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Given a finite subset A of a distributive lattice, its total orderization to(A) is a natural transformation of A into a totally ordered set. Recently, the author showed that multivariate maps on distributive lattices which remain invariant under total orderizations generalize various maps on vector lattices, including bounded orthosymmetric multilinear maps and finite sums of bounded orthogonally additive polynomials. Therefore, a study of total orderization invariant maps on distributive lattices provides new perspectives for maps widely researched in vector lattice theory. However, the unwieldy notation of total orderizations can make calculations extremely long and difficult. In this paper we resolve this complication by providing considerably simpler characterizations of total orderization maps. Utilizing these easier representations, we then prove that a lattice multi-homomorphism on a distributive lattice is total orderization invariant if and only if it is symmetric, and we show that the diagonal of a symmetric lattice multi-homomorphism is a lattice homomorphism, extending known results for orthosymmetric vector lattice homomorphisms. [ABSTRACT FROM AUTHOR]

Given a separation oracle SO for a convex function f defined on Rn that has an integral minimizer inside a box with radius R, we show how to find an exact minimizer of f using at most • O(n(n log log(n)/ log(n) + log(R))) calls to SO and poly(n, log(R)) arithmetic operations, or • O(n log(nR)) calls to SO and exp(O(n)) · poly(log(R)) arithmetic operations. When the set of minimizers of f has integral extreme points, our algorithm outputs an integral minimizer of f . This improves upon the previously best oracle complexity of O(n² (n + log(R))) for polynomial time algorithms and O(n² log(nR)) for exponential time algorithms obtained by [Grötschel, Lovász and Schrijver, Prog. Comb. Opt. 1984, Springer 1988] over thirty years ago. Our improvement on Grötschel, Lovász and Schrijver’s result generalizes to the setting where the set of minimizers of f is a rational polyhedron with bounded vertex complexity. For the Submodular Function Minimization problem, our result immediately implies a strongly polynomial algorithm that makes at most O(n³ log log(n)/ log(n)) calls to an evaluation oracle, and an exponential time algorithm that makes at most O(n2 log(n)) calls to an evaluation oracle. These improve upon the previously best O(n³ log² (n)) oracle complexity for strongly polynomial algorithms given in [Lee, Sidford and Wong, FOCS 2015] and [Dadush, Végh and Zambelli, SODA 2018], and an exponential time algorithm with oracle complexity O(n³ log(n)) given in the former work. Our result is achieved via a reduction to the Shortest Vector Problem in lattices. We show how an approximately shortest vector of an auxiliary lattice can be used to effectively reduce the dimension of the problem. Our analysis of the oracle complexity is based on a potential function that simultaneously captures the size of the search set and the density of the lattice, which we analyze via tools from convex geometry and lattice theory. [ABSTRACT FROM AUTHOR]

The statistically unbounded p-convergence is an abstraction of the statistical order, unbounded order, and p-convergences. We investigate the concept of the statistically unbounded convergence on lattice-normed Riesz spaces with respect to statistical p-decreasing sequences. Also, we get some relations between this concept and the other kinds of statistical convergences on Riesz spaces. [ABSTRACT FROM AUTHOR]

A "Bochner-type" integral for vector lattice-valued functions with respect to (possibly infinite) vector lattice-valued measures is presented with respect to abstract convergences, satisfying suitable axioms, and some fundamental properties are studied. Moreover, by means of this integral, some convergence results on operators in vector lattice-valued modulars are proved. Some applications are given to moment kernels and to the Brownian motion. [ABSTRACT FROM AUTHOR]

We prove interpolating estimates providing a bound for the oscillation of a function in terms of two Lp norms of its gradient. They are based on a pointwise bound of a function on cones in terms of the Riesz potential of its gradient. The estimates hold for a general class of domains, including, e.g., Lipschitz domains. All the constants involved can be explicitly computed. As an application, we show how to use these estimates to obtain stability for Alexandrov's Soap Bubble Theorem and Serrin's overdetermined boundary value problem. The new approach results in several novelties and benefits for these problems. [ABSTRACT FROM AUTHOR]

The Shannon-Whittaker Sampling Theorem states that a frequency bounded signal can be completely determined by its sampled values at a countable number of points. Thus, the theorem allows us to convert analog signals to digital signals by sampling (or evaluating) the signal at these points. In prior work, it was shown that if a signal is oversampled, and if some of the sampled values are lost when transmitting the signal, then it is still possible to reconstruct the signal. However, in certain situations, the reconstruction algorithm is very unstable. In this paper, we provide stability bounds on the reconstruction algorithm and determine when it is not feasible to perform the reconstruction. [ABSTRACT FROM AUTHOR]

In this article, we introduce and study a new class of operators defined on a Cartesian product of ideal spaces of measurable functions. We use the general approach of the theory of vector lattices. We say that an operator T : E × F ⟶ W defined on a Cartesian product of vector lattices E and F and taking values in a vector lattice W is orthogonally biadditive if all partial operators T y : E ⟶ W and T x : F ⟶ W are orthogonally additive. In the first part of the article, we prove that, under some mild conditions, a vector space of all regular orthogonally biadditive operators O B A r E , F ; W is a Dedekind complete vector lattice. We show that the set of all horizontally-to-order continuous regular orthogonally biadditive operators is a projection band in O B A r E , F ; W . In the last section of the paper, we investigate orthogonally biadditive operators on a Cartesian product of ideal spaces of measurable functions. We show that an integral Uryson operator which depends on two functional variables is orthogonally biadditive and obtain a criterion of the regularity of an orthogonally biadditive Uryson operator. [ABSTRACT FROM AUTHOR]

The article introduces the scale‐deviating operator. The scale‐deviating differential operator comprises the parameters to designate the operator order and the parameters to define the dimension of space. The operator order depends on the characteristic length κ. There are two types of linear scale‐deviating operators. For the distances r, which are much less than κ, the scale‐deviating operator of the first type A reduces to the common operators. For the distances, which exceed the length κ, this operator reduces to the fractional Riesz operator. The second type of the scale‐deviating operator B behaves oppositely. For the distances r, which are much higher than κ, the scale‐deviating operator of the second type reduces to the common operators. Finally, for the distances, which below the length κ, this operator lessens to the fractional Riesz operator. These linear, isotropic operators possess the order, less than two. The solutions of new scale‐deviating equations and the shell theorem for these operators are provided closed form. [ABSTRACT FROM AUTHOR]

We prove an analog of the Faber-Krahn inequality for the Riesz potential operator. The proof is based on Riesz's inequality under Steiner symmetrization and the continuity of the first eigenvalue of the Riesz potential operator with respect to the convergence, in the complementary Hausdorff distance, of a family of uniformly bounded non-empty convex open sets. [ABSTRACT FROM AUTHOR]

In this paper, we generate an extended result by Bor and Seyhan concerning absolute Riesz summability factors. Further, we develop some well-known results from our main result. [ABSTRACT FROM AUTHOR]

A mixed lattice group is a generalization of a lattice ordered group. The theory of mixed lattice semigroups dates back to the 1970s, but the corresponding theory for groups and vector spaces has been relatively unexplored. In this paper we investigate the basic structure of mixed lattice groups, and study how some of the fundamental concepts in Riesz spaces and lattice ordered groups, such as the absolute value and other related ideas, can be extended to mixed lattice groups and mixed lattice vector spaces. We also investigate ideals and study the properties of mixed lattice group homomorphisms and quotient groups. Most of the results in this paper have their analogues in the theory of Riesz spaces. [ABSTRACT FROM AUTHOR]

We discuss and compare a number of notions of modulars appeared in literature, among which there is a selection of the well known ones. We highlight the connections between the various definitions and provide several examples, taken from existing literature, recalling known results and completing the picture with some original considerations. [ABSTRACT FROM AUTHOR]

In this work, we present the ORPIT Matlab toolbox. ORPIT applies the theory of vector lattices to solve (a) the problem of option replication and (b) the cost minimization problem of portfolio insurance as well as related sub problems. The key point is that we use the theory of lattice-subspaces and the theory of positive bases, in Riesz spaces, so ORPIT does not require the presence of linear programming methods. This is a great advantage of this approach that allows us to find multiple, if any, solutions of the corresponding problems after one call of the proposed methods. We illustrate the diverse features of the ten Matlab functions inside this toolbox through a representative collection of examples. To the best of our knowledge, there is no such solver to apply in problems of portfolio optimization or option replication. [ABSTRACT FROM AUTHOR]

Highlights • A method for generating vector-value integrable analogies of integrable lattice systems is presented. • Some new vector lattice systems are obtained. • The zero-curvature representations and Hamiltonian structures of the resulting vector lattice systems are formulated. Abstract A method for generating vector-value integrable analogies of integrable lattice systems or integrable differential-difference equations is presented. The basic ingredient of the method is to insert permutation matrices. We formulate the zero-curvature representations and Hamiltonian structures of the resulting vector lattice systems. The effectiveness of the method is illustrated using some examples such as the Volterra lattice, the Belov–Chaltikian lattice, the Ablowitz–Ladik lattice and the Heisenberg ferromagnet lattice. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to introduce the concept of a Menger probabilistic normed Riesz space. We study some properties of these spaces and compare normed Riesz spaces with Menger probabilistic normed Riesz spaces. Next, we investigate the Hyers-Ulam stability of lattice homomorphisms in Menger probabilistic normed Riesz spaces. [ABSTRACT FROM AUTHOR]

A sequence {φn} in a Hilbert space H with inner product <·,·> is called a generalized Riesz system if there exist an ONB e={en} in H and a densely defined closed operator T in H with densely defined inverse such that {en}⊂D(T)∩D((T-1)⁎) and Ten=φn, n=0,1,⋯, and (e,T) is called a constructing pair for {φn} and T is called a constructing operator for {φn}. The main purpose of this paper is to investigate under what conditions the ordered set Cφ of all constructing operators for a generalized Riesz system {φn} has maximal elements, minimal elements, the largest element, and the smallest element in order to find constructing operators fitting to each of the physical applications. [ABSTRACT FROM AUTHOR]

Let rk(n) denote the number of lattice points on a k-dimensional sphere of radius n. The generating functionζks=∑n=1∞rknn−s,k≥2,is Epstein’s zeta function. The paper considers the Riesz mean of the typeDρxζ3=1Γρ+1∑n≤xx−nρr3n,where ρ > 0; the error term Δρ(x; ζ3) is defined byDρxζ3=π3/2xρ+3/2Γρ+5/2+xρΓρ+1ζ30+Δρxζ3.K. Chandrasekharan and R. Narasimhan (1962, MR25#3911) proved thatΔρxζ3={O(x1/2+ρ/2)ρ>1,Ω±x1/2+ρ/2ρ≥0.In the present paper, it is proved thatΔρxζ3={Oxlogxρ=1,Ox2/3+ρ/3+ε1/2<ρ<1,Ox3/4+ρ/4+ε0<ρ≤1/2,and the Riesz means of the coefficients of ζk(s), k ≥ 4, are studied. [ABSTRACT FROM AUTHOR]

We study a nonlinear parabolic equation with measurable nonlinearity and measure data: u t − d i v a (D u, u, x, t) = μ i n Ω T. We obtain an optimal point-wise estimate for the gradient of a solution in terms of a certain Riesz potential of the right-hand measure in a very general setting that the nonlinearity is not only allowed to be measurable in one of spatial variables, but also depends on the variable of the possibly unbounded solution itself. [ABSTRACT FROM AUTHOR]

In this paper, we introduce and study some strongly almost convergent double sequence spaces by Riesz mean associated with four-dimensional bounded regular matrix and a Musielak–Orlicz function over n -normed spaces. We make an effort to study some topological and algebraic properties of these sequence spaces. We also study some inclusion relations between the spaces. Finally, we establish some relation between weighted lacunary statistical sequence spaces and Riesz lacunary almost statistical convergent sequence spaces over n -normed spaces. [ABSTRACT FROM AUTHOR]

We consider the Dirac operator on the interval [0, π] with an integrable potential P = (pij (x))i,j=12 and strongly regular boundary conditions U. It is well known that for any integrable potential P the system {yn}n∈Z of root functions of the strongly regular operator LP,U is a Riesz basis in the space H = L2[0, π] × L2[0, π]. We obtain estimates, uniform on every compact set of potentials, of the Riesz constants ||T||||T−1||, where T is the operator of transition to an orthonormal basis. [ABSTRACT FROM AUTHOR]

We prove the existence of Riesz bases of exponentials of L2(Ω), provided that Ω � Rd is a measurable set of finite and positive measure, not necessarily bounded, that satisfies a multi-tiling condition and an arithmetic property that we call admissibility. This property is satisfied for any bounded domain, so our results extend the known case of bounded multi-tiles. We also extend known results for submulti-tiles and frames of exponentials to the unbounded case. [ABSTRACT FROM AUTHOR]

Continuing the analysis undertaken in previous articles, we discuss some features of non-self-adjoint operators and sesquilinear forms which are defined starting from two biorthogonal families of vectors, like the so-called generalized Riesz systems, enjoying certain properties. In particular, we discuss what happens when they forms two D -quasi-bases. [ABSTRACT FROM AUTHOR]

We use a landmark result in the theory of Riesz spaces – Freudenthal’s 1936 spectral theorem – to canonically represent any Archimedean lattice-ordered group G with a strong unit as a (non-separating) lattice-group of real-valued continuous functions on an appropriate G-indexed zero-dimensional compactification w G ⁢ Z G {w_{G}Z_{G}} of its space Z G {Z_{G}} of minimal prime ideals.The two further ingredients needed to establish this representation are the Yosida representation of G on its space X G {X_{G}} of maximal ideals, and the well-known continuous surjection of Z G {Z_{G}} onto X G {X_{G}}.We then establish our main result by showing that the inclusion-minimal extension of this representation of G that separates the points of Z G {Z_{G}} – namely, the sublattice subgroup of C ⁡ ( Z G ) {\operatorname{C}(Z_{G})} generated by the image of G along with all characteristic functions of clopen (closed and open) subsets of Z G {Z_{G}} which are determined by elements of G – is precisely the classical projectable hull of G.Our main result thus reveals a fundamental relationship between projectable hulls and minimal spectra, and provides the most direct and explicit construction of projectable hulls to date. Our techniques do require the presence of a strong unit. [ABSTRACT FROM AUTHOR]

Let Y be a sublattice of a vector lattice X. We consider the problem of identifying the smallest order closed sublattice of X containing Y. It is known that the analogy with topological closure fails. Let ... be the order closure of Y consisting of all order limits of nets of elements from Y. Then ... need not be order closed. We show that in many cases the smallest order closed sublattice containing Y is in fact the second order closure .... Moreover, if X is a σ-order complete Banach lattice, then the condition that ... is order closed for every sublattice Y characterizes order continuity of the norm of X. The present paper provides a general approach to a fundamental result in financial economics concerning the spanning power of options written on a financial asset. [ABSTRACT FROM AUTHOR]

In this paper we show that if $$\mu $$ is a Borel measure in $${{\mathbb {R}}}^{n+1}$$ with growth of order n, such that the n-dimensional Riesz transform $${{\mathcal {R}}}_\mu $$ is bounded in $$L^2(\mu )$$ , and $$B\subset {{\mathbb {R}}}^{n+1}$$ is a ball with $$\mu (B)\approx r(B)^n$$ such that: then there exists a uniformly n-rectifiable set $$\Gamma $$ , with $$\mu (\Gamma \cap B)\gtrsim \mu (B)$$ , and such that $$\mu |_\Gamma $$ is absolutely continuous with respect to $${{\mathcal {H}}}^n|_\Gamma $$ . This result is an essential tool to solve an old question on a two phase problem for harmonic measure in subsequent papers by Azzam, Mourgoglou, Tolsa, and Volberg. [ABSTRACT FROM AUTHOR]

In this note, we investigate the existence of frames of exponentials for L 2 ( Ω ) in the setting of LCA groups. Our main result shows that sub-multitiling properties of Ω ⊂ G ˆ with respect to a uniform lattice Γ of G ˆ guarantee the existence of a frame of exponentials with frequencies in a finite number of translates of the annihilator of Γ. We also prove the converse of this result and provide conditions for the existence of these frames. These conditions extend recent results on Riesz bases of exponentials and multitilings to frames. [ABSTRACT FROM AUTHOR]

An explicit description of the algebraic and vector lattice operations on the Isbell-convex hull of an asymmetrically normed real vector space is provided. Connections between the concepts of Isbell-convexity, injectivity and Dedekind-completeness of asymmetrically normed real vector spaces are also considered. [ABSTRACT FROM AUTHOR]

Using some recent results of the Riesz decomposition method for sharp estimates of certain boundary value problems of harmonic functions in (St. Cer. Mat. 27:323-328, 1975), the boundary behaviors of upper and lower superharmonic multifunctions are studied. Several fundamental properties of these new classes of these functions are shown. A new technique is proposed to find the exact boundary behaviors by using Levin's type boundary behaviors for harmonic functions admitting certain lower bounds in (Pacific J. Math. 15:961-970, 1965). Finally, some examples are given to illustrate the applications of our results. [ABSTRACT FROM AUTHOR]

We study the nonlocal Choquard equation − ε 2 Δ u ε + V u ε = ( I α ∗ | u ε | p ) | u ε | p − 2 u ε in R N where N ≥ 1 , I α is the Riesz potential of order α ∈ ( 0 , N ) and ε > 0 is a parameter. When the nonnegative potential V ∈ C ( R N ) achieves 0 with a homogeneous behavior or on the closure of an open set but remains bounded away from 0 at infinity, we show the existence of groundstate solutions for small ε > 0 and exhibit the concentration behavior as ε → 0 . [ABSTRACT FROM AUTHOR]

This paper offers a new perspective to look at the Riesz potential. On the one hand, it is shown that not only contains under the conditions , , , , but also exists as an associate space under the condition , where and are the Morrey-Sobolev and Campanato spaces on respectively. On the other hand, a nonnegative Radon measure μ is completely characterized to produce a continuous map under the condition or , where and are the -Lorentz and -Lebesgue spaces on respectively. [ABSTRACT FROM AUTHOR]

In this paper, we consider a Riesz-Feller space-fractional backward diffusion problem with a time-dependent coefficient [ABSTRACT FROM AUTHOR]

In this article, we study the space-fractional wave equation with Riesz fractional derivative. The continuation of the solution of this space-fractional equation to the solution of the corresponding integer order equation is proved. The series solution is obtained based on properties of Riesz fractional derivative operator and utilizing the optimal homotopy analysis method (OHAM). Numerical simulations are presented to validate the method and to show the effect of changing the fractional derivative parameter on the solution behavior. [ABSTRACT FROM AUTHOR]

Let Λ be a lattice in Rd with positive co-volume. Among Λ-periodic N-point configurations, we consider the minimal renormalized Riesz s-energy εs,Λ(N). While the dominant term in the asymptotic expansion of εs,Λ(N) as N goes to infinity in the long-range case that 0 < s < d (or s=log) can be obtained from classical potential theory, the next order term(s) require a different approach. Here we derive the form of the next order term or terms, namely for s>0 they are of the form Cs,d|Λ|−s/dN1+s/d and −2/dNlogN+(Clog,d−2ζ′Λ(0))N where we show that the constants Cs,d and Clog,d are independent of the lattice Λ. [ABSTRACT FROM AUTHOR]

In the setting of a Dedekind complete Riesz space on which a conditional expectation is defined, we study the Itô integral. This integral obtains the integration of a stochastic process of a continuous parameter relative to a martingale. A considerable part of the paper is devoted to aspects of integration of a vector-valued function with respect to a vector measure. Here we use the Dobrakov integral that is, in our case, a generalization of the well known Bartle integral. We discuss natural and predictable processes to prove a general version of the Doob–Meyer decomposition of a submartingale. This decomposition provides an essential tool used in the definition of the stochastic integral. We derive the properties of the stochastic integral that are useful in developing the theory of stochastic processes in Riesz spaces. [ABSTRACT FROM AUTHOR]

The space-time-fractional diffusion equation with the Caputo time-fractional derivative and Riesz fractional Laplacian is considered in the case of axial symmetry. Mass absorption (mass release) is described by a source term proportional to concentration. The integral transform technique is used. Different particular cases of the solution are studied. The numerical results are illustrated graphically. [ABSTRACT FROM AUTHOR]

We give the characterization of the differentiability and nondifferentiability points of a generalization of the Riesz-N'agy-Tak'acs (RNT) singular function, namely, the generalized RNT (GRNT) singular function. A particular characterization generalizes recentmultifractal and metric-number-theoretical results associated with the RNT singular function. Furthermore, we compute the moments of the GRNT singular function. [ABSTRACT FROM AUTHOR]

In this paper we establish new L1-type estimates for the classical Riesz potentials of order a α ∈ (0,N): ‖ Iɑu ‖LN/(N-α)(RN) ⩽ C ‖ Ru ‖L1(RN;RN). This sharpens the result of Stein and Weiss on the mapping properties of Riesz potentials on the real Hardy space H1(RN) and provides a new family of L1-Sobolev inequalities for the Riesz fractional gradient. [ABSTRACT FROM AUTHOR]

This is a survey of some recent applications of Boolean-valued models of set theory to the study of order-bounded operators in vector lattices. [ABSTRACT FROM AUTHOR]

In this paper, we study some important properties of positive solutions for a nonlinear integral system. With the help of the method of moving planes in an integral form, we show that under certain integrable conditions, all of positive solutions to this system are radially symmetric and decreasing with respect to the origin. Meanwhile, using the regularity lifting lemma, which was recently introduced by Chen and Li in, we obtain the optimal integrable intervals and sharp asymptotic behaviors for such positive solutions, which characterize the closeness of system to some extent. [ABSTRACT FROM AUTHOR]

In the present paper, we shall give a characterization for the Spanne and Adams type boundedness of the Riesz potential and its commutators on the generalized Orlicz-Morrey spaces, respectively. Also we give criteria for the weak versions of Spanne and Adams type boundedness of the Riesz potential on the generalized Orlicz-Morrey spaces. In all the cases the conditions for the boundedness are given in terms of Zygmund type integral inequalities involving the Young function $\Phi(u)$ and the function $\varphi(x,r)$ defining the space. [ABSTRACT FROM AUTHOR]

We study completions of Archimedean vector lattices relative to any nonempty set of positively homogeneous functions on finite-dimensional real vector spaces. Examples of such completions include square mean closed and geometric mean closed vector lattices, amongst others. These functional completions also lead to a universal definition of the complexification of any Archimedean vector lattice and a theory of tensor products and powers of complex vector lattices in a companion paper. [ABSTRACT FROM AUTHOR]

This paper deals with the boundary feedback stabilization of a SCOLE model, namely a flexible beam hinged at one end, and free at the other end where a rigid body is attached, using the Riesz basis approach, it is shown that the closed-loop system is a Riesz spectral system and as consequences, the asymptotic expression of eigenvalues and the expoenential stability are concluded. To verify the theoritical developments, numerical study of the spectrum is performed. [ABSTRACT FROM AUTHOR]

Projections onto several special subsets in the Dedekind complete vector lattice of orthogonally additive, order bounded (called abstract Uryson) operators between two vector lattices E and F are considered and some new formulas are provided. [ABSTRACT FROM AUTHOR]

In this paper we will study the boundary values properties of the functions in the Hardy spaces; generalize the F. and M. Riesz theorem to higher dimensions; discuss the existence of boundary values of the functions in Hp(Dn) on non-distinguished boundary ∂ Dn \ Tn and the intersection of the spaces Hpu (Dn). [ABSTRACT FROM AUTHOR]

The aim of this paper is to investigate Hyers-Ulam stability of the following lattice preserving functional equation on Riesz space with fixed point method: ||F(τx V ηy) -- τF(x) V ηF(y)|| ≤ φ(τx V ηy, τx Λ ηy), where X is a Banach lattice and φ : X × X ↑ X is a mapping such that φ(x,y)≤(τη)α/2 φ(x/τ,y/η) for all τ,η ≥ 1 and α ∈[0,1/2). [ABSTRACT FROM AUTHOR]

We establish conditions when a certain type of the Riesz decomposition property (RDP) holds in the lexicographic product of two po-groups. It is well known that the resulting product is an $$\ell $$ -group if and only if the first one is linearly ordered and the second one is an $$\ell $$ -group. This can be equivalently studied as po-groups with a special type of the RDP. In the paper we study three different types of RDPs. RDPs of the lexicographic products are important for the study of pseudo effect algebras where infinitesimal elements play an important role both for algebras as well as for the first-order logic of valid but not provable formulas. [ABSTRACT FROM AUTHOR]

In this paper we introduce variable exponent bounded variation spaces in the Riesz sense, prove some embedding results and finally we show that a type of Riesz representation lemma is valid in the newly introduced spaces. [ABSTRACT FROM AUTHOR]

We prove a law of large numbers and a central limit theorem with respect to the order convergence topology in vector lattices. [ABSTRACT FROM AUTHOR]