fourier integral operators fractal sets and the regular value theorem we prove that if e r2d for d 2 is an ahlfors david regular product set of sufficiently large hausdorff dimension denoted by dimh e and is a sufficiently regular function then the upper minkowski dimension of the set w e l w tl 1 l m does not exceed dimh e m in line with the regular value theorem from the elementary differential geometry our arguments are based on the mapping properties of the underlying fourier integral operators and are intimately connected with the falconer distance conjecture in geometric measure theory we shall see that our results are in general sharp in the sense that if the hausdorff dimension is smaller than a certain threshold then the dimensional inequality fails in a quantifiable way the constructions used to demonstrate this are based on the distribution of lattice points on convex surfaces and have connections with combinatorial geometry

product matrix msr codes with bandwidth adaptive exact repair in the distributed storage systems dsss with k systematic nodes robustness against node failure is commonly provided by storing redundancy in a number of other nodes and performing repair mechanism to reproduce the content of the failed nodes efficiency is then achieved by minimizing the storage overhead and the amount of data transmission required for data reconstruction and repair provided by coding solutions such as regenerating codes common explicit regenerating code constructions enable efficient repair through accessing a predefined number d of arbitrary chosen available nodes namely helpers in practice however the state of the system dynamically changes based on the request load the link traffic and so on and the parameters which optimize system s performance vary accordingly it is then desirable to have coding schemes which are able to operate optimally under a range of different parameters simultaneously specifically adaptivity in the number of helper nodes for repair is of interest while robustness requires capability of performing repair with small number of helpers it is desirable to use as many helpers as available to reduce the transmission delay and total repair traffic in this paper we focus on the minimum storage regenerating msr codes where each of the n nodes in the network is supposed to store alpha information units and the source data of size k alpha could be recovered from any arbitrary set of k nodes we introduce a class of msr codes that realize the optimal repair bandwidth simultaneously with a set of different choices for the number of helpers namely d d 1 ldots d delta our coding scheme follows the product matrix pm framework introduced by rashmi et al and could be considered as a generalization of the pm msr code presented by rashmi et al such that any d i i 1 k 1 helpers can perform an optimal repair as a result the coding rate in our construction is limited by k n leq 1 2 however similar to the original design of pm msr codes our solution can realize practical values of the parameter alpha recently ye and barg have presented another explicit msr coding scheme which is capable of performing optimal repair for various number of helpers the solution presented by ye and barg works for any arbitrary set of parameters k and d and can achieve high coding rates but the required alpha for this code is exponentially large we show that the required value for alpha in the coding scheme presented in this paper is exponentially smaller when compared with the work of ye and barg for the same set of other parameters particularly for a dss with n nodes and k systematic nodes the required value for alpha is reduced from s n to sk where s mathrm l cm d 1 k 1 cdots d delta k 1 we also show the required field size in the presented coding scheme is equal to n

geometric incidence theorems via fourier analysis we show that every non trivial sobolev bound for generalized radon transforms which average functions over families of curves and surfaces yields an incidence theorem for suitably regular discrete sets of points and curves or surfaces in euclidean space this mechanism allows us to deduce geometric results not readily accessible by combinatorial methods

the triangle averaging operator abstract we examine the averaging operator t corresponding to the manifold in r 2 d of pairs of points u v satisfying u v u v 1 so that 0 u v is the set of vertices of an equilateral triangle we establish l p l q l r boundedness for t for 1 p 1 q 1 r in the convex hull of the set of points 0 0 0 1 0 1 0 1 1 1 p d 1 p d 2 p d where p d 5 d 3 d 2

from harmonic analysis to arithmetic combinatorics arithmetic combinatorics or additive combinatorics is a fast developing area of research combining elements of number theory combinatorics harmonic analysis and ergodic theory its arguably best known result and the one that brought it to global prominence is the proof by ben green and terence tao of the long standing conjecture that primes contain arbitrarily long arithmetic progressions there are many accounts and expositions of the green tao theorem including the articles by kra 119 and tao 182 in the bulletin the purpose of the present article is to survey a broader highly interconnected network of questions and results built over the decades and spanning several areas of mathematics of which the green tao theorem is a famous descendant an old geometric problem lies at the heart of key conjectures in harmonic analysis a major result in partial differential equations invokes combinatorial theorems on intersecting lines and circles an unexpected argument points harmonic analysts towards additive number theory with consequences that could have hardly been anticipated we will not try to give a comprehensive survey of harmonic analysis combinatorics or additive number theory we will not even be able to do full justice to our specific areas of focus instead referring the reader to the more complete expositions and surveys listed in section 7 our goal here is to emphasize the connections between these areas we will thus concentrate on relatively few problems chosen as much for their importance to their fields as for their links to each other the article is written from the point of view of an analyst who in the course of her work was gradually introduced to the questions discussed here and found them fascinating we hope that the reader will enjoy a taste of this experience

communication over mimo x channels interference alignment decomposition and performance analysis in a multiple antenna system with two transmitters and two receivers a scenario of data communication known as the x channel is studied in which each receiver receives data from both transmitters in this scenario it is assumed that each transmitter is unaware of the other transmitter s data noncooperative scenario this system can be considered as a combination of two broadcast channels from the transmitters points of view and two multiple access channels from the receivers points of view taking advantage of both perspectives two signaling schemes for such a scenario are developed in these schemes some linear filters are employed at the transmitters and at the receivers which decompose the system into either two noninterfering multiple antenna broadcast subchannels or two noninterfering multiple antenna multiple access subchannels the main objective in the design of the filters is to exploit the structure of the channel matrices to achieve the highest multiplexing gain mg it is shown that the proposed noncooperative signaling schemes outperform other known noncooperative schemes in terms of the achievable mg in particular it is shown that in some specific cases the achieved mg is the same as the mg of the system if full cooperation is provided either between the transmitters or between the receivers in the second part of the paper it is shown that by using mixed design schemes rather than decomposition schemes and taking the statistical properties of the interference terms into account the power offset of the system can be improved the power offset represents the horizontal shift in the curve of the sum rate versus the total power in decibels

covers of algebraic varieties iii the discriminant of a cover of degree 4 and the trigonal construction in c e a general structure theorem for gorenstein covers of degrree d has been proved such a result has been used for characterizing covers of degree 3 4 and 5 see c e and cs respectively of algebraic varieties more precisely in section 4 resp 3 of c e it is proved that the total space x of each gorenstein cover e x y of degree 4 esp 3 is obta ned inside a suit bl p ojective bundle 7r 11 y of rank 2 resp 1 as the b se locus of a pencil relativ conics resp

on the tradeoff region of secure exact repair regenerating codes we consider the n k d ell secure exact repair regenerating code problem which generalizes the n k d exact repair regenerating code problem with the additional constraint that the stored file needs to be kept information theoretically secure against an eavesdropper who can access the data transmitted to regenerate a total of ell different failed nodes for all known results on this problem the achievable tradeoff regions between the normalized storage capacity and repair bandwidth have a single corner point achieved by a scheme proposed by shah rashmi and kumar the srk point since the achievable tradeoff regions of the exact repair regenerating code problem without any secrecy constraints are known to have multiple corner points in general these existing results suggest a phase change like behavior i e enforcing a secrecy constraint ell geq 1 immediately reduces the tradeoff region to one with a single corner point in this paper we first show that when the secrecy parameter ell is sufficiently large the srk point is indeed the only corner point of the tradeoff region however when ell is small we show that the tradeoff region can in fact have multiple corner points in particular we establish a precise characterization of the tradeoff region for the 7 6 6 1 problem which has exactly two corner points thus a smooth transition instead of a phase change type of transition should be expected as the secrecy constraint is gradually strengthened

asymptotic interference alignment for optimal repair of mds codes in distributed storage the high repair bandwidth cost of n k maximum distance separable mds erasure codes has motivated a new class of codes that can reduce repair bandwidth over that of conventional mds codes in this paper we address n k d exact repair mds codes which allow for any single failed node to be repaired exactly with access to any arbitrary set of d survivor nodes we show the existence of exact repair mds codes that achieve minimum repair bandwidth matching the cut set lower bound for arbitrary admissible n k d i e k d n 1 moreover we extend our results to show the optimality of our codes for multiple node failure scenarios in which an arbitrary set of r n k failed nodes needs to repaired our approach is based on asymptotic interference alignment proposed by cadambe and jafar as a byproduct we also characterize the capacity of a class of multisource nonmulticast networks

explicit constructions of optimal access mds codes with nearly optimal sub packetization an n k l maximum distance separable mds array code of length n dimension k n r and sub packetization l is formed of l times n matrices over a finite field f with every column of the matrix stored on a separate node in the distributed storage system and viewed as a coordinate of the codeword repair of a failed node recovery of one erased column can be performed by accessing a set of d le n 1 surviving helper nodes the code is said to have the optimal access property if the amount of data accessed at each of the helper nodes meets a lower bound on this quantity for optimal access mds codes with d n 1 the sub packetization l satisfies the bound l ge r k 1 r in our previous work ieee trans inf theory vol 63 no 4 2017 for any n and r we presented an explicit construction of optimal access mds codes with sub packetization l r n 1 in this paper we take up the question of reducing the sub packetization value l to make it to approach the lower bound we construct an explicit family of optimal access codes with l r lceil n r rceil which differs from the optimal value by at most a factor of r 2 these codes can be constructed over any finite field f as long as f ge r lceil n r rceil and afford low complexity encoding and decoding procedures we also define a version of the repair problem that bridges the context of regenerating codes and codes with locality constraints lrc codes which we call group repair with optimal access in this variation we assume that the set of n sm nodes is partitioned into m repair groups of size s and require that the amount of accessed data for repair is the smallest possible whenever the d s k 1 helper nodes include all the other s 1 nodes from the same group as the failed node for this problem we construct a family of codes with the group optimal access property these codes can be constructed over any field f of size f ge n and also afford low complexity encoding and decoding procedures

constructions of high rate minimum storage regenerating codes over small fields a novel technique for construction of minimum storage regenerating msr codes is presented based on this technique three explicit constructions of msr codes are given the first two constructions provide access optimal msr codes with two and three parities respectively which attain the sub packetization bound for access optimal codes the third construction provides longer msr codes with three parities i e codes with larger number of systematic nodes this improvement is achieved at the expense of the access optimality and the field size in addition to a minimum storage in a node all three constructions allow the entire data to be recovered from a minimal number of storage nodes that is given storage ell in each node the entire stored data can be recovered from any 2 log 2 ell for two parity nodes and either 3 log 3 ell or 4 log 3 ell for three parities second in the first two constructions a helper node accesses the minimum number of its symbols for repair of a failed node access optimality the goal of this paper is to provide a construction of such optimal codes over the smallest possible finite fields the generator matrix of these codes is based on perfect matchings of complete graphs and hypergraphs and on a rational canonical form of matrices for two parities the field size is reduced by a factor of two for access optimal codes compared to previous constructions for three parities in the first construction a field size of at least 6 log 3 ell 1 or 3 log 3 ell 1 for fields with characteristic 2 is sufficient and in the second construction the field size is larger yet linear in log 3 ell both constructions with three parities provide a significant improvement over previous works due to either decreased field size or lower subpacketization

hilbert integrals singular integrals and radon transforms i on introduit deux nouvelles classes d operateurs on etudie leur relation et on montre comment on peut les appliquer a l etude des problemes aux valeurs limites

pliable fractional repetition codes for distributed storage systems design and analysis a distributed storage system dss is one of the most vital components of a cloud computing system used for storing and sharing big data among authorized users a typical dss consists of n storage nodes each with a storage capacity of alpha units of data such that the entire file stored on the dss can be recovered by accessing any k nodes the objective of this paper is to construct fractional repetition fr codes with applications in dsss fr codes are the key to constructing a class of distributed storage codes with exact repair by transfer i e upon failure a failed storage node is exactly regenerated through simple downloading a major drawback of existing fr codes is that they are insufficiently flexible to adapt adequately to system changes in dsss to address this problem this paper introduces a new type of fr codes called pliable fr codes in which both the per node storage alpha and repetition degree rho can easily and simultaneously be adjusted in addition this paper presents several constructions of pliable fr codes and provides a relatively comprehensive analysis of the constructed pliable fr codes which reveals the following versatile properties of the codes 1 the transposed codes of the constructed codes are also pliable fr codes 2 the constructed codes attain a tight upper bound on the file size with equality at least for 1 le k where k is the smallest value of k such that k alpha k 1 frac n 1 rho 1 3 the constructed codes also meet a singleton like bound on the minimum distance at least for 1 le k which demonstrates their optimality 4 the computational complexity necessary for determining the file size or the minimum distance of the constructed codes can be greatly reduced when it is hard to exactly determine them and 5 the constructed codes can be used as fractional repetition batch codes to provide load balancing in dsss for which the batch size i e the number of symbols that can be read in parallel can be exactly determined

on positivity of line bundles on enriques surfaces we study linear systems on enriques surfaces we prove rationality of seshadri constants of ample line bundles on enriques surfaces and provide lower bounds on these numbers we show the nonexistence of k very ample line bundles on enriques surfaces of degree 4k 4 for k 1 thus answering an old question of ballico and sommese introduction in this note we answer a question posed by ballico and sommese 1 see also terakawa 14 concerning the existence of k very ample line bundles of degree 4k 4 on enriques surfaces we show that such line bundles exist if and only if k 0 and that in fact the degree of a k very ample line bundle on an enriques surface is subject to a much more restrictive bound theorem 2 4 this has strong consequences for the local positivity of line bundles on enriques surfaces we address this questions in section 3 in particular we show that seshadri constants of ample line bundles on enriques surfaces are always rational 1 auxiliary material and notation the basic reference for this paper is the monograph 6 by cossec and dolgachev here we briefly recall properties of enriques surfaces needed for our considerations first of all an enriques surface is a surface y of kodaira dimension zero with no differential forms in particular ho ky 0 and the canonical bundle ky is a 2torsion element in pico y i e 2ky oy to alleviate notation it is convenient to use the following convention in the sequel we make frequent use of vanishing theorems and statements on adjoint line bundles since the canonical divisor ky is numerically trivial the adjoint line bundle ky l has the same numerical properties as l in particular if l is nef and big ample then so is ky l by the nakai moishezon criterion so in order to get vanishing statements for l we apply vanishing theorems to ky l and use the fact that 2ky is trivial in the sequel we do so without comment and hope to cause no confusion this way received by the editors june 1 2000 2000 mathematics subject classification primary 14j28 secondary 14c20 14e25

explicit constructions of high rate mds array codes with optimal repair bandwidth maximum distance separable mds codes are optimal error correcting codes in the sense that they provide the maximum failure tolerance for a given number of parity nodes suppose that an mds code with k information nodes and r n k parity nodes is used to encode data in a distributed storage system it is known that if h out of the n nodes are inaccessible and d surviving helper nodes are used to recover the lost data then we need to download at least h d h k fraction of the data stored in each of the helper nodes dimakis et al 2010 and cadambe et al 2013 if this lower bound is achieved for the repair of any h erased nodes from any d helper nodes we say that the mds code has the h d optimal repair property we study high rate mds array codes with the optimal repair property also known as minimum storage regenerating codes or msr codes explicit constructions of such codes in the literature are only available for the cases where there are at most three parity nodes and these existing constructions can only optimally repair a single node failure by accessing all the surviving nodes in this paper given any r and n we present two explicit constructions of mds array codes with the h d optimal repair property for all h le r and k le d le n h simultaneously codes in the first family can be constructed over any base field f as long as f ge sn where s text lcm 1 2 dots r the encoding decoding repair of failed nodes and update procedures of these codes all have low complexity codes in the second family have the optimal access property and can be constructed over any base field f as long as f ge n 1 moreover both code families have the optimal error resilience capability when repairing failed nodes we also construct several other related families of mds codes with the optimal repair property

degrees of freedom region of the mimo x channel we provide achievability as well as converse results for the degrees of freedom region of a multiple input multiple output mimo x channel i e a system with two transmitters two receivers each equipped with multiple antennas where independent messages need to be conveyed over fixed channels from each transmitter to each receiver the inner and outer bounds on the degrees of freedom region are tight whenever integer degrees of freedom are optimal for each message with m 1 antennas at each node we find that the total sum rate degrees of freedom are bounded above and below as 1 les eta x les 4 3 if m 1 and channel matrices are nondegenerate then the precise degrees of freedom eta x 4 3 m thus the mimo x channel has noninteger degrees of freedom when m is not a multiple of 3 simple zero forcing without dirty paper encoding or successive decoding suffices to achieve the 4 3 m degrees of freedom if the channels vary with time frequency then the channel with single antennas m 1 at all nodes has exactly 4 3 degrees of freedom the key idea for the achievability of the degrees of freedom is interference alignment i e signal spaces are aligned at receivers where they constitute interference while they are separable at receivers where they are desired we also explore the increase in degrees of freedom when some of the messages are made available to a transmitter or receiver in the manner of cognitive radio

hilbert integrals singular integrals and radon transforms ii etude des transformations de radon singulieres et des integrales de hilbert appliquees aux problemes aux valeurs limites probleme de neumann

degrees of freedom of rank deficient mimo interference channels we characterize the degrees of freedom dof of multiple input and multiple output mimo interference channels with rank deficient channel matrices for the two user rank deficient mimo interference channel we provide a tight outer bound to show that the previously known achievable dof in the symmetric case is optimal and generalize the result to fully asymmetric settings for the k user rank deficient interference channel we improve the previously known achievable dof and provide a tight outer bound to establish optimality in symmetric settings in particular we show that for the k user rank deficient interference channel when all nodes have m antennas all direct channels have rank d0 all cross channels are of rank d and the channels are otherwise generic the optimal dof value per user is min d0 m min m k 1 d 2 notably for interference channels the rank deficiency of direct channels does not help and the rank deficiency of cross channels does not hurt the main technical challenge is to account for the spatial dependences introduced by rank deficiencies in the interference alignment schemes that typically rely on the independence of channel coefficients

exact repair mds code construction using interference alignment the high repair cost of n k maximum distance separable mds erasure codes has recently motivated a new class of mds codes called repair mds codes that can significantly reduce repair bandwidth over conventional mds codes in this paper we describe n k d exact repair mds codes which allow for any failed node to be repaired exactly with access to d survivor nodes where k d n 1 we construct exact repair mds codes that are optimal in repair bandwidth for the cases of k n 1 2 and d 2k 11 b k 3 our codes are deterministic and require a finite field size of at most 2 n k our constructive codes are based on interference alignment techniques

on the picard group of enriques surfaces an enriques surface over an algebraically closed field k of characteristic 4 2 is a non singular projective surface s with h s d s h2 s cs 0 and 2ks 0 the unramitied double cover defined by ks is a k3 surface r a non singular projective surface with hi r d r 0 kr 0 illusie has shown i that the group of divisors modulo numerical equivalence is isomorphic to the enriques lattice u e 8 1 where u and e s 1 denote respectively the unique even unimodular lattices of index of inertia 1 1 and 0 8 the purpose of this note is to use this isomorphism to study the picard group of s we prove the existence of certain configurations of irreducible curves of arithmetic genus 0 or 1 and deduce from them the existence of certain projective models for s and r for example we prove the following results

maximal averages over surfaces letmf x supt 0 f t d x denote the maximal operator associated with surface measured on a smooth surfaces we prove that ifsis convex and has finite order contact with its tangent lines then m is bounded onlp rn p 2 if and only ifd x h 1 l1 loc s for all tangent planeshnot passing through the origin letm f x supt 0 f t d x be the maximal operator associated with a nonisotropic dilation tof surface measured we prove that m often behaves far better than m due to a rotational curvature in the time parametert

reye constructions for nodal enriques surfaces a classical reye congruence x is an enriques surface of rational equivalence class 3 7 in the grassmannian g 1 3 of lines of p 3 x is the locus of lines of p 3 which are included in two quadrics of w web of quadrics a generaiization to g 1 t is given 1 for each t 2 there exist enriques surfaces x of class t 3t 2 in g 1 t 2 the determinant of the dual of the universal bundle on x is o x 2e r k x with e isolated elliptic curve r 2 2 e r t 3 x parameterizes lines of p t which are included in a codimension 2 subsystem of w w linear system of quadrics of dimension 2 t the paper includes a description of the variety of trisecant lines to a smooth enriques surface of degree 10 p 5

on continuum incidence problems related to harmonic analysis abstract we consider averaging operators over curves and surfaces satisfying the rotational curvature condition of phong and stein using combinatorial arguments restricted weak type versions of the standard estimates are obtained under weaker assumptions on the regularity of these surfaces we also consider maximal functions over circles in the plane and obtain a simpler proof of wolff s endpoint result

unit distance problems we study some discrete and continuous variants of the following problem of erd h os given a finite subset p of bbb r 2 or bbb r 3 what is the maximum number of pairs p 1 p 2 with p 1 p 2 in p and p 1 p 2 1