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In this paper author was proved the boundedness of the multidimensional Hardy type operator in weighted Lebesgue spaces with a variable exponent. As an application we prove the boundedness of certain sublinear operators on the weighted variable Lebesgue space. Note that the proof of multidimensional Hardy type operator in weighted Lebesgue spaces with a variable exponent don't contained any mistakes. But at the proving of the boundedness of certain sublinear operators on the weighted variable Lebesgue space Georgian colleagues discovered a small but significant error in this paper. This result is assigned as Theorem 5 in noted paper. In other words, sufficient conditions for general weights ensuring the validity of the two-weight strong type inequalities for some sublinear operator was found. In this theorem the inequality (9) isn't true. In this note we give the details of the correct argument. We presume that the reader is familiar with the contents and notation of our original paper. At the heart of our correction is the following Theorem which replaces Theorem 5.

We prove a new effective Chebotarev density theorem for Galois extensions $L/\mathbb{Q}$ that allows one to count small primes (even as small as an arbitrarily small power of the discriminant of $L$); this theorem holds for the Galois closures of "almost all" number fields that lie in an appropriate family of field extensions. Previously, applying Chebotarev in such small ranges required assuming the Generalized Riemann Hypothesis. The error term in this new Chebotarev density theorem also avoids the effect of an exceptional zero of the Dedekind zeta function of $L$, without assuming GRH. We give many different "appropriate families," including families of arbitrarily large degree. To do this, we first prove a new effective Chebotarev density theorem that requires a zero-free region of the Dedekind zeta function. Then we prove that almost all number fields in our families yield such a zero-free region. The innovation that allows us to achieve this is a delicate new method for controlling zeroes of certain families of non-cuspidal $L$-functions. This builds on, and greatly generalizes the applicability of, work of Kowalski and Michel on the average density of zeroes of a family of cuspidal $L$-functions. A surprising feature of this new method, which we expect will have independent interest, is that we control the number of zeroes in the family of $L$-functions by bounding the number of certain associated fields with fixed discriminant. As an application of the new Chebotarev density theorem, we prove the first nontrivial upper bounds for $\ell$-torsion in class groups, for all integers $\ell \geq 1$, applicable to infinite families of fields of arbitrarily large degree., Comment: 52 pages. This shorter version aligns with the published paper. Note that portions of Section 8 of the longer v1 have been developed as a separate paper with identifier arXiv:1902.02008

In the authors' first paper, Beurling-Rudin-Korenbljum type characterization of the closed ideals in a certain algebra of holomorphic functions was used to describe the lattice of invariant subspaces of the shift plus a complex Volterra operator. Current work is an extension of the previous work and it describes the lattice of invariant subspaces of the shift plus a positive integer multiple of the complex Volterra operator on the Hardy space. Our work was motivated by a paper by Ong who studied the real version of the same operator., We deleted a proposition and a corollary from section 4, and simplified the proof of the main theorem. **The article has been published in Archiv der Mathematik**

This paper addresses the sharpness of a weighted $L^{2}$-estimate for the Fourier extension operator associated to the circle, obtained by J. Bennett, A. Carbery, F. Soria and A. Vargas in 2006. A point left open in their paper was the necessity of a certain $\log R$-factor in the bound. Here, I show that the factor is necessary for all $1/2$-Ahlfors-David regular weights on the circle, but it can be removed for $s$-Ahlfors-David regular weights with $s \neq 1/2$., 12 pages

In this paper we investigate quasi-Monte Carlo (QMC) integration using digital nets over Z b in reproducing kernel Hilbert spaces. The tent transformation (previously called baker’s transform) was originally used for lattice rules by Hickernell (2002) to achieve higher order convergence of the integration error for smooth non-periodic integrands, and later, has been successfully applied to digital nets over Z 2 by Cristea et al. (2007) and Goda (2015). The aim of this paper is to generalize the latter two results to digital nets over Z b for an arbitrary prime b . For this purpose, we introduce the b -adic tent transformation for an arbitrary positive integer b greater than 1, which is a generalization of the original (dyadic) tent transformation. Further, again for an arbitrary positive integer b greater than 1, we analyze the mean square worst-case error of QMC rules using digital nets over Z b which are randomly digitally shifted and then folded using the b -adic tent transformation in reproducing kernel Hilbert spaces. Using this result, for a prime b , we prove the existence of good higher order polynomial lattice rules over Z b among a smaller number of candidates as compared to the result by Dick and Pillichshammer (2007), which achieve almost the optimal convergence rate of the mean square worst-case error in unanchored Sobolev spaces of smoothness of arbitrary high order.

Let $\Omega$ be either $\mathbb{R}^n$ or a strongly Lipschitz domain of $\mathbb{R}^n$, and $\omega\in A_{\infty}(\mathbb{R}^n)$ (the class of Muckenhoupt weights). Let $L$ be a second order divergence form elliptic operator on $L^2 (\Omega)$ with the Dirichlet or Neumann boundary condition, and assume that the heat semigroup generated by $L$ has the Gaussian property $(G_1)$ with the regularity of their kernels measured by $\mu\in(0,1]$. Let $\Phi$ be a continuous, strictly increasing, subadditive, positive and concave function on $(0,\infty)$ of critical lower type index $p_{\Phi}^-\in(0,1]$. In this paper, the authors introduce the "geometrical" weighted local Orlicz-Hardy spaces $h^{\Phi}_{\omega,\,r}(\Omega)$ and $h^{\Phi}_{\omega,\,z}(\Omega)$ via the weighted local Orlicz-Hardy spaces $h^{\Phi}_{\omega}(\mathbb{R}^n)$, and obtain their two equivalent characterizations in terms of the nontangential maximal function and the Lusin area function associated with the heat semigroup generated by $L$ when $p_{\Phi}^-\in(n/(n+\mu),1]$. As applications, the authors prove that the operators $\nabla^2{\mathbb G}_D$ are bounded from $h^{\Phi}_{\omega,\,r}(\Omega)$ to the weighted Orlicz space $L^{\Phi}_{\omega}(\Omega)$, and from $h^{\Phi}_{\omega,\,r}(\Omega)$ to itself when $\Omega$ is a bounded semiconvex domain in $\mathbb{R}^n$ and $p_{\Phi}^-\in(\frac{n}{n+1},1]$, and the operators $\nabla^2{\mathbb G}_N$ are bounded from $h^{\Phi}_{\omega,\,z}(\Omega)$ to $L^{\Phi}_{\omega}(\Omega)$, and from $h^{\Phi}_{\omega,\,z}(\Omega)$ to $h^{\Phi}_{\omega,\,r}(\Omega)$ when $\Omega$ is a bounded convex domain in $\mathbb{R}^n$ and $p_{\Phi}^-\in(\frac{n}{n+1},1]$, where ${\mathbb G}_D$ and ${\mathbb G}_N$ denote, respectively, the Dirichlet Green operator and the Neumann Green operator., Comment: This paper has been withdrawn by the authors

Let v be a planar Lipschitz vector field. We prove that the r-th variation-norm Hilbert transform along v, composed with a standard Littlewood-Paley projection operator P_k, is bounded from L^2 to L^{2, \infty}, and from L^p to itself for all p>2. Here r>2 and the operator norm is independent of k\in \Z. This generalises Lacey and Li's result for the case of the Hilbert transform. However, their result only assumes measurability for vector fields. In contrast to that, we need to assume vector fields to be Lipschitz., To appear in the Proc. of the AMS. Part of a companion paper by Hickman, Lie, Roos and the author, relies on the result in the present paper

The Naruse hook-length formula is a recent general formula for the number of standard Young tableaux of skew shapes, given as a positive sum over excited diagrams of products of hook-lengths. In 2015 we gave two different $q$-analogues of Naruse's formula: for the skew Schur functions, and for counting reverse plane partitions of skew shapes. In this paper we give an elementary proof of Naruse's formula based on the case of border strips. For special border strips, we obtain curious new formulas for the Euler and $q$-Euler numbers in terms of certain Dyck path summations., Comment: 33 pages, 10 figures. This is the second paper of the series "Hook formulas for skew shapes". Most of Sections 8 and 9 in this paper used to be part of arxiv:1512.08348 (v1,v2); v2 fixed several typos; v3 made precision in definition of flagged tableaux in Section 3.2 and fixed typo in Example 3.3; v4 fixed small typos in proof of Corollary 7.6

In this paper we discuss a problem of generalization of binomial distributed triangle, that is sequence A287326 in OEIS. The main property of A287326 that it returns a perfect cube n as sum of n-th row terms over k, 0 or 1 , by means of its symmetry. In this paper we have derived a similar triangles in order to receive powers m=5,7 as row items sum and generalized obtained results in order to receive every odd-powered monomial n2m+1, m as sum of row terms of corresponding triangle. This version might be out of date, proceed by the link to see actual version., 16 pages, 8 figures, 2 tables, typos revised, added missing references, results generalized and shortened, derivations detailed.

Let $\Omega$ be either $\mathbb{R}^n$ or an unbounded strongly Lipschitz domain of $\mathbb{R}^n$, and $\Phi$ be a continuous, strictly increasing, subadditive and positive function on $(0,\infty)$ of upper type 1 and of strictly critical lower type $p_{\Phi}\in(n/(n+1),1]$. Let $L$ be a divergence form elliptic operator on $L^2 (\Omega)$ with the Neumann boundary condition and the heat semigroup generated by $L$ have the Gaussian property $(G_{\infty})$. In this paper, the authors introduce the Orlicz-Hardy space $H_{\Phi,\,L}(\Omega)$ via the nontangential maximal function associated with $\{e^{-t\sqrt{L}}\}_{t\ge0}$, and establish its equivalent characterization in terms of the Lusin area function associated with $\{e^{-t\sqrt{L}}\}_{t\ge0}$. The authors also introduce the "geometrical" Orlicz-Hardy space $H_{\Phi,\,z}(\Omega)$ via the classical Orlicz-Hardy space $H_{\Phi}(\mathbb{R}^n)$, and prove that the spaces $H_{\Phi,\,L}(\Omega)$ and $H_{\Phi,\,z}(\Omega)$ coincide with equivalent norms, from which, characterizations of $H_{\Phi,\,L}(\Omega)$, including the vertical and the nontangential maximal function characterizations associated with $\{e^{-tL}\}_{t\ge0}$, and the Lusin area function characterization associated with $\{e^{-tL}\}_{t\ge0}$, are deduced. All the above results generalize the well-known results of P. Auscher and E. Russ by taking $\Phi(t)\equiv t$ for all $t\in(0,\infty)$., Comment: This paper has been withdrawn by the authors

In this paper we provide a first realization of an idea of Jacques Tits from a 1956 paper, which first mentioned that there should be a field of characteristique une, which is now called $${\mathbb{F}_1}$$ , the field with one element. This idea was that every split reductive group scheme over $${\mathbb{Z}}$$ should descend to $${\mathbb{F}_1}$$ , and its group of $${\mathbb{F}_1}$$ -rational points should be its Weyl group. We connect the notion of a torified scheme to the notion of $${\mathbb{F}_1}$$ -schemes as introduced by Connes and Consani. This yields models of toric varieties, Schubert varieties and split reductive group schemes as $${\mathbb{F}_1}$$ -schemes. We endow the class of $${\mathbb{F}_1}$$ -schemes with two classes of morphisms, one leading to a satisfying notion of $${\mathbb{F}_1}$$ -rational points, the other leading to the notion of an algebraic group over $${\mathbb{F}_1}$$ such that every split reductive group is defined as an algebraic group over $${\mathbb{F}_1}$$ . Furthermore, we show that certain combinatorics that are expected from parabolic subgroups of GL(n) and Grassmann varieties are realized in this theory.

Let $G$ be a simple, undirected, finite graph with vertex set $V(G)$ and edge set $E(G)$. A $k$-dimensional box is a Cartesian product of closed intervals $[a_1,b_1]\times [a_2,b_2]\times\cdots\times [a_k,b_k]$. The boxicity of $G$, box$(G)$, is the minimum integer $k$ such that $G$ can be represented as the intersection graph of $k$-dimensional boxes; i.e., each vertex is mapped to a $k$-dimensional box and two vertices are adjacent in $G$ if and only if their corresponding boxes intersect. Let $\mathcal{P}=(S,P)$ be a poset, where $S$ is the ground set and $P$ is a reflexive, antisymmetric and transitive binary relation on $S$. The dimension of $\mathcal{P}$, $\dim(\mathcal{P})$, is the minimum integer $t$ such that $P$ can be expressed as the intersection of $t$ total orders. Let $G_{\mathcal{P}}$ be the underlying comparability graph of $\mathcal{P}$; i.e., $S$ is the vertex set and two vertices are adjacent if and only if they are comparable in $\mathcal{P}$. It is a well-known fact that posets with the same underlying comparability graph have the same dimension. The first result of this paper links the dimension of a poset to the boxicity of its underlying comparability graph. In particular, we show that for any poset $\mathcal{P}$, box$(G_{\mathcal{P}})/(\chi(G_{\mathcal{P}})-1) \le \dim(\mathcal{P})\le 2\mbox{box}(G_{\mathcal{P}})$, where $\chi(G_{\mathcal{P}})$ is the chromatic number of $G_{\mathcal{P}}$ and $\chi(G_{\mathcal{P}})\ne1$. It immediately follows that if $\mathcal{P}$ is a height-$2$ poset, then box$(G_{\mathcal{P}})\le \dim(\mathcal{P})\le 2\mbox{box}(G_{\mathcal{P}})$ since the underlying comparability graph of a height-$2$ poset is a bipartite graph. The second result of the paper relates the boxicity of a graph $G$ with a natural partial order associated with the extended double cover of $G$, denoted as $G_c$: Note that $G_c$ is a bipartite graph with partite sets $A$ and $B$ which are copies of $V(G)$ such that, corresponding to every $u\in V(G)$, there are two vertices $u_A\in A$ and $u_B\in B$ and $\{u_A,v_B\}$ is an edge in $G_c$ if and only if either $u=v$ or $u$ is adjacent to $v$ in $G$. Let $\mathcal{P}_c$ be the natural height-$2$ poset associated with $G_c$ by making $A$ the set of minimal elements and $B$ the set of maximal elements. We show that $\frac{\mbox{box}(G)}{2} \le \dim(\mathcal{P}_c) \le 2\mbox{box}(G)+4$. These results have some immediate and significant consequences. The upper bound $\dim(\mathcal{P})\le 2\mbox{box}(G_\mathcal{P})$ allows us to derive hitherto unknown upper bounds for poset dimension such as $\dim(\mathcal{P})\le 2\mbox{ tree width }(G_{\mathcal{P}})+4$, since boxicity of any graph is known to be at most its tree width $+\; 2$. In the other direction, using the already known bounds for partial order dimension we get the following: (1) The boxicity of any graph with maximum degree $\Delta$ is $O(\Delta\log^2\Delta)$, which is an improvement over the best-known upper bound of $\Delta^2+2$. (2) There exist graphs with boxicity $\Omega(\Delta\log\Delta)$. This disproves a conjecture that the boxicity of a graph is $O(\Delta)$. (3) There exists no polynomial-time algorithm to approximate the boxicity of a bipartite graph on $n$ vertices with a factor of $O(n^{0.5-\epsilon})$ for any $\epsilon>0$ unless $NP=ZPP$.

We say that the polynomial sequence $(Q^{(\lambda)}_n)$ is a semiclassical Sobolev polynomial sequence when it is orthogonal with respect to the inner product $$ _S= +\lambda , $$ where ${\bf u}$ is a semiclassical linear functional, ${\mathscr D}$ is the differential, the difference or the $q$--difference operator, and $\lambda$ is a positive constant. In this paper we get algebraic and differential/difference properties for such polynomials as well as algebraic relations between them and the polynomial sequence orthogonal with respect to the semiclassical functional $\bf u$. The main goal of this article is to give a general approach to the study of the polynomials orthogonal with respect to the above nonstandard inner product regardless of the type of operator ${\mathscr D}$ considered. Finally, we illustrate our results by applying them to some known families of Sobolev orthogonal polynomials as well as to some new ones introduced in this paper for the first time., Comment: 23 pages, special issue dedicated to Professor Guillermo Lopez lagomasino on the occasion of his 60th birthday, accepted in Journal of Approximation Theory

In this paper we classify, up to analytic isomorphism, the family of almost stretched Artinian complete intersection A=R/I with a given Hilbert function, in the case R is a power series ring with an arbitrary number of variables., 20 pages. This paper generalizes a previous version where the result was proven for a power series ring in two variables

This is a paper in mathematics, specifically in set theory. On the example of the measure recognition problem (MRP), the paper highlights the phenomenon of the utility of a multidisciplinary mathematical approach to a single mathematical problem, in particular, the value of a set-theoretic analysis. MRP asks if for a given Boolean algebra, , and a property, Φ , of measures, one can recognize by purely combinatorial means if supports a strictly positive measure with property Φ . The most famous instance of this problem is MRP (countable additivity), and in the first part of the paper, we survey the known results on this and some other problems. We show how these results naturally lead to asking about two other specific instances of the problem MRP, namely MRP (non-atomic) and MRP (separable). Then, we show how our recent work gives an easy solution to the former of these problems and some partial information about the latter. The long-term goal of this line of research is to obtain a structure theory of Boolean algebras that support a finitely additive strictly positive measure, along the lines of Maharam theorem, which gives such a structure theorem for measure algebras.

We develop the theory of multiresolutions in the context of Hausdorff measure of fractional dimension between 0 and 1. While our fractal wavelet theory has points of similarity that it shares with the standard case of Lebesgue measure on the line, there are also sharp contrasts. These are stated in our main result, a dichotomy theorem. The first section is the case of the middle-third Cantor set. This is followed by a review of the essentials on Hausdorff measure. The remaining sections of the paper cover multiresolutions in the general context of affine iterated function systems., We have followed all the suggestions and corrections made by the referee. In the introduction, we added an overview of the three themes in the paper, and added some motivation. This was motivated by the comments and suggestions of the referee. We also rearranged the material as suggested by the referee. This makes the three themes in our paper more clear, we believe

A paradigm that was successfully applied in the study of both pure and algorithmic problems in graph theory can be colloquially summarized as stating that any graph is close to being the disjoint union of expanders. Our goal in this paper is to show that in several of the instantiations of the above approach, the quantitative bounds that were obtained are essentially best possible. Three examples of our results are the following: A classical result of Lipton, Rose and Tarjan from 1979 states that if F is a hereditary family of graphs and every graph in F has a vertex separator of size n/(log⁡n)1+o(1), then every graph in F has O(n) edges. We construct a hereditary family of graphs with vertex separators of size n/(log⁡n)1−o(1) such that not all graphs in the family have O(n) edges. Trevisan and Arora-Barak-Steurer have recently shown that given a graph G, one can remove only 1% of its edges to obtain a graph in which each connected component has good expansion properties. We show that in both of these decomposition results, the expansion properties they guarantee are essentially best possible, even when one is allowed to remove 99% of G's edges. Sudakov and the second author have recently shown that every graph with average degree d contains an n-vertex subgraph with average degree at least (1−o(1))d and vertex expansion 1/(log⁡n)1+o(1). We show that one cannot guarantee a better vertex expansion even if allowing the average degree to be O(1). The above results are obtained as corollaries of a new family of graphs which we construct in this paper. These graphs have a super-linear number of edges and nearly logarithmic girth, yet each of their subgraphs has (optimally) poor expansion properties.

Let (\Pi,\Sigma) be a Coxeter system. An ordered list of elements in \Sigma and an element in \Pi determine a {\em subword complex}, as introduced in our paper on Gr\"obner geometry of Schubert polynomials (math.AG/0110058). Subword complexes are demonstrated here to be homeomorphic to balls or spheres, and their Hilbert series are shown to reflect combinatorial properties of reduced expressions in Coxeter groups. Two formulae for double Grothendieck polynomials, one of which is due to Fomin and Kirillov, are recovered in the context of simplicial topology for subword complexes. Some open questions related to subword complexes are presented., Comment: 14 pages. Final version, to appear in Advances in Mathematics. This paper was split off from math.AG/0110058v2, whose version 3 is now shorter

We prove that the symmetric convexified Tsirelson space is of weak cotype 2 but not of cotype 2. Introduction Weak type 2 and weak cotype 2 spaces were originally introduced and investigated by V.D. Milman and G. Pisier in [10] and weak Hilbert spaces by Pisier in [13]. A further detailed investigation can be found in Pisier’s book [14]. The first example of a weak Hilbert space which is not isomorphic to a Hilbert space is the 2-convexified Tsirelson space (called the convexified Tsirelson space in this paper). This follows from the results of W.B. Johnson in [4]. For a detailed study of the original Tsirelson space we refer to [3]. Let X be a Banach space with a symmetric basis. It was proved in [14] that if X is a weak Hilbert space, then it is isomorphic to a Hilbert space and this has lead to the belief that if X is just of weak cotype 2, then it is of cotype 2. However, this turns out not necessarily to be the case. The main result of this paper states that the symmetric convexified Tsirelson space is of weak cotype 2 but not of cotype 2. We now wish to discuss the arrangement of this paper in greater detail. In Section 1 we give some basic facts on properties related to weak type 2 and weak cotype 2 while Section 2 is devoted to a review of some results on the convexified Tsirelson space which we need for our main result. Most of these results are stated without proofs since they can be proved in a similar manner as the corresponding results for the original Tsirelson space. In Section 3 we make the construction of the symmetric convexified Tsirelson space, investigate its basic properties and prove our main result stated above. 02000 Mathematics Subject Classification: 46B03, 46B07. ∗Supported by NSF grants DMS 9706108 and 0102686. †Supported by the Danish Natural Science Research Council, grant 9801867.

Generalized Zeckendorf decompositions are expansions of integers as sums of elements of solutions to recurrence relations. The simplest cases are base-$b$ expansions, and the standard Zeckendorf decomposition uses the Fibonacci sequence. The expansions are finite sequences of nonnegative integer coefficients (satisfying certain technical conditions to guarantee uniqueness of the decomposition) and which can be viewed as analogs of sequences of variable-length words made from some fixed alphabet. In this paper we present a new approach and construction for uniform measures on expansions, identifying them as the distribution of a Markov chain conditioned not to hit a set. This gives a unified approach that allows us to easily recover results on the expansions from analogous results for Markov chains, and in this paper we focus on laws of large numbers, central limit theorems for sums of digits, and statements on gaps (zeros) in expansions. We expect the approach to prove useful in other similar contexts., Comment: Version 1.0, 25 pages. Keywords: Zeckendorf decompositions, positive linear recurrence relations, distribution of gaps, longest gap, Markov processes

A function f:R -> R is approximately continuous iff it is continuous in the density topology, i.e., for any ordinary open set U the set E=f^{-1}(U) is measurable and has Lebesgue density one at each of its points. Denjoy proved that approximately continuous functions are Baire 1., i.e., pointwise For any f:R^2 -> R define f_x(y) = f^y(x) = f(x,y). A function f:R^2 -> R is separately continuous if f_x and f^y are continuous for every x,y in R. Lebesgue in his first paper proved that any separately continuous function is Baire 1. Sierpinski showed that there exists a nonmeasurable f:R^2 -> R which is separately Baire 1. In this paper we prove: Thm 1. Let f:R^2 -> R be such that f_x is approximately continuous and f^y is Baire 1 for every x,y in R. Then f is Baire 2. Thm 2. Suppose there exists a real-valued measurable cardinal. Then for any function f:R^2 -> R and countable ordinal i, if f_x is approximately continuous and f^y is Baire i for every x,y in R, then f is Baire i+1 as a function of two variables. Thm 3. (i) Suppose that R can be covered by omega_1 closed null sets. Then there exists a nonmeasurable function f:R^2 -> R such that f_x is approximately continuous and f^y is Baire 2 for every x,y in R. (ii) Suppose that R can be covered by omega_1 null sets. Then there exists a nonmeasurable function f:R^2 -> R such that f_x is approximately continuous and f^y is Baire 3 for every x,y in R. Thm 4. In the random real model for any function f:R^2 -> R if f_x is approximately continuous and f^y is measurable for every x,y in R, then f is measurable as a function of two variables.

We prove that if the system of integer translates of a square integrable function is $\ell^2$-linear independent then its periodization function is strictly positive almost everywhere. Indeed we show that the above inference is true for any square integrable function since the following statement on Fourier analysis is true: For any (Lebesgue) measurable subset A of [0,1], with positive measure, there exists a non trivial square summable function, with support in A, whose partial sums of Fourier series are uniformly bounded., Comment: This paper has been withdrawn by the author due to a crucial error in the main result of the first version. The abstract has been changed accordingly. A new, corrected version of this paper has been published elsewhere

In an earlier paper we introduced a notion of Markov automaton, together with parallel operations which permit the compositional description of Markov processes. We illustrated by showing how to describe a system of n dining philosophers, and we observed that Perron-Frobenius theory yields a proof that the probability of reaching deadlock tends to one as the number of steps goes to infinity. In this paper we add sequential operations to the algebra (and the necessary structure to support them). The extra operations permit the description of hierarchical systems, and ones with evolving geometry.

In this paper, we compute the regularity and Hilbert series of symbolic powers of the cover ideal of a graph $G$ when $G$ is either a crown graph or a complete multipartite graph. We also compute the multiplicity of symbolic powers of cover ideals in terms of the number of edges., 14 pages. Suggestions and comments are welcome

International audience; In the context of Mather's theory of Lagrangian systems, we study the decomposition in chain-transitive classes of the Mather invariant sets. As an application, we prove, under appropriate hypotheses, the semi-continuity of the so-called Aubry set as a function of the Lagrangian. In the study of Lagrangian systems, John Mather introduced several invariant sets composed of globally minimizing extremals. He developed methods to construct several orbits undergoing interesting behaviors in phase space under some assumptions on these invariant sets, see [14]. In order to pursue this theory and to apply it on examples, it is necessary to have tools to describe precisely the invariant sets. At least two points of view can be adopted. One can study the invariant set from a purely topological point of view in the style of Conley as compact metric spaces with flows, and study their transitive components. One can also study these set from the point of view of action minimization, and decompose them in invariant subsets that have been called static classes. These points of view are very closely related, but each of them has specific features. For example, understanding the decomposition in static classes is necessary for the variational construction of interesting orbits, while the topological decomposition behaves well under perturbations. Our goal in the present paper is to explicit the links between these two decompositions. We explain that the topological decomposition is finer than the variational one, and that they coincide for most (but not all) systems. As an application, we prove a result of semi-continuity of the so-called Aubry set as a function of the Lagrangian, under certain non-degeneracy hypotheses. The semi-continuity of the Aubry set is a subtle problem, which has remained open for several years, until John Mather gave a counter example, see §18 in [16]. In the same paper, he also states without proof that semi-continuity holds under appropriate hypotheses. Our result extends the one of Mather. The methods we use are inspired from the recent work of Fathi, Figalli and Rifford, [9].

One of the main goals in this paper is to establish convolution sums of functions for the divisor sums $\widetilde{\sigma}_s(n)=\sum_{d|n}(-1)^{d-1}d^s$ and $\widehat{\sigma}_s(n)=\sum_{d|n}(-1)^{\frac{n}{d}-1}d^s$, for certain $s$, which were first defined by Glaisher. We first introduce three functions $\mathcal{P}(q)$, $\mathcal{E}(q)$, and $\mathcal{Q}(q)$ related to $\widetilde{\sigma}(n)$, $\widehat{\sigma}(n)$, and $\widetilde{\sigma}_3(n)$, respectively, and then we evaluate them in terms of two parameters $x$ and $z$ in Ramanujan's theory of elliptic functions. Using these formulas, we derive some identities from which we can deduce convolution sum identities. We discuss some formulae for determining $r_s(n)$ and $\delta_s(n)$, $s=4,$ $8$, in terms of $\widetilde{\sigma}(n)$, $\widehat{\sigma}(n)$, and $\widetilde{\sigma}_3(n)$, where $r_s(n)$ denotes the number of representations of $n$ as a sum of $s$ squares and $\delta_s(n)$ denotes the number of representations of $n$ as a sum of $s$ triangular numbers. Finally, we find some partition congruences by using the notion of colored partitions., Comment: This is an old paper uploaded for archival purposes

Fomin and Kirillov initiated a line of research into the realization of the cohomology and K-theory of generalized flag varieties G/B as commutative subalgebras of certain noncommutative algebras. This approach has several advantages, which we discuss. This paper contains the most comprehensive result in a series of papers related to the mentioned line of research. More precisely, we give a model for the T -equivariant K-theory of a generalized flag variety KT (G/B) in terms of a certain braided Hopf algebra called the Nichols-Woronowicz algebra. Our model is based on the Chevalley-type multiplication formula for KT (G/B) due to the first author and Postnikov; this formula is stated using certain operators defined in terms of so-called alcove paths (and the corresponding affine Weyl group). Our model is derived using a type-independent and concise approach. Dedicated to Professor Kenji Ueno on the occasion of his sixtieth birthday

In foundational papers, Gross, Zagier, and Kohnen established two formulas for arithmetic intersection numbers of certain Heegner divisors on integral models of modular curves. In [GZ1], only one imaginary quadratic discriminant plays a role. In [GZ2] and [GKZ], two quadratic discriminants play a role. In this paper we generalize the two-discriminant formula from the modular curves X0(N) to certain Shimura curves defined over Q. Our intersection formula was stated in [Ro], but the proof was only outlined there. Independently, the general formula was given, in a weaker and less explicit form, in [Ke2]; there it was proved completely. This paper is thus a synthesis of parts of [Ro] and [Ke2]. The intersection multiplicities computed here were used in [Ku] to derive a relation between height pairings and special values of the derivatives of certain Eisenstein series. We note also that Zhang [Zh] has generalized all of [GZ1] from ground field Q to general totally real ground fields F , working with general Shimura curves. So we certainly expect that all of [GKZ] should generalize similarly. Our work here can be viewed as a step in this direction. We are happy to thank B. Gross and S. Kudla for their encouragement during the long period in which this work was done.

Let V be a finite dimensional vector space over a local field. Let us say that a complex function on V is elementary if it is a product of the additive character of a rational function Q on V and multiplicative characters of polynomials on V. In this paper we study when the Fourier transform of an elementary function is elementary. If Q has a nonzero Hessian, a necessary condition for this is that the Legendre transform Q_* of Q is rational. The basic example is a nondegenerate quadratic form. We study such functions Q, give examples, and find all of them such that both Q and Q_* are of the form f(x)/t, where f is a cubic form in many variables (the simplest case after quadratic forms). It turns out that this classification is closely related to Zak's classification of Severi varieties. The second half of the paper is devoted to finding and classifying elementary functions with elementary Fourier transforms when Q is a fixed function with rational Q_*. We consider the simplest case when Q is a monomial, and classify combinations of multiplicative characters that can arise. The answer (for real and complex fields) is given in terms of exact covering systems. We also describe examples related to prehomogeneous vector spaces. Finally, we consider examples over p-adic fields, and in particular give a local proof of an integral formula of D.K. that could previously be proved only by a global method., Comment: 31 pages, latex; the proof of Theorem 3.10 given in the previous version was found to be incomplete. We have added a remark about this and a reference to a paper by Chaput and Sabatino that gives a complete proof

The main goal of the present paper is two-fold. First we extend the theory of toroidal embeddings introduced by Kempf, Knudsen, Mumford and Saint-Donat to the class of toroidal varieties with stratifications (which is the main body of the paper). Second we give a proof of the following weak factorization theorem as an application and illustration of the theory: A birational map between complete nonsingular varieties over an algebraically closed field K of characteristic zero is a composite of blow ups and blow downs with smooth centers. Another proof of the weak factorization theorem appeared in a joint paper with Abramovich, Karu and Matsuki (math.AG/9904135) In that paper the theorem is stated and proven in general for proper algebraic and analytic spaces., Comment: 69 pages, a revised version with conceptual simplifications. The present version is self-contained and it does not rely on weak factorization theorem for toric varieties

It is known that every semigroup of normal completely positive maps $P = {P_t: t\geq 0}$ of $B(H)$, satisfying $P_t(1) = 1$ for every $t\geq 0$, has a minimal dilation to an E_0-semigroup acting on $B(K)$ for some Hilbert space K containing H. The minimal dilation of P is unique up to conjugacy. In a previous paper a numerical index was introduced for semigroups of completely positive maps and it was shown that the index of P agrees with the index of its minimal dilation to an E_0-semigroup. However, no examples were discussed, and no computations were made. In this paper we calculate the index of a unital completely positive semigroup whose generator is a bounded operator $ L: B(H)\to B(H) $ in terms of natrual structures associated with the generator. This includes all unital CP semigroups acting on matrix algebras. We also show that the minimal dilation of the semigroup $P={\exp{tL}: t\geq 0}$ to an \esg\ is is cocycle conjugate to a CAR/CCR flow., Comment: 31 pp. AMS-TeX 2.0

Let g be a symmetrisable Kac-Moody algebra, and U_h(g) the corresponding quantum group. We showed in arXiv:1610.09744 and arXiv:1610.09741 that the braided quasi-Coxeter structure on integrable, category O representations of U_h(g) which underlies the R-matrix actions arising from the Levi subalgebras of U_h(g) and the quantum Weyl group action of the generalised braid group B_g can be transferred to integrable, category O representations of g. We prove in this paper that, up to unique equivalence, there is a unique such structure on the latter category with prescribed restriction functors, R--matrices, and local monodromies. This extends, simplifies and strengthens a similar result of the second author valid when g is semisimple, and is used in arXiv:1512.03041 to describe the monodromy of the rational Casimir connection of g in terms of the quantum Weyl group operators of U_h(g). Our main tool is a refinement of Enriquez's universal algebras, which is adapted to the PROP describing a Lie bialgebra graded by the non-negative roots of g., Expanded Introduction and Sec. 5 to discuss convolution product (5.11), cosimplicial structure on basis elements (5.13) and module structure on coinvariants (5.15). Minor revisions in Sec. 7.1 (gradings), 7.4 (deformation DY modules), 9.7 (exposition), 15.7 (Drinfeld double) and 15.15 (rigidity for diagrammatic KM algebras). Final version, to appear in Adv. Math. 81 pages

Let $G$ be an $n$-vertex graph with $n\ge 3$. A classic result of Dirac from 1952 asserts that $G$ is hamiltonian if $\delta(G)\ge n/2$. Dirac's theorem is one of the most influential results in the study of hamiltonicity and by now there are many related known results\,(see, e.g., J. A. Bondy, Basic Graph Theory: Paths and Circuits, Chapter 1 in: {\it Handbook of Combinatorics Vol.1}). A {\it Halin graph} is a planar graph consisting of two edge-disjoint subgraphs: a spanning tree of at least 4 vertices and with no vertex of degree 2, and a cycle induced on the set of the leaves of the spanning tree. Halin graphs possess rich hamiltonicity properties such as being hamiltonian, hamiltonian connected, and almost pancyclic. As a continuous "generalization" of Dirac's theorem, in this paper, we show that there exists a positive integer $n_0$ such that any graph $G$ with $n\ge n_0$ vertices and $\delta(G)\ge (n+1)/2$ contains a spanning Halin subgraph. In particular, it contains a spanning Halin subgraph which is also pancyclic., Comment: 25 pages, 3 figures

A Boolean function $g$ is said to be an optimal predictor for another Boolean function $f$ , if it minimizes the probability that $f(X^{n})\neq g(Y^{n})$ among all functions, where $X^{n}$ is uniform over the Hamming cube and $Y^{n}$ is obtained from $X^{n}$ by independently flipping each coordinate with probability $\delta$ . This paper is about self-predicting functions, which are those that coincide with their optimal predictor.

Whitney proved in 1931 that 4-connected planar triangulations are Hamiltonian. Hakimi, Schmeichel, and Thomassen conjectured in 1979 that if $G$ is a 4-connected planar triangulation with $n$ vertices then $G$ contains at least $2(n-2)(n-4)$ Hamiltonian cycles, with equality if and only if $G$ is a double wheel. On the other hand, a recent result of Alahmadi, Aldred, and Thomassen states that there are exponentially many Hamiltonian cycles in 5-connected planar triangulations. In this paper, we consider 4-connected planar $n$-vertex triangulations $G$ that do not have too many separating 4-cycles or have minimum degree 5. We show that if $G$ has $O(n/{\log}_2 n)$ separating 4-cycles then $G$ has $\Omega(n^2)$ Hamiltonian cycles, and if $\delta(G)\ge 5$ then $G$ has $2^{\Omega(n^{1/4})}$ Hamiltonian cycles. Both results improve previous work. Moreover, the proofs involve a "double wheel" structure, providing further evidence to the above conjecture., Comment: 20 pages, 2 figures

For any positive integer $k$, define $f(k)$ (respectively, $f_3(k)$) to be the minimal integer $\ge k$ such that every 3-connected planar graph $G$ (respectively, 3-connected cubic planar graph $G$) of circumference $\ge k$ has a cycle whose length is in the interval $[k, f(k)]$ (respectively, $[k, f_3(k)]$). Merker showed that $f_3(k) \le 2k + 9$ for any $k \ge 2$, and $f_3(k) \ge 2k + 2$ for any even $k \ge 4$. He conjectured that $f_3(k) \le 2k + 2$ for any $k \ge 2$. This conjecture was disproved by Zamfirescu, who gave an infinite family of counterexamples for every even $k \ge 6$ whose graphs have no cycle length in $[k, 2k + 2]$, i.e. $f_3(k) \ge 2k + 3$ for any even $k \ge 6$. However, the exact value of $f_3(k)$ was only known for $k \le 4$, and it was left open to determine $f_3(k)$ for $k \ge 5$. In this paper we improve Merker's upper bound, and give the exact value of $f_3(k)$ for every $k \ge 5$. We show that $f_3(5) = 10$, $f_3(7) = 15$, $f_3(9) = 20$, and $f_3(k) = 2k + 3$ for any $k = 6, 8$ or $\ge 10$. For general 3-connected planar graphs, Merker conjectured that there exists some positive integer $c$ such that $f(k) \le 2k + c$ for any positive integer $k$. We give a complete positive answer to this conjecture. We prove that $f(k) = 5$ for any $k \le 3$, $f(4) = 10$, and $f(k) = 2k + 3$ for any $k \ge 5$.

In this paper we study the weak Roman domination number and the secure domination number of a graph. In particular, we obtain general bounds on these two parameters and, as a consequence of the study, we derive new inequalities of Nordhaus-Gaddum type involving secure domination and weak Roman domination. Furthermore, the particular case of Cartesian product graphs is considered.

A classical P��lya urn scheme is a Markov process whose evolution is encoded by a replacement matrix $(R_{i,j})_{1\leq i,j\leq d}$. At every discrete time-step, we draw a ball uniformly at random, denote its colour $c$, and replace it in the urn together with $R_{c,j}$ balls of colour $j$ (for all $1\leq j\leq d$). We are interested in multi-drawing P��lya urns, where the replacement rule depends on the random drawing of a set of $m$ balls from the urn (with or without replacement). This generalisation has already been studied in the literature, in particular by Kuba & Mahmoud (ArXiv:1503.09069 and 1509.09053), where second order asymptotic results are proved for $2$-colour urns under the balanced and the affinity assumptions. The main idea of this work is to apply stochastic approximation methods to this problem, which enables us to remove the affinity hypothesis of Kuba & Mahmoud and generalise the result to more-than-two-colour urns. We also give some partial results in the two-colour non-balanced case., This new arxiv version (v6) corrects a mistake that we discovered in the previous versions of this paper (v1-5). The mistake was in Theorem 1$(a)$ and in the last sentence of Theorem 4. In this new version, Theorem 1$(a)$ has been corrected, and Theorem 4 has been deleted

Let $G$ be a finite connected graph, and let $\rho$ be the spectral radius of its universal cover. For example, if $G$ is $k$-regular then $\rho=2\sqrt{k-1}$. We show that for every $r$, there is an $r$-covering (a.k.a. an $r$-lift) of $G$ where all the new eigenvalues are bounded from above by $\rho$. It follows that a bipartite Ramanujan graph has a Ramanujan $r$-covering for every $r$. This generalizes the $r=2$ case due to Marcus, Spielman and Srivastava (2013). Every $r$-covering of $G$ corresponds to a labeling of the edges of $G$ by elements of the symmetric group $S_{r}$. We generalize this notion to labeling the edges by elements of various groups and present a broader scenario where Ramanujan coverings are guaranteed to exist. In particular, this shows the existence of richer families of bipartite Ramanujan graphs than was known before. Inspired by Marcus-Spielman-Srivastava, a crucial component of our proof is the existence of interlacing families of polynomials for complex reflection groups. The core argument of this component is taken from a recent paper of them (2015). Another important ingredient of our proof is a new generalization of the matching polynomial of a graph. We define the $r$-th matching polynomial of $G$ to be the average matching polynomial of all $r$-coverings of $G$. We show this polynomial shares many properties with the original matching polynomial. For example, it is real rooted with all its roots inside $\left[-\rho,\rho\right]$., Comment: 38 pages, 4 figures, journal version (minor changes from previous arXiv version). Shortened version appeared in STOC 2016

Dvo\v{r}\'ak, Kr\'al' and Thomas gave a description of the structure of triangle-free graphs on surfaces with respect to 3-coloring. Their description however contains two substructures (both related to graphs embedded in plane with two precolored cycles) whose coloring properties are not entirely determined. In this paper, we fill these gaps., Comment: 15 pages, 1 figure; corrections from the review process. arXiv admin note: text overlap with arXiv:1509.01013

In this paper we find the optimal error bound (smallest possible estimate, independent of the starting point) for the linear convergence rate of the simultaneous projection method applied to closed linear subspaces in a real Hilbert space. We achieve this by computing the norm of an error operator which we also express in terms of the Friedrichs number. We compare our estimate with the optimal one provided for the alternating projection method by Kayalar and Weinert (1988). Moreover, we relate our result to the alternating projection formalization of Pierra (1984) in a product space. Finally, we adjust our results to closed affine subspaces and put them in context with recent dichotomy theorems., Accepted for publication in the Journal of Approximation Theory

Macaulay's Inverse System gives an effective method to construct Artinian Gorenstein k-algebras. To date a general structure for Gorenstein k-algebras of any dimension (and codimension) is not understood. In this paper we extend Macaulay's correspondence characterizing the submodules of the divided power ring in one-to-one correspondence with Gorenstein d-dimensional k-algebras. We discuss effective methods for constructing Gorenstein graded rings. Several examples illustrating our results are given., 19 pages, to appear in Advances in Mathematics

Let $X$ be a $d\times d$ symmetric random matrix with independent but non-identically distributed Gaussian entries. It has been conjectured by Lata��{a} that the spectral norm of $X$ is always of the same order as the largest Euclidean norm of its rows. A positive resolution of this conjecture would provide a sharp understanding of the probabilistic mechanisms that control the spectral norm of inhomogeneous Gaussian random matrices. This paper establishes the conjecture up to a dimensional factor of order $\sqrt{\log\log d}$. Moreover, dimension-free bounds are developed that are optimal to leading order and that establish the conjecture in special cases. The proofs of these results shed significant light on the geometry of the underlying Gaussian processes., 18 pages, 1 figure; final version, to appear in Trans. Amer. Math. Soc

In this paper, we proved that for generic Tonelli Lagrangian, there always exists a residual set $\mathcal{G}\subset H^1(M,\mathbb{R})$ such that \[ \widetilde{\mathcal{M}}(c)=\widetilde{\mathcal{A}}(c)=\widetilde{\mathcal{N}}(c),\quad \forall c\in\mathcal{G} \] with $\widetilde{\mathcal{M}}(c)$ supports on a uniquely ergodic measure., Comment: 8 pages, 1 figure

Let $\unicode[STIX]{x1D6FD}\in (1,2)$ be a real number. For a function $\unicode[STIX]{x1D6F9}:\mathbb{N}\rightarrow \mathbb{R}_{\geq 0}$, define $W_{\unicode[STIX]{x1D6FD}}(\unicode[STIX]{x1D6F9})$ to be the set of $x\in \mathbb{R}$ such that for infinitely many $n\in \mathbb{N},$ there exists a sequence $(\unicode[STIX]{x1D716}_{i})_{i=1}^{n}\in \{0,1\}^{n}$ satisfying $0\leq x-\sum _{i=1}^{n}(\unicode[STIX]{x1D716}_{i}/\unicode[STIX]{x1D6FD}^{i})\leq \unicode[STIX]{x1D6F9}(n)$. In Baker [Approximation properties of $\unicode[STIX]{x1D6FD}$-expansions. Acta Arith. 168 (2015), 269–287], the author conjectured that for Lebesgue almost every $\unicode[STIX]{x1D6FD}\in (1,2)$, the condition $\sum _{n=1}^{\infty }2^{n}\unicode[STIX]{x1D6F9}(n)=\infty$ implies that $W_{\unicode[STIX]{x1D6FD}}(\unicode[STIX]{x1D6F9})$ is of full Lebesgue measure within $[0,1/(\unicode[STIX]{x1D6FD}-1)]$. In this paper we make a significant step towards proving this conjecture. We prove that given a sequence of positive real numbers $(\unicode[STIX]{x1D714}_{n})_{n=1}^{\infty }$ satisfying $\lim _{n\rightarrow \infty }\unicode[STIX]{x1D714}_{n}=\infty$, for Lebesgue almost every $\unicode[STIX]{x1D6FD}\in (1.497,\ldots ,2)$, the set $W_{\unicode[STIX]{x1D6FD}}(\unicode[STIX]{x1D714}_{n}\cdot 2^{-n})$ is of full Lebesgue measure within $[0,1/(\unicode[STIX]{x1D6FD}-1)]$. We also study the case where $\sum _{n=1}^{\infty }2^{n}\unicode[STIX]{x1D6F9}(n) in which the set $W_{\unicode[STIX]{x1D6FD}}(\unicode[STIX]{x1D6F9})$ has Lebesgue measure zero. Applying the mass transference principle developed by Beresnevich and Velani in [A mass transference principle and the Duffin–Schaeffer conjecture for Hausdorff measures. Ann. of Math. (2) 164(3) (2006), 971–992], we obtain some results on the Hausdorff dimension and the Hausdorff measure of $W_{\unicode[STIX]{x1D6FD}}(\unicode[STIX]{x1D6F9})$.

This paper presents the Gaussian subordination framework to generate optimal one-sided approximations to multidimensional real-valued functions by functions of prescribed exponential type. Such extremal problems date back to the works of Beurling and Selberg and provide a variety of applications in analysis and analytic number theory. Here we majorize and minorize (on $\mathbb{R}^N$) the Gaussian ${\bf x} \mapsto e^{-\pi \lambda |{\bf x}|^2}$, where $\lambda >0$ is a free parameter, by functions with distributional Fourier transforms supported on Euclidean balls, optimizing weighted $L^1$-errors. By integrating the parameter $\lambda$ against suitable measures, we solve the analogous problem for a wide class of radial functions. Applications to inequalities and periodic analogues are discussed. The constructions presented here rely on the theory of de Branges spaces of entire functions and on new interpolations tools derived from the theory of Laplace transforms of Laguerre-P\'{o}lya functions., Comment: 31 pages. To appear in Amer. J. Math

This paper is devoted to a detailed study of nowhere-zero flows on signed eulerian graphs. We generalise the well-known fact about the existence of nowhere-zero $2$-flows in eulerian graphs by proving that every signed eulerian graph that admits an integer nowhere-zero flow has a nowhere-zero $4$-flow. We also characterise signed eulerian graphs with flow number $2$, $3$, and $4$, as well as those that do not have an integer nowhere-zero flow. Finally, we discuss the existence of nowhere-zero $A$-flows on signed eulerian graphs for an arbitrary abelian group~$A$., 17 pages, 1 figure

We develop the intersection theory at relative chain-cochain level, and apply it along with the use of Seifert disks for an oriented link to give a combinatorial algorithm to compute Massey's higher order linking numbers. It is subtle to compute higher-order linking numbers, and it has been a folklore to use the intersection theory in the process, which was first suggested by W. Massey. Massey introduced the higher-order linking as an application of his higher-order cohomology operations defined in terms of suitable cochains, and he calculated the third-order linking numbers for some 3-component links by shifting from cohomology and cup product to homology and intersection theory via duality theorems for manifolds. Later several works along this direction gave methods in various forms for computing the higher-order linking numbers, and others computed Milnor's invariants by using their well-known connection to Massey invariants; but a formal derivation of the general formulae in intersection theory which are used for the computation has not yet been given, and a concrete algorithm for the computation is hence lacking. In this paper we complete Massey's original approach by developing systematically the intersection theory at the relative chain-cochain level in the simplicial category, and use it to derive recursive combinatorial formulae for computing all higher-order linking numbers., LaTeX document, 27 pages, 12 labeled figures

We prove that the discrete harmonic function corresponding to smooth Dirichlet boundary conditions on orthodiagonal maps, that is, plane graphs having quadrilateral faces with orthogonal diagonals, converges to its continuous counterpart as the mesh size goes to 0. This provides a convergence statement for discrete holomorphic functions, similar to the one obtained by Chelkak and Smirnov for isoradial graphs. We observe that by the double circle packing theorem, any finite, simple, 3-connected planar map admits an orthodiagonal representation. Our result improves the work of Skopenkov and Werness by dropping all regularity assumptions required in their work and providing effective bounds. In particular, no bound on the vertex degrees is required. Thus, the result can be applied to models of random planar maps that with high probability admit orthodiagonal representation with mesh size tending to 0. In a companion paper, we show that this can be done for the discrete mating-of-trees random map model of Duplantier, Gwynne, Miller and Sheffield., Comment: 44 pages, 11 figures

We prove novel results on interpolation of Fredholm operators including an abstract factorization theorem. The main result of this paper provides sufficient conditions on the parameters $\theta \in (0,1)$ and $q\in \lbrack 1,\infty ]$ under which an operator $A$ is a Fredholm operator from the real interpolation space $(X_{0},X_{1})_{\theta ,q}$ to $(Y_{0},Y_{1})_{\theta ,q} $ for a given operator $A\colon (X_{0},X_{1})\rightarrow (Y_{0},Y_{1})$ between compatible pairs of Banach spaces such that its restrictions to the endpoint spaces are Fredholm operators. These conditions are expressed in terms of the corresponding indices generated by the $K$-functional of elements from the kernel of the operator $A$ in the interpolation sum $X_{0}+X_{1}$. If in addition $q\in \lbrack 1,\infty )$ and $A$ is invertible operator on endpoint spaces, then these conditions are also necessary. We apply these results to obtain and present an affirmative solution of the famous Lions-Magenes problem on the real interpolation of closed subspaces. We also discuss some applications to the spectral theory of operators as well as to perturbation of the Hardy operator by identity on weighted $L_{p}$-spaces.