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In this paper, we investigate contractions in a rational form in the context of the supermetric space, which is a very interesting generalization of the metric space. We consider an illustrative example to support this new result on supermetric space. [ABSTRACT FROM AUTHOR]

We consider mappings of domains of Riemannian manifolds that admit branch points and satisfy a certain condition regarding the distortion of the modulus of families of paths. We have established logarithmic estimates of distance distortion under such mappings. A separate study relates to the situation when the mappings are defined in metric spaces, and one of them is the Loewner space. We also studied the question of equicontinuity of the families of the indicated mappings in the closure of the domain. In addition, we have established the possibility of continuous extension of these mappings to an isolated point of the boundary. [ABSTRACT FROM AUTHOR]

Let ([n],d) be a metric space with diameter Δ and with average distance r ̄ , where [n] ≡{1,2,...,n}. We show that if Δ = o (n r ̄) , then ∑ i = 1 ⌊ n / 2 ⌋ d (π (2 i − 1) , π (2 i)) is (1 / 2 ± o (1)) n r ̄ with probability 1 − o(1) over a uniformly random permutation π: [n] → [n]. In particular, a uniformly random perfect matching in ([n],d) has size concentrating around n r ̄ / 2 when n is even and Δ = o (n r ̄) . Our result has implications for finding maximum travelling salesman tours in metric spaces. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to introduce and prove some fixed point theorems for two maps that satisfy (φ—ψ)-contractive conditions with rational expression in partially ordered metric spaces, our results improve and unify a multitude of fixed point theorems and generalize some recent results in ordered partially metric space. [ABSTRACT FROM AUTHOR]

As in [6, 3, 4, 5], we discuss published assertions concerning fixed points in "digital metric spaces" - assertions that are incorrect or incorrectly proven, or reduce to triviality. [ABSTRACT FROM AUTHOR]

In this paper, we will investigate the existence and uniqueness of common fixed points of certain mappings in the frame of a metric space. The given results cover a number of unique common fixed point theorems especially a result of Phaneendra and Swatmaram [12]. We will also display two examples to illustrate our theorems. [ABSTRACT FROM AUTHOR]

We show that the standard partition of unity subordinate to an open cover of a metric space has Lipschitz constant max ⁡ (1 , M − 1) / L , where L is the Lebesgue number and M is the multiplicity of the cover. If the metric space satisfies the approximate midpoint property, as length spaces do, then the upper bound improves to (M − 1) / (2 L). These Lipschitz estimates are optimal. We also address the Lipschitz analysis of ℓ p -generalizations of the standard partition of unity, their partial sums, and their categorical products. Lastly, we characterize metric spaces with Assouad–Nagata dimension n as exactly those metric spaces for which every Lebesgue cover admits an open refinement with multiplicity n + 1 while reducing the Lebesgue number by at most a constant factor. [ABSTRACT FROM AUTHOR]

In the commenced work, we establish some best proximity point results for multivalued generalized contractions on partially ordered complete metric spaces along with the tactic of altering distance function. Furthermore, we deliver some examples to elaborate and explain the usability of the attained results. To arouse further interest in the subject and to show its efficacy, we devote this work to recent applications of fractional calculus and also invoke our findings to the equation of motion modeling to differential equations. [ABSTRACT FROM AUTHOR]

In the context of a complete metric space, we will consider the common fixed point problem for two self operators. The operators are assumed to satisfy a general contraction type condition inspired by the Ćirić fixed point theorems. Under some appropriate conditions we establish existence, uniqueness and approximation results for the common fixed point. In the same framework, the second problem is to study various stability properties. More precisely, we will obtain sufficient conditions assuring that the common fixed point problem is well-posed and has the Ulam–Hyers stability, as well as the Ostrowski property for the considered problem. Some examples and applications are finally given in order to illustrate the abstract theorems proposed in the first part of the paper. Our results extend and complement some theorems in the recent literature. [ABSTRACT FROM AUTHOR]

In the present paper we study the relationships between different concepts of convex structures in metric spaces that that are related to the works of K. Menger [Menger, K. Untersuchungen über allgemeine Metrik. Math. Ann. 100 (1928), 75-163], H. Busemann [Busemann, H. The geometry of geodesics, Academic Press, 1955], I. N. Herstein; J. Milnor [Herstein, I. N.; Milnor, J. An axiomatic approach to measurable utility. Econometrica 21 (1953), 291-297], E. Michael [Michael, E. Convex structures and continuous selections. Canad. J. Math. 11 (1959), 556-575], A. Nijenhuis [Nijenhuis, A. A note on hyperconvexity in Riemannian manifolds. Canad. J. Math. 11 (1959), 576-582.], W. Takahashi; T. Shimizu [Shimizu, T.; Takahashi, W. Fixed points of multivalued mappings in certain metric spaces. Topol. Methods Nonlinear Anal. 8 (1996), no. 1, 197-203 and Takahashi, W. A convexity in metric space and nonexpansive mappings I. Kodai Math. Sem. Rep. 22 (1970), 142-149], M. Taskovi'c [Taskovi'c, M. General convex topological spaces and fixed points. Math. Moravica 1 (1997), 127-134], Yu. A. Aminov [Aminov, Yu. A. Two-Dimensional Surfaces in 3-Dimensional and 4-Dimensional Euclidean Spaces. Results and Unsolved Problems. Ukr. Math. J. 71 (2019), no. 1, 1-38.], H. V. Machado [Machado, H. V. A characterization of convex subsets of normed spaces. Kodai Math. Sem. Rep. 25 (1973), 307-320], and many other papers. Some well known examples of concrete convex structures are reexamined and the possibilities of different embeddings of metric spaces with convex structures are also studied. Corollary 6.1 states that the Bolyai-Lobachevskii plane and the Bolyai-Lobachevskii half-plane are not isometrically embedding in some strictly convex normed space. A characteristic of the invariant metric generated by a norm is presented (Proposition 4.6). [ABSTRACT FROM AUTHOR]

We introduce a new pair of mappings (S , T) on D * -metric spaces called D S * -W.C. and D R S * -W.C. Many examples are presented to show the difference between these mappings and other types of mappings in the literature. Moreover, we obtain several common fixed point results by using these types of mappings and the (E.A) property. We then employ the fixed point results to establish the existence and uniqueness of a solution for a class of nonlinear integral equations. [ABSTRACT FROM AUTHOR]

In this article, we introduce the concept of tripled controlled V -metric spaces and study Kannan type contractions in this new structure. Also, we extend the concept of Kannan type contractions to Kannan-Presic type contractions. Our results modify some known ones in the literature. To support our main results, an example and an application in integral equations are presented. [ABSTRACT FROM AUTHOR]

Any information is (transported as) a selection from an ordered set, which is the "domain" of the information. For example, any piece of digital information is a number sequence that represents such a selection. Its senders and receivers (with software) should know the format and domain of the number sequence in a uniform way worldwide. So far, this is not guaranteed. However, it can be guaranteed after the introduction of the new "Domain Vector" (DV) data structure: "UL plus number sequence". Thereby "UL" is a "Uniform Locator", which is an efficient global pointer to the machine-readable online definition of the number sequence. The online definition can be adapted to the application so that the DV represents the application-specific, reproducible features in a precise (one-to-one), comparable, and globally searchable manner. The systematic, nestable online definition of domains of digital information (number sequences) and the globally defined DV data structure have great technical potential and are recommended as a central focus of future computer science. [ABSTRACT FROM AUTHOR]

In 2014, Jleli and Samet [6] introduced a contraction mapping called θ-contraction. After this, many authors had given their contribution on φ − θ-contraction (see [7, 8]). Motivated by [[6, 9]], we introduce the notion of generalized φ − θ contraction and establish some new fixed point theorems for this contraction in the setting of complete partial metric spaces. [ABSTRACT FROM AUTHOR]

We study mappings that satisfy the so-called inverse Poletsky inequality. Provided that the given domain has a weakly flat boundary, and the mapped domain is locally connected on it, we have established a continuous boundary extension of the indicated mappings. Under some additional conditions, the corresponding class of mappings is equicontinuous at the inner and boundary points of a given domain. In particular, we have established a result on the equicontinuity of families of mappings with the inverse Poletskii inequality and one normalization condition. As an example, we have constructed a family of corresponding mappings acting between quotient spaces along two fixed groups of Möbius mappings. [ABSTRACT FROM AUTHOR]

The purpose of this manuscript is to demonstrate the existence and uniqueness of triple fixed-point results for Geraghty-type contractions in ordinary metric spaces with binary relations. Moreover, the well-posedness of the tripled fixed point problem is investigated. Consequently, α -dominated mapping on such space is discussed. Ultimately, to promote our paper, some illustrative examples are derived, and the existence of the solution to a system of differential equations is obtained as an application. [ABSTRACT FROM AUTHOR]

This paper deals with the existence of positive radial solutions to the iterative system of nonlinear elliptic equations of the form ▵ z j + (N - 2) 2 r 0 2 N - 2 | x | 2 N - 2 z j + φ (| x |) g j (z j + 1) = 0 , R 1 < | x | < R 2 , where j ∈ { 1 , 2 , 3 , · · · , ℓ } , z 1 = z ℓ + 1 , ▵ z = div (▿ z) , N > 2 , 0 < r 0 < π / 2 , φ = ∏ i = 1 n φ i , each φ i : (r 0 , + ∞) → (0 , + ∞) is continuous, r N - 1 φ is integrable, and g j : [ 0 , + ∞) → R is continuous, by an application of various fixed point theorems in a Banach space. Further, we also establish uniqueness of the solution for the addressed system by using Rus's theorem in a complete metric space. [ABSTRACT FROM AUTHOR]

Sufficient conditions are given for the existence of solutions of bifunction-set optimization problems, whose underlying sets are complete metric spaces. The main result of this paper is formulated in Theorem 3.1, with the help of some new data. The advantage of introducing the new data is that, by choosing suitable new data, we can obtain existence results with assumptions, that are imposed directly on the objective maps, or that are formulated via scalarization. The main result is applied successfully to Kuroiwa set optimization problems and Ky Fan vector inequality problems. We also discuss equivalences between the existence of solutions of the bifunction-set optimization problem, an Ekeland-type variational principle and a Caristi-type fixed point theorem. Examples are provided. [ABSTRACT FROM AUTHOR]

The aim of this manuscript is to propose a contraction to pursue the existence of g-best proximity point results. The finding of this manuscript generalize and unify the results of Rohen and Mlaiki by using the new contraction with P-property and prove the existence and uniqueness of quadruple best proximity point alongwith an example. [ABSTRACT FROM AUTHOR]

This paper surveys regarding solutions of linear and nonlinear integral equations through fixed point theorem. Banach’s contraction mapping principle is the most widely applied fixed point theorem in all of analysis with special applications to the theory of integral equations. Over a decade researchers have generalized the definition of a metric space and using the technique of fixed point have found solutions to many linear and nonlinear Volterra-Fredholm integral equations. [ABSTRACT FROM AUTHOR]

In this paper we establish a coupled fixed point theorem of a coupled multivalued mapping defined on a complete metric space. In our result we use a new contractive inequality. There are rational terms in the expression of the inequality. The contractive condition on the nonlinear map is assumed only to hold on the elements that are related by a binary relation. The main theorem has several corollaries and is illustrated with example. The main result is deduced in metric spaces. Its consequences are discussed for α-admissible mapping as well as in metric spaces with partial order and graph respectively. [ABSTRACT FROM AUTHOR]

In this paper, unique common fixed point theorems for pairs of subcompatible and reciprocally continuous mappings or compatible and subsequentially continuous mappings are obtained. This unique common fixed point is guaranteed under a new concept named a weak altering distance function. Our theorems improve some results especially the ones of [2]. [ABSTRACT FROM AUTHOR]

In this paper, some common fixed point theorems for pairs of subcompatible and reciprocally continuous mappings or compatible and subsequentially continuous mappings satisfying integral type are obtained. Our results improve several results especially the results of [3], Pathak et al. [13], Djoudi and Aliouche [6], Mbarki [10] and others. [ABSTRACT FROM AUTHOR]

In this study, using a lacunary sequence we introduce the concepts of lacunary d-statistically convergent sequences of order α and lacunary d-statistically bounded sequences of order α in general metric spaces. [ABSTRACT FROM AUTHOR]

In this paper, we equip the partition lattice of the positive integers with a metric structure, and study the properties of this metric. We show, that this new space is complete, compact and separable. We also investigate how continuous functions look like in this structure. [ABSTRACT FROM AUTHOR]

This paper is a brief review of results in the theory of covering mappings of metric spaces and vector metric spaces and its applications to implicit differential equations. For the Cauchy problem and boundary-value problems, we obtain existence conditions, estimates of solutions, and conditions of the continuous dependence of solutions on the parameters of the equation and initial and boundary conditions. [ABSTRACT FROM AUTHOR]

We study the density of the set SNA (M, Y) of those Lipschitz maps from a (complete pointed) metric space M to a Banach space Y which strongly attain their norm (i.e., the supremum defining the Lipschitz norm is actually a maximum). We present new and somehow counterintuitive examples, and we give some applications. First, we show that SNA (T, Y) is not dense in Lip0 (T, Y) for any Banach space Y, where T denotes the unit circle in the Euclidean plane. This provides the first example of a Gromov concave metric space (i.e., every molecule is a strongly exposed point of the unit ball of the Lipschitz-free space) for which the density does not hold. Next, we construct metric spaces M satisfying that SNA (M,Y ) is dense in Lip0 (M, Y) regardless Y but which contain isometric copies of [0, 1] and so the Lipschitz-free space F(M) fails the Radon--Nikodým property, answering in the negative a question posed by Cascales et al in 2019 and by Godefroy in 2015. Furthermore, an example of such M can be produced failing all the previously known sufficient conditions for the density of strongly norm attaining Lipschitz maps. Finally, among other applications, we show that if M is a boundedly compact metric space for which SNA (M,R) is dense in Lip0 (M, R), then the unit ball of the Lipschitz-free space on M is the closed convex hull of its strongly exposed points. Further, we prove that given a compact metric space M which does not contain any isometric copy of [0, 1] and a Banach space Y, if SNA (M,Y) is dense, then SNA (M,Y) actually contains an open dense subset. [ABSTRACT FROM AUTHOR]

The aim of this paper is to investigate the interpolative contractions involving rational forms in the framework of b-metric spaces. We prove the existence of a fixed point of such a mapping with different combinations of the rational forms. A certain example is considered to indicate the validity of the observed result. [ABSTRACT FROM AUTHOR]

Distributed consensus in the Wasserstein metric space of probability measures on the real line is introduced in this work. Convergence of each agent's measure to a common measure is proven under a weak network connectivity condition. The common measure reached at each agent is one minimizing a weighted sum of its Wasserstein distance to all initial agent measures. This measure is known as the Wasserstein barycenter. Special cases involving Gaussian measures, empirical measures, and time-invariant network topologies are considered, where convergence rates and average-consensus results are given. This work has possible applicability in computer vision, machine learning, clustering, and estimation. [ABSTRACT FROM AUTHOR]

The large intersection classes in real Euclidean spaces have been introduced by Falconer in 1985. First, we generalize Falconer's large intersection classes to compact metric spaces and show that these classes are closed under countable intersections. Next, we prove that the limsup sets generated by rectangles in a compact metric space enjoy this intersection property under certain full Hausdorff measure assumptions. We also apply our results to high-dimensional Diophantine approximation on fractals. [ABSTRACT FROM AUTHOR]

Often, straightforward notions from one mathematical domain, when altered even slightly, can become rich and rewarding investigations involving numerous additional domains -- particularly when the investigation includes rigorous proof. This study begins with a familiar high school geometry problem (namely finding the circumcentre of a triangle), extends it to the study of equidistance and analyses the problem using geometry in concert with a number of other mathematical areas to construct a new region called the Circumregion. Altogether, the beautiful and intricate interconnectedness of mathematics is both employed and demonstrated. [ABSTRACT FROM AUTHOR]

• We propose the novel concept of metric hulls. • We define the covered operator using mutual distances only. • We evaluate the proposal on synthetic and real-life data. • Our method achieves online response times. • We show that a few hull objects provide high coverage in comparison to the related work. [Display omitted] Similarity searching has become widely utilized in many online services processing unstructured and complex data, e.g., Google Images. Metric spaces are often applied to model and organize such data by their mutual similarity. As top-k queries provide only a local view on data, a data analyst must pose multiple requests to observe the entire dataset. Thus, group-by operators for metric data have been proposed. These operators identify groups by respecting a given similarity constraint and produce a set of objects per group. The analyst can then tediously browse these sets directly, but representative members may provide better insight. In this paper, we focus on concise representations of metric datasets. We propose a novel concept of a metric hull which encompasses a given set by selecting a few objects. Testing an object to be part of the set is then made much faster. We verify this concept on synthetic Euclidean data and real-life image and text datasets and show its effectiveness and scalability. The metric hulls provide much faster and more compact representations when compared with commonly used ball representations. [ABSTRACT FROM AUTHOR]

In this paper, we establish weighted higher order exponential type inequalities in the geodesic space (X, d, μ) by proposing an abstract higher order Poincaré inequality. These are also new in the non-weighted case. As applications, we obtain a weighted Trudinger's theorem in the geodesic setting and weighted higher order exponential type estimates for functions in Folland–Stein type Sobolev spaces defined on stratified Lie groups. A higher order exponential type inequality in a connected homogeneous space is also given. [ABSTRACT FROM AUTHOR]

We verify that the Künneth and Mayer-Vietoris formulae for magnitude homology of graphs, proven by Hepworth and Willerton, generalise naturally to the metric setting. Similarly, we extend the notion of diagonality of graphs to metric spaces, and verify its stability under products, retracts, and filtrations. As an application, we show that median spaces are diagonal; in particular any Menger convex median space has vanishing magnitude homology. [ABSTRACT FROM AUTHOR]

The existence of a coupled fixed point for a multi-valued mapping satisfying a certain admissibility condition is established. Requirement that the mapping be a contraction is met implicitly through a function of several variables. Further, we study the stability of coupled fixed point sets for such mappings. Several illustrative examples are presented. [ABSTRACT FROM AUTHOR]

The capacitated vehicle routing problem (CVRP) is a classical combinatorial optimization problem having a wide range of practically important applications in operations research. As most combinatorial problems, CVRP is strongly NP-hard (even on the Euclidean plane). A metric instance of CVRP is APX-complete, so it cannot be approximated to arbitrary prescribed accuracy in the class of polynomial time algorithms (assuming that ). Nevertheless, in the case of finite-dimensional Euclidean spaces, a quasi-polynomial or even polynomial time approximation scheme can be found for the problem by applying an approach based on works by S. Arora, A. Das, and C. Mathieu. Below, this approach has been extended for the first time to a significantly larger class of metric spaces of fixed doubling dimension. It is shown that CVRP formulated in such a space has a quasi-polynomial time approximation scheme whenever the number of routes in its optimal solution is bounded above by a polynomial in the logarithm of the input size. [ABSTRACT FROM AUTHOR]

We establish a coupled coincidence and common coupled fixed point theorem for hybrid pair of mappings under φ - ψ contraction on a noncomplete metric space, which is not partially ordered. It is to be noted that to find coupled coincidence point, we do not employ the condition of continuity of any mapping involved therein. We also give an example to validate our result. We improve and generalize several known results. [ABSTRACT FROM AUTHOR]

Let R be a commutative ring with identity. In this paper, for a given monotone decreasing positive sequence and an increasing sequence of subsets of R, we will define a metric on R using them. Then, we will use this kind of metric to obtain a variant of the proof of Van der Waerden’s theorem by Furstenberg [3]. [ABSTRACT FROM AUTHOR]

In his 1990 Inventiones paper, P. Jones characterized subsets of rectifiable curves in the plane, using a multiscale sum of what is now known as Jones β-numbers, numbers measuring flatness in a given scale and location. This work was generalized to Rn by Okikiolu, to Hilbert space by the second author, and has many variants in a variety of metric settings. Notably, in 2005, Hahlomaa gave a sufficient condition for a subset of a metric space to be contained in a rectifiable curve. We prove the sharpest possible converse to Hahlomaa's theorem for doubling curves, and then deduce some corollaries for subsets of metric and Banach spaces, as well as the Heisenberg group. [ABSTRACT FROM AUTHOR]

The main goal of this paper is to study a new dispersive order in an arbitrary metric space. More precisely, given two random variables on a metric space, the order decides which one has a more concentrated distribution. It is motivated by the wish to be able to make new comparisons that the currently defined orderings cannot make. Related to this problem, several statistical parameters of a random variable, giving different kinds of information about the distribution, are studied. It is proved that the new dispersive order also compares some of these parameters. Special attention is paid to Riemannian manifolds, a natural class of metric spaces. Here, the new order behaves in a distinguished way: the differentiability axioms allow obtaining several equations which facilitate the decision. The principal properties of the order are presented, as well as several applications. Finally, the independence of this order from those already known is proved. • A new dispersive order for an arbitrary metric space. • Distinguished properties in Riemannian manifolds. • Better behaviour than other dispersive stochastic orders. [ABSTRACT FROM AUTHOR]

Let (M,d) be a metric space, X⊂ M be a nonempty closed subset and K⊂ M be a nonempty compact subset. By definition, an upper semi-continuous multivalued operator F:X→ P(X) is said to be a strong Frum-Ketkov type operator if there exists α ∈ ]0,1[ such that ed(F(x),K)\≤ α Dd(x,K), for every x→ X, where ed is the excess functional generated by d and Dd is the distance from a point to a set. In this paper, we will study the fixed points of strong Frum-Ketkov type multivalued operators. [ABSTRACT FROM AUTHOR]

The Banach Contraction Mapping Principle has many generalizations and extensions in different directions. Here we define -Uniformly Local Weak Contractions and show that these mappings admit unique fixed points. We obtain a generalization of two existing results. We construct an example which illustrates of our main result in which the mapping is neither a -weak contraction nor a local contraction. [ABSTRACT FROM AUTHOR]

We classify countable metrically homogeneous graphs of diameter 3. [ABSTRACT FROM AUTHOR]

In this paper, we investigate sufficient conditions for the existence of solutions to the system T x = x , α i (x) = 0 E , i = 1 , 2 , ... , r , where 0 E is the zero vector of E, and α i : E → E i = 1 , 2 , ... , r are mappings, T is a mapping satisfying the Pachpatte-contraction. [ABSTRACT FROM AUTHOR]

We introduce a metric on the set of permutations of given order, which is a weighted generalization of Kendall's τ rank distance and study its properties. Using the edge graph of a permutohedron, we give a criterion which guarantees that a permutation lies metrically between another two fixed permutations. In addition, the conditions under which four points from the resulting metric space form a pseudolinear quadruple were found. [ABSTRACT FROM AUTHOR]

The 'the' in the title hides a subtlety. A metric space induces not one but four topologies - by means of open sets, closed sets, closure, and interior - they just so happen to coincide. The agreement between these four structures arising from a metric function d : X × X → 0 , ∞ is due to a combination of the metric axioms and the lattice structure of 0 , ∞. Further motivation materializes from Lawvere's observation from 1973 to the effect that a (slightly generalized) metric space is a category enriched in 0 , ∞. Metric spaces taking values in structures other than 0 , ∞ are relevant for generalizations of metric spaces and find a natural home in Lawvere's categorical setting. In particular, in recent years quantales emerged as structures occupying an important niche in between 0 , ∞ and arbitrary monoidal categories. Since a category enriched in a quantale Q is the same thing as a metric space taking values in Q one may ask whether such a thing belongs to algebra or geometry. Further, does the quadruplet of topologies associated to a Q -valued space/category still consist of identical siblings? We propose a litmus test for the geometricity of Q -valued spaces as we investigate these issues. [ABSTRACT FROM AUTHOR]

Faults are unwanted events in any industrial production system. Early detection and diagnosis of faults in automated systems is important in order to prevent equipment damage, loss of performance and profits. For this purpose, more and more sophisticated and complex systems for observation and monitoring of basic characteristics in automated processes are being built. Preconditions for increasing their efficiency are processing and analysis of process information is obtained through a significant number of sensors. For pneumatic systems in addition to the identification of certain faults that may affect the normal production process, it is important to consider the possibilities to improve their energy efficiency. In this regards, the work focuses on the detection of leaks. The fault detection is based on the measurement of the compressed air consumption at the inlet of the pneumatic module and synchronization with signal from the PLC to the valve, and controlled the pneumatic cylinder. The experimental study aims to develop methods for automatic detection and classification of leaks that may be used in machine learning algorithms. [ABSTRACT FROM AUTHOR]

For the definition of diseases and symptoms, medical classifications use linguistic descriptions. The present work is a follow-up of the investigations concerning the presentation of the research area "Psychiatric and Behavioural Disorders in Epilepsy" by numerical values. A metric space was developed and allowed for the evaluation of the distances between psychiatric and behavioural disorders in epilepsy. [ABSTRACT FROM AUTHOR]