. Each lexeme is characterized by a lemma and a part of speech, and some of them (nouns and pronouns) by grammatical gender and number. The lexeme could be specialized by POS category: Noun, Pronoun, and Other_Part_Of_Speech. Special cases of Other_Part_Of_Speech are Comma and Conjunction, which are required for the description of conditions of some coreference resolution rules.The main concepts of coreference resolution are Named_Entity concept defines an object to whom pronouns or certain nouns can refer. NER algorithms usually recognize three types of entities: a person (Person_NE), an organization (Organization_NE), and a location (Location_NE). The named entities of a person type require special attention a person can be mentioned not only using pronouns but also using a position he/she holds (Position_Held) and a professional name (Profession). Additional information about a person could help resolving such coreferences more precisely. As an example, source of such information could be a Database of Public Person, which includes Known_Person—a well-known person mentioned as Person_NE in the text. The output of a coreference resolution algorithm is a Coreference—a relationship between coreferents. For each coreference, its type , subtype (relative pronoun and noun repetition), position , and group  are specified. Each referent refers at least to one coreferent (a concept Mention). Each Mention starts at a certain position in the text, is of a certain length, and fits at least one Lexeme. Some of them can fit a certain Named_Entity.A The decision table with guidelines for the application of the certain resolution algorithm is shown as A1: specific rules resolution algorithm for resolution of certain usage of pronounsA2: general pronoun resolution algorithm which focuses on the cases where pronouns refer to nouns (or noun phrases) that are recognized as named entities of “person” classA3: PRA  resolution algorithm for resolution of nouns recognized as named entities and their repeated usage in the same textA4: HHS  resolution algorithm for resolution of nouns recognized as profession names including their synonyms and hypernyms/hyponymsA5: feature resolution algorithm for resolution of nouns that represent certain feature (at the moment only public position being held) of the named entity of a personFor resolution of a specific type of references, we propose the following algorithms:The coreference resolution starts from the sequential analysis of each lexeme looking for a certain type of pronoun and noun. Depending on identified features of lexeme, a decision about further analysis is taken. The decision table  summarizThe idea is that the pronoun-related coreferences should be solved first sequentially by checking the conditions C1, C2, and C3. Then a noun-related coreference resolution should start by sequentially checking the conditions C4, C6, and C7.First-order logic (FOL) formulas are employed to define the main conditions the algorithms should check when resolving coreferences. The concepts of the coreference resolution model  became tThe algorithms follow the grammar rules of the Lithuanian language which are based on the analysis of morphological features of lexemes and their order in the sentence and text. Examples of Lithuanian language sentences were translated into English as closely as possible. All proper names were changed to generic abbreviations to comply with GDPR.vyras [noun], kuris [pronoun] skundėsi nugaros skausmu.[LT] Šiandien buvo atėjęs man [noun] who [pronoun] had a backache came today.[EN] A vyras [noun], su [preposition] kuriuo [pronoun] aptarėme nugaros skausmą.[LT] Šiandien buvo atėjęs man [noun] with [preposition] whom [pronoun] we discussed a backache came today.[EN] A In some cases, there exists a rather rigid structure for pronoun usage and it can be easily defined by using specific rules, for example,Both examples are similar in their construction: [noun] [comma] [optional preposition] [specific pronoun]. In both cases, pronoun “kuriuo” refers to the noun “vyras.” In the first example, we do not have an optional preposition “su,” while we have it in the second one.A condition for the existence of such reference formally is defined as follows:For every sentence s of text t and for every “Relative” type pronoun p, which is contained in the sentence s and has a start position sp1, is of length ln1, follows comma c or follows prepositional lexeme l1, which follows comma c, and for every noun l2, which has a start position sp2, is of length ln2, precedes comma c, is of the same gender gand of the same number n as the pronoun p, the only one coreference relation r, which is resolved in text t, is of “Pronominal” type, “Relative” subtype, “Backward” position and “Single” group between the pronoun p and the noun n, its referent starts at position sp1 and has length ln1, and which fits only one lexeme p and refers to only one mention m, which starts at position sp2, has length ln2, and fits only one lexeme l2, exists (Rule 1).  Rule 1: ∀t, s, p, l1, c, l2, g, n, sp1, sp2, ln2. [Text(t) ⋀ Sentence(s) ⋀ consists_of ⋀ Pronoun(p) ⋀ contains ⋀ has_type ⋀ has_start_position ⋀ has_length ⋀ Comma(c) ⋀  ⋁ (Lexeme(l1) ⋀ has_pos ⋀ follows ⋀ follows) ⋀ Noun(l2) ⋀ follows ⋀ has_gender  ⋀ has_gender  ⋀ has_number ⋀ has_number ⋀ has_start_position ⋀ has_length ⟶ ∃!r ∃!m. [Coreference(r) ⋀ resolved_in ⋀ has_type ⋀ has_subtype ⋀ has_position ⋀ has_ group ⋀ has_start_position ⋀ has_length ⋀ fits ⋀ Mention(m) ⋀ refers_to ⋀ has_start_position ⋀ has_length ⋀ fits]]Tomo [noun], Lino [noun], Petro [noun] ir [conjunction] Eglės [noun], kurių [pronoun] sužalojimai atrodė panašūs.[LT] Komisija nerado panašumų tarp Tom [noun], Linas [noun], Peter [noun] and [conjunction] Eglė [noun], who [pronoun] shared similar injuries.[EN] The committee did not find The relative pronoun might be plural and refer to multiple singular  nouns:In this case, a plural pronoun “kurių” is referring to four singular nouns that have different genders. The previous rule would not be able to solve such coreference. For this case, the construction would be: [noun] [comma] [noun] [comma] [noun] [conjunction] [noun] [comma] [optional preposition] [specific pronoun].For such case, a special condition must be defined:For every sentence s in text t and for every “Relative” type pronoun p of “Plural” number, which is contained in the sentence s and has a start position sp1, is of length ln1, follows comma c1 or follows prepositional lexeme l, which follows comma c1, and for every noun n1, which precedes comma c1, has a start position sp2, is of length ln2, follows conjunction j, and for every noun n2, which precedes conjunction j, has a start position sp3, is of length ln3, and for every existing noun n3, which follows comma c2, and for every existing noun n4, which precedes comma c2, has a start position sp4, is of length ln4, the only one coreference relation r, which is resolved in text t, is of “Pronominal” type, “Relative” subtype, “Backward” position and “Multiple” group, its referent starts at position sp1 and has length ln1, fits only one lexeme p, refers to only one mention m1, which starts at position sp2, has length ln2, and fits noun n1, refers to only one mention m2, which starts at position sp3, has length ln3, and fits only one noun n2, and refers at least to one mention m3, which starts at position sp4, has length ln4, and fits noun n4, exists (Rule 2).  Rule 2: ∀t, s, p, l, c1, n1, sp1, ln1, sp2, ln2, j, n2, sp3, ln3.[Text(t) ⋀ Sentence(s) ⋀ consists_of ⋀ Pronoun(p) ⋀ contains ⋀ has_number ⋀ has_type ⋀ has_start_position ⋀ has_length ⋀ Comma(c1) ⋀  ⋁ (Lexeme(l) ⋀ has_pos ⋀ follows ⋀ follows) ⋀ Noun(n1) ⋀ follows ⋀ has_start_position ⋀ has_length ⋀ Conjunction(j) ⋀ follows ⋀ Noun(n2) ⋀ follows ⋀ has_start_position ⋀ has_length ⋀  ⋀ Comma(c2) ⋀ Noun(n4) ⋀ follows ⋀ follows ⋀ has_start_position ⋀ has_length) ⟶ ∃!r ∃!m1 ∃!m2 ∃m3. [Coreference(r) ⋀ resolved_in ⋀ has_type ⋀ has_subtype ⋀ has_position ⋀ has_ group ⋀ has_start_position ⋀ has_length ⋀ fits ⋀ Mention(m1) ⋀ refers_to ⋀ has_start_position ⋀ has_length ⋀ fits ⋀ Mention(m2) ⋀ refers_to ⋀ has_start_position ⋀ has_length ⋀ fits ⋀ Mention(m3) ⋀ refers_to ⋀ has_start_position ⋀ has_length ⋀ fits]]  [PL] Dzisiaj przychodził mężczyzna [noun], który [pronoun] skarżył się na ból pleców.  [RU] Сегодня приходил мужчина [noun], который [pronoun] жаловался на боль в спине.Though examples illustrating the certain case of coreference are given in Lithuanian and English only, rules for resolution of such coreferences could be applied for other languages as well. For example, Rule 1 could be successfully applied for coreference resolution in Polish or Russian languages. Let us take the same example of a sentence in Polish and Russian:We can see that a structure of the sentence (number and order of lexemes) is similar, a pronoun goes after the comma and it refers to a noun, and compatibility of morphological features  of noun and pronoun is retained. From the given example, we understand that the coreference relation between pronoun and noun exists and conditions for such existence are the same as specified in Rule 1.Jonas Jonaitis [person noun phrase] skambino į registratūrą. Jis [pronoun] skundėsi galvos skausmu.[LT] Jonas Jonaitis [person noun phrase] called a reception. He [pronoun] complained about headache.[EN] This algorithm focuses on the cases where pronouns refer to nouns (or noun phrases) that are recognized as named entities of “person” class by NER. The algorithm starts from the identification of not demonstrative pronoun. In a given example below, such a pronoun is in the second sentence—“Jis” (“He”)If the pronoun is in the relative clause, the algorithm moves backwards analysing words going before the pronoun. In a given example, the pronoun is at the beginning of the sentence, so remaining parts of the sentence are not analysed, and the algorithm moves one sentence backwards.s1 before pronoun p:The conditions for the existence of such reference formally could be defined as three alternatives. The first one describes conditions for reference existing in the same sentence For each text's t sentence s1 and pronoun p not of Demonstrative type that is contained in sentence s1 and has gender g, number n, start position sp1 and length of ln1, and named entity e1 that is in the same sentence s1, is expressed by lexeme l, and has gender g, number n, start position sp2 and is of length ln2, and is before pronoun p (sp2 is lower than sp1), but closer to pronoun p than possible named entities e2 and e3 (sp2 higher than sp3 and sp4), the only one coreference relation r, which is resolved in text t, is of “Pronominal” type, “Relative” subtype, “Backward” position and “Single” group between the pronoun p and the named entity e1, its referent starts at position sp1 and has length ln1, and which fits only one pronoun p and refers to only one mention m, which starts at position sp2, has length ln2, and fits only one named entity e1, exists (Rule 3).  Rule 3: ∀t, s1, p, l, e1, g, n, sp1, ln1, sp2, ln2.[Text(t) ⋀ Sentence(s1) ⋀ consists_of ⋀ Pronoun(p) ⋀ contains ⋀ ¬has_type ⋀ has_gender ⋀ has_number ⋀ has_start_position ⋀ has_length ⋀ Person_NE(e1) ⋀ includes ⋀ Lexeme(l) ⋀ expressed_by ⋀ has_gender ⋀ has_number ⋀ has_start_position ⋀ has_length ⋀ sp2<sp1 ⋀ ¬ ⋀ includes ⋀ has_gender ⋀ has_number ⋀ has_start_position ⋀ Person_NE(p3) ⋀ includes ⋀ has_gender ⋀ has_number ⋀ has_start_position ⋀ sp2>sp3 ⋀ sp4>sp2)) ⟶ ∃!r ∃!m. [Coreference(r) ⋀ resolved_in ⋀ has_type  ⋀ has_subtype  ⋀ has_position ⋀ has_group ⋀ has_start_position ⋀ has_length ⋀ fits ⋀ Mention(m) ⋀ refers_to ⋀ has_start_position ⋀ has_length ⋀ fits ⋀ fits]]p refers to the named entity in the previous sentence s2:The second alternative describes a case when a pronoun For each text's t sentence s1, s2, where s1 follows s2, and pronoun p not of Demonstrative type that is contained in sentence s1 and has gender g, number n, start position sp1 and length of ln1, and named entity e1 that is contained in sentence s2, is expressed by lexeme l, and has gender g, number n, start position sp2 and is of length ln2, and is closer to pronoun p than possible named entities e2 and e3 (sp2 higher than sp3 and sp4), the only one coreference relation r, which is resolved in text t, is of “Pronominal” type, “Relative” subtype, “Backward” position and “Single” group between the pronoun p and the named entity e1, its referent starts at position sp1 and has length ln1, and which fits only one pronoun p and refers to only one mention m, which starts at position sp2, has length ln2, and fits only one named entity e1, exists (Rule 4).  Rule 4: ∀t, s1, s2, p, l, e1, g, n, sp1, ln1, sp2, ln2.[Text(t) ⋀ Sentence(s1) ⋀ Sentence(s2) ⋀ consists_of ⋀ consists_of ⋀ follows  ⋀ Pronoun(p) ⋀ contains ⋀ ¬has_type ⋀ has_gender ⋀ has_number ⋀ has_start_position ⋀ has_length ⋀ Person_NE(e1) ⋀ includes ⋀ Lexeme(l) ⋀ expressed_by ⋀ has_gender ⋀ has_number ⋀ has_start_position ⋀ has_length ⋀ ¬ ⋀ includes ⋀ has_gender ⋀ has_number ⋀ has_start_position ⋀ Person_NE(p3) ⋀ includes ⋀ has_gender ⋀ has_number ⋀ has_start_position ⋀ sp2>sp3 ⋀ sp4>sp2)) ⟶ ∃!r ∃!m. [Coreference(r) ⋀ resolved_in ⋀ has_type  ⋀ has_subtype  ⋀ has_position ⋀ has_group ⋀ has_start_position ⋀ has_length ⋀ fits ⋀ Mention(m) ⋀ refers_to ⋀ has_start_position ⋀ has_length ⋀ fits ⋀ fits]]p in the sentence s1 refers to the named entity in the sentence s3, preceding sentences s2 and s1:The third alternative describes a case when a pronoun For each text's t sentence s1, s2, s3, where s1 follows s2 and s2 follows s3, and pronoun p not of Demonstrative type that is contained in sentence s1 and has gender g, number n, start position sp1 and length of ln1, and named entity e1 that is contained in sentence s3, is expressed by lexeme l, and has gender g, number n, start position sp2 and is of length ln2, and is closer to pronoun p than possible named entities e2 and e3 (sp2 higher than sp3 and sp4), the only one coreference relation r, which is resolved in text t, is of “Pronominal” type, “Relative” subtype, “Backward” position and “Single” group between the pronoun p and the named entity e1, its referent starts at position sp1 and has length ln1, and which fits only one pronoun p and refers to only one mention m, which starts at position sp2, has length ln2, and fits only one named entity e1, exists (Rule 5).  Rule 5: ∀t, s1, s2, s3, p, l, e1, g, n, sp1, ln1, sp2, ln2.[Text(t) ⋀ Sentence(s1) ⋀ Sentence(s2) ⋀ Sentence(s3) ⋀ consists_of ⋀ consists_of ⋀ consists_of ⋀ follows  ⋀ follows  ⋀ Pronoun(p) ⋀ contains ⋀ ¬has_type ⋀ has_gender ⋀ has_number ⋀ has_start_position ⋀ has_length ⋀ Person_NE(e1) ⋀ Lexeme(l) ⋀ expressed_by ⋀ includes ⋀ has_gender ⋀ has_number ⋀ has_start_position ⋀ has_length ⋀ ¬ ⋀ includes ⋀ has_gender ⋀ has_number ⋀ has_start_position ⋀ Person_NE(e3) ⋀ includes ⋀ has_gender ⋀ has_number ⋀ has_start_position ⋀ sp2>sp3 ⋀ sp4>sp2)) ⟶ ∃!r ∃!m. [Coreference(r) ⋀ resolved_in ⋀ has_type  ⋀ has_subtype  ⋀ has_position ⋀ has_group ⋀ has_start_position ⋀ has_length ⋀ fits ⋀ Mention(m) ⋀ refers_to ⋀ has_start_position ⋀ has_length ⋀ fits ⋀ fits]]man [pronoun] teko analizuoti, padidinus skaitmenines paslaugas tik nedidelė dalis Lietuvos [location noun] ligoninių sumažino etatų ar atleido darbuotojus, o tai lėmė nemažą papildomą indėlį į paslaugos kokybę” teigė J. Jonaitis [person noun phrase].[LT] Pastebėtina, kad ligoginėse apsilankė 10 mln. pacientų, nepaisant to, kad 2016 m. jų buvo 4% mažiau . Tai labiausiai lėmė skaitmeninių paslaugų padidinimas: „Kiek I [pronoun] had analysed, only a small part of Lithuanian [location noun] hospitals have reduced their posts or dismissed employees, but digitization of services has led to a significant additional contribution to the quality of service,” said J. Jonaitis [person noun phrase].[EN] It is noteworthy that the total hospital patient count has reached 10 million, even though in 2016 the number was less than 4% (around 9.5 million patients). This was influenced by the digitization of e-health services. “As far as Another example presents a case when a coreferent of the pronoun “man”  is in the following sentence:s1 after pronoun was mentioned:If the algorithm does not find any named entities moving backwards, it moves back to pronoun and proceeds forward. The algorithm continues moving forward until it locates “J. Jonaitis” entity, which is recognized as a person. Since the gender of the pronoun “man” is ambiguous , only their grammatical numbers are compared. Both are singular; therefore, the algorithm picks “J. Jonaitis” as a postcedent of the corefering object “man.” Conditions for the existence of such reference formally could be defined as two alternatives. The first one describes the conditions for reference existing in the same sentence For each text's t sentence s1 and pronoun p not of Demonstrative type that is contained in sentence s1 and has gender g, number n, start position sp1 and length of ln1, and named entity e1 that is in the same sentence s1, is expressed by lexeme l, and has gender g, number n, start position sp2 and is of length ln2, and is after pronoun p (sp2 is higher than sp1), but closer to pronoun p than possible named entities e2 and e3 (sp2 higher than sp3 and sp4), the only one coreference relation r, which is resolved in text t, is of “Pronominal” type, “Relative” subtype, “Backward” position and “Single” group between the pronoun p and the named entity e1, its referent starts at position sp1 and has length ln1, and which fits only one pronoun p and refers to only one mention m, which starts at position sp2, has length ln2, and fits only one named entity e1, exists (Rule 6).  Rule 6: ∀t, s1, p, l, e1, g, n, sp1, ln1, sp2, ln2.[Text(t) ⋀ Sentence(s1) ⋀ consists_of ⋀ Pronoun(p) ⋀ contains ⋀ ¬has_type ⋀ has_gender ⋀ has_number ⋀ has_start_position ⋀ has_length ⋀ Person_NE(e1) ⋀ includes ⋀ Lexeme(l) ⋀ expressed_by ⋀ has_gender ⋀ has_number ⋀ has_start_position ⋀ has_length ⋀ sp1<sp2 ⋀ ¬ ⋀ includes ⋀ has_start_position ⋀ Person_NE(e3) ⋀ includes ⋀ has_start_position ⋀ sp2<sp3 ⋀ sp4<sp2)) ⟶ ∃!r ∃!m. [Coreference(r) ⋀ resolved_in ⋀ has_type  ⋀ has_subtype  ⋀ has_position ⋀ has_group ⋀ has_start_position ⋀ has_length ⋀ fits ⋀ Mention(m) ⋀ refers_to ⋀ has_start_position ⋀ has_length ⋀ fits ⋀ fits]]p refers to the named entity in the following sentence s4:The second alternative describes the case when the pronoun For each text's t sentence s1, s4, where s4 follows s1, and pronoun p not of Demonstrative type that is contained in sentence s1 and has gender g, number n, start position sp1 and length of ln1, and named entity e1 that is contained in sentence s2, is expressed by lexeme l, and has gender g, number n, start position sp2 and is of length ln2, and is closer to pronoun p than possible named entities e2 and e3 ( sp2 higher than sp3 and sp4 ), the only coreference relation r, which is resolved in text t, is of “Pronominal” type, “Relative” subtype, “Backward” position and “Single” group between the pronoun p and the named entity e1, its referent starts at position sp1 and has length ln1, and which fits only one pronoun p and refers to only one mention m, which starts at position sp2, has length ln2, and fits only one named entity e1, exists (Rule 7).  Rule 7: ∀t, s1, s2, p, l, e1, g, n, sp1, ln1, sp2, ln2.[Text(t) ⋀ Sentence(s1) ⋀ Sentence(s2) ⋀ consists_of ⋀ consists_of ⋀ follows  ⋀ Pronoun(p) ⋀ contains ⋀ ¬has_type ⋀ has_gender ⋀ has_number ⋀ has_start_position ⋀ has_length ⋀ Person_NE(e1) ⋀ includes ⋀ Lexeme(l) ⋀ expressed_by ⋀ has_gender ⋀ has_number ⋀ has_start_position ⋀ has_length ⋀ ¬ ⋀ includes ⋀ has_start_position ⋀ Person_NE(e3) ⋀ includes ⋀ has_start_position ⋀ sp2<sp3 ⋀ sp4<sp2)) ⟶ ∃!r ∃!m. [Coreference(r) ⋀ resolved_in ⋀ has_type  ⋀ has_subtype  ⋀ has_position ⋀ has_group ⋀ has_start_position ⋀ has_length ⋀ fits ⋀ Mention(m) ⋀ refers_to ⋀ has_start_position ⋀ has_length ⋀ fits ⋀ fits]]The algorithm ignores demonstrative pronouns because they are often used to refer to entities that are not present in the written text. Such pronouns do not carry any additional semantic information and do not refer to any noun phrase. They are used mostly for syntactic reasons and due to that are usually ignored in coreference resolution.Tomaitis [named entity] pateko į avariją. Po pietų Tomaitį [named entity] išvežė į operacinę.[LT] Tomaitis [named entity] got into a car accident. In the afternoon, Tomaitis [named entity] has been taken to a surgery room.[EN] This algorithm is based on exact  string matches and several rules for acronyms. Once a first named entity that can be matched with an initial named entity is found, then the algorithm stops to keep annotations simple: B ⟶ A, C ⟶ B and D ⟶ C. This allows the formation of the coreference chains linking all mentions of the same entity in a text that can be later reused for semantic analysis, for example,In this example, two mentions of the same entity are made: “Tomaitis” and “Tomaitį.” They are of different cases, but their lemmas are identical. A condition for the existence of such reference formally is defined as follows:For each text's t sentence s1 that includes named entity e1, that has start position sp1 and is of length ln1, which is expressed by lexeme l1 that has lemma l and for each same text's t sentence s2 that includes named entity e2, that has a start position sp1 and is of length ln1, which is expressed by lexeme l2 that has lemma l, the only one coreference relation r, which is resolved in text t, is of “Nominal” type, “Repetition” subtype, “Irrelevant” position and “Single” group between the noun n1 and the noun n2, its referent starts at position sp1 and has length ln1, and which fits only one noun n1 and refers to only one mention m, which starts at position sp2, has length ln2, and fits only one noun n2, exists (Rule 8).  Rule 8: ∀t, s1, s2, e1, e2, sp1, ln1, sp2, ln2.[Text(t) ⋀ Sentence(s1) ⋀ Sentence(s2) ⋀ consists_of ⋀ consists_of ⋀ Named_Entity(e1) ⋀ includes ⋀ has_start_position ⋀ has_length ⋀ Named_Entity(e2) ⋀ includes ⋀ has_start_position ⋀ has_length ⋀ e1≠e2 ⋀ (∃l1 ∃l2 ∃l.(Lexeme(l1) ⋀ Lexeme(l2) ⋀ expressed_by ⋀ expressed_by ⋀ has_lemma ⋀ has_lemma) ⟶ ∃!r ∃!m. [Coreference(r) ⋀ resolved_in ⋀ has_type  ⋀ has_subtype  ⋀ has_position ⋀ has_group ⋀ has_start_position ⋀ has_length ⋀ fits ⋀ fits ⋀ Mention(m) ⋀ refers_to ⋀ has_start_position ⋀ has_length ⋀ fits ⋀ fits]]Acronym rules vary depending on the type of named entity .Gydytojai [noun referring to profession] skundžiasi dideliu darbo krūviu. Chirurgų [noun referring to profession] darbo krūvis pats didžiausias.[LT] Doctors [noun referring to profession] complain about heavy workload. Surgeons' [noun referring to profession] workload is the greatest.[EN] This algorithm is based on profession classification. It attempts to resolve the use of synonyms and hypernyms/hyponyms.The algorithm determines that “Doctor” in professions classification is a hyponym of “Surgeon,” they also agree in gender and number; therefore, the algorithm adds their pair to annotations. Conditions for the existence of such reference formally are defined as follows:For each text's t sentence s1 that has profession p1, which is either broader or narrower than profession p2, name v1 expressing noun n1, which has gender g, number m, start position sp1 and is of length ln1, and for each same text's t sentence s2 that has profession p2, which is either broader or narrower than profession p1, name v2 expressing noun n2, which has gender g, number m, start position sp2 and is of length ln2, the only one coreference relation r, which is resolved in text t, is of “Nominal” type, “Hypernym_hyponym” subtype, “Irrelevant” position and “Single” group between the noun n1 and the noun n2, its referent starts at position sp1 and has length ln1, and which fits only one noun n1 and refers to only one mention m, which starts at position sp2, has length ln2, and fits only one noun n2, exists (Rule 9).  Rule 9: ∀t, s1, s2, n1, n2, sp1, ln1, sp2, ln2, v1, v2, p1, p2.[Text(t) ⋀ Sentence(s1) ⋀ Sentence(s2) ⋀ consists_of ⋀ consists_of ⋀ Noun(n1) ⋀ contains ⋀ has_start_position ⋀ has_length ⋀ Noun(n2) ⋀ contains ⋀ has_start_position ⋀ has_length ⋀ n1≠n2 ⋀ Profession(p1) ⋀ Profession(p2) ⋀ p1≠p2 ⋀ Profession_Name(v1) ⋀ Profession_Name(v2) ⋀ express ⋀ express ⋀ describes ⋀ describes ⋀  ⋁ broadens) ⋀ has_gender ⋀ has_gender ⋀ has_number ⋀ has_number ⟶ ∃!r ∃!m. [Coreference(r) ⋀ resolved_in ⋀ has_type  ⋀ has_subtype  ⋀ has_position ⋀ has_group ⋀ has_start_position ⋀ has_length ⋀ fits ⋀ Mention(m) ⋀ refers_to ⋀ has_start_position ⋀ has_length ⋀ fits]]vadovu [noun referring to profession]. Deja, vyr. chirurgo [noun referring to profession] vykdoma operacija buvo nesėkminga.[LT] J. Jonaitis buvo operacijos head surgeon [noun referring to profession] of operation. Unfortunately, the last operation of chief surgeon [noun referring to profession] was unsuccessful.[EN] From now J. Jonaitis was the An example of synonym is given as follows:Both “head surgeon” and “chief surgeon” are synonymous; therefore, the condition for the existence of such reference formally could be defined as follows:For each text's t sentence s1 that has a profession's p name v1, which is expressed by noun n1 that has gender g, number m, start position sp1 and is of length ln1, and for each same text's t sentence s2 that has same profession's p name v2 expressed by noun n2 that has gender g, number m, start position sp2 and is of length ln2, the only one coreference relation r, which is resolved in text t, is of “Nominal” type, “Synonym” subtype, “Irrelevant” position and “Single” group between the noun n1 and the noun n2, its referent starts at position sp1 and has length ln1, and which fits only one noun n1 and refers to only one mention m, which starts at position sp2, has length ln2, and fits only one noun n2, exists (Rule 10).  Rule 10: ∀t, s1, s2, n1, n2, sp1, ln1, sp2, ln2, v1, v2, p, g, n.[Text(t) ⋀ Sentence(s1) ⋀ Sentence(s2) ⋀ consists_of ⋀ consists_of ⋀ Noun(n1) ⋀ contains ⋀ has_start_position ⋀ has_length ⋀ Noun(n2) ⋀ contains ⋀ has_start_position ⋀ has_length ⋀ n1≠n2 ⋀ Profession_name(v1) ⋀ Profession_name(v2) ⋀ Profession(p) ⋀ express ⋀ express ⋀ describes ⋀ describes ⋀ has_gender ⋀ has_gender ⋀ has_number ⋀ has_number ⋀ n1≠n2 ⟶ ∃!r ∃!m. [Coreference(r) ⋀ resolved_in ⋀ has_type  ⋀ has_subtype ⋀ has_position ⋀ has_group ⋀ has_start_position ⋀ has_length ⋀ fits ⋀ Mention(m) ⋀ refers_to ⋀ has_start_position ⋀ has_length ⋀ fits]][LT] Ką rekomenduoja S. Suskelis [person noun phrase]? Aptarkime kardiologo [noun referring to held position] siūlomą gydymo planą.[EN] What does S. Suskelis [person noun phrase] recommend? Let's discuss treatment plan proposed by the cardiologist [noun referring to held position]. Here a noun “cardiologist” is selected, the algorithm moves backwards till it reaches “S. Suskelis” and checks the knowledge base if at the time of the publication of the medical record he has held the position of the cardiologist.. Since “he holds it” the algorithm checks if “S. Suskelis” and “cardiologist” agree in gender and number. They agree, and their pair is added to annotation as a feature reference. A condition for the existence of such reference formally is defined as follows:This algorithm at the time attempts to resolve only those cases when a person is being referred to by his public post (feature) that he holds, other types of features are not currently resolved, for example,For each text's t sentence s1 that has known person k, who during publication date d had certain position h (publication date d is same or later than position h start date fd and same or earlier than position h end date td), mention as named entity e, that has a start position sp1 and is of length ln1, and for each same text's t sentence s2 mentioned noun n, that has a start position sp2 and is length ln2, which is mentioned after named entity e (noun n has a higher start position sp2 than named entity's sp1), whose lemma l matches with position's h lemma l, number is Singular and gender g matches known person's k gender g, the only one coreference relation r, which is resolved in text t, is of “Nominal” type, “Feature” subtype, “Backward” position and “Single” group between the noun n and the named entity e, its referent starts at position sp2 and has length ln2, and which fits only one noun n and refers to only one mention m, which starts at position sp1, has length ln1, and fits only one named entity e, exists (Rule 11).  Rule 11: ∀t, s1, s2, n, k, h, l, d, fd, td, g, sp1, sp2, ln1, ln2.[Text(t) ⋀ Sentence(s1) ⋀ Sentence(s2) ⋀ consists_of ⋀ consists_of ⋀ Person_NE(e) ⋀ includes ⋀ Noun(n) ⋀ contains ⋀ Known_Person(k) ⋀ mentioned_as ⋀ Position_held(h) ⋀ holds ⋀ has_lemma ⋀ has_lemma ⋀ has_publication_date ⋀ has_from_date ⋀ has_to_date ⋀ fd≤d ⋀ td≥d ⋀ has_gender ⋀ has_gender ⋀ has_number ⋀ has_start_position ⋀ has_start_position ⋀ sp1<sp2 ⋀ has_length ⋀ has_length ⟶ ∃!r ∃!m. [Coreference(r) ⋀ resolved_in ⋀ has_type  ⋀ has_subtype  ⋀ has_position ⋀ has_ group ⋀ has_start_position ⋀ has_length ⋀ fits ⋀ Mention(m) ⋀ refers_to ⋀ has_start_position ⋀ has_length ⋀ fits]]šeimos gydytojas [noun referring to held position]? T. Tomaitis [person noun phrase] yra labai patyręs jūsų šeimos gydytojas.[LT] Koks yra mano family doctor [noun referring to held position]? T. Tomaitis [person noun phrase] is very experienced doctor assigned to you.[EN] Who is my In this case, it is also relevant to track if coreference is pointing backward or forward. We can rewrite the same example and switch a known person with his positions:sp1 is higher than sp2 and coreference has different position constant values:As a result, For each text's t sentence s1 that has known person k, who during publication date d had certain position h (publication date d is same or later than position h start date fd and same or earlier than position h end date td), mention as named entity e, that has a start position sp1 and is of length ln1, and for each same text's t sentence s2 mentioned noun n, that has a start position sp2 and is length ln2, which is mentioned after named entity e (noun n has a lower start position sp2 than named entity's sp1), whose lemma l matches with position's h lemma l, number is Singular and gender g matches known person's k gender g, the only one coreference relation r, which is resolved in text t, is of “Nominal” type, “Feature” subtype, “Forward” position and “Single” group between the noun n and the named entity e, its referent starts at position sp2 and has length ln2, and which fits only one noun n and refers to only one mention m, which starts at position sp1, has length ln1, and fits only one named entity e, exists (Rule 12).  Rule 12: ∀t, s1, s2, n, k, h, l, d, fd, td, g, sp1, sp2, ln1, ln2.[Text(t) ⋀ Sentence(s1) ⋀ Sentence(s2) ⋀ consists_of ⋀ consists_of ⋀ Person_NE(e) ⋀ includes ⋀ Noun(n) ⋀ contains ⋀ Known_Person(k) ⋀ mentioned_as ⋀ Position_held(h) ⋀ holds ⋀ has_lemma ⋀ has_lemma ⋀ has_publication_date ⋀ has_from_date ⋀ has_to_date ⋀ fd≤d ⋀ td≥d ⋀ has_gender ⋀ has_gender ⋀ has_number ⋀ has_start_position ⋀ has_start_position ⋀ sp1>sp2 ⋀ has_length ⋀ has_length ⟶ ∃!r ∃!m. [Coreference(r) ⋀ resolved_in ⋀ has_type  ⋀ has_subtype  ⋀ has_position ⋀ has_ group ⋀ has_start_position ⋀ has_length ⋀ fits ⋀ Mention(m) ⋀ refers_to ⋀ has_start_position ⋀ has_length ⋀ fits]]The coreference resolution algorithms and rules presented in Coreference resolution for Lithuanian was implemented using Java programming language and JSON data format for annotation storage. But the proposed approach is not technology dependent, and for other languages, it can be implemented on any other platform.The evaluation was performed by analysing 100 articles that have been preannotated and are available in our Lithuanian Language Coreference Corpus , in addiF1 metrics. Recall R is the ratio of correctly resolved anaphoric expressions C to the total number of anaphoric expressions T. Precision P is the ratio of correctly resolved anaphoric expressions C to the number of resolved anaphoric expressions F. F1 is a harmonic mean of P and R:For evaluation, we used precision, recall, and Five experiments were performed with different combinations of coreferencing algorithms presented in The database of public persons must be constantly updated as new information becomes available. Otherwise, recall will get noticeably lower when annotating newer texts.In the case where plural pronouns and nouns are used, they are difficult to be identified because of many variations possible that often ignore grammatical compatibility rules.Note the following threats to validity of our results:Linking the named entity to the position held taking into account the date of the publication of the text is limited considering that the text might be published today but written about things that happened in the past. There are no tools, which can identify the timeframe of a certain part of the text.Medical entity recognition and coreferencing are difficult tasks in Lithuanian natural language processing (NLP). We proposed the coreference resolution approach for the Lithuanian language. The coreference resolution algorithm depends on morphological and named entity recognition (NER) annotations and preexisting databases. Due to the proposed approach being detached from specific implementation and rules being formalized, it would not be difficult to adapt it for grammatically similar languages. Our novelty is the ability to process coreferences with minimal linguistic resources, which are very important to consider in linguistic applications for under-resourced and endangered languages. While the proposed method provides encouraging results, when analysing transcribed medical records and other corpora, and they are comparable to the results achieved by other authors applying different resolution approaches on other languages, it has certain limitations: it is domain specific and is able to resolve only a subset of coreference types, while the relatively small dataset was used for experiments. Nevertheless, we hope that our method can contribute to the sustainable development of the NLP-powered online healthcare services in Lithuania.


Following publication of this article  we foundKa ≈ 1 × 105 M−1 as well as a binding site size of n ≈ 2.5 base pairs for mitoxantrone. An unwinding angle of mitoxantrone-intercalation of ϑ ≈ 16° was determined. The original article has been updated.Using this method, we were able to estimate an equilibrium constant of association