eigenvalue assignment by minimal state feedback gain in lti multivariable systems in this article an improved method for eigenvalue assignment via state feedback in the linear time invariant multivariable systems is proposed this method is based on elementary similarity operations and involves mainly utilisation of vector companion forms and thus is very simple and easy to implement on a digital computer in addition to the controllable systems the proposed method can be applied for the stabilisable ones and also systems with linearly dependent inputs moreover two types of state feedback gain matrices can be achieved by this method 1 the numerical one which is unique and 2 the parametric one in which its parameters are determined in order to achieve a gain matrix with minimum frobenius norm the numerical examples are presented to demonstrate the advantages of the proposed method

branch and price column generation for solving huge integer programs we discuss formulations of integer programs with a huge number of variables and their solution by column generation methods i e implicit pricing of nonbasic variables to generate new columns or to prove lp optimality at a node of the branch and bound tree we present classes of models for which this approach decomposes the problem provides tighter lp relaxations and eliminates symmetry we then discuss computational issues and implementation of column generation branch and bound algorithms including special branching rules and efficient ways to solve the lp relaxation we also discuss the relationship with lagrangian duality

the derivative of the maximal function in this note we show that the local hardy littlewood maximal operator is bounded in the sobolev space thus the maximal function often has partial derivatives we also show that the maximal operator preserves the zero boundary values in sobolev s sense

the price of robustness a robust approach to solving linear optimization problems with uncertain data was proposed in the early 1970s and has recently been extensively studied and extended under this approach we are willing to accept a suboptimal solution for the nominal values of the data in order to ensure that the solution remains feasible and near optimal when the data changes a concern with such an approach is that it might be too conservative in this paper we propose an approach that attempts to make this trade off more attractive that is we investigate ways to decrease what we call the price of robustness in particular we flexibly adjust the level of conservatism of the robust solutions in terms of probabilistic bounds of constraint violations an attractive aspect of our method is that the new robust formulation is also a linear optimization problem thus we naturally extend our methods to discrete optimization problems in a tractable way we report numerical results for a portfolio optimization problem a knapsack problem and a problem from the net lib library

regularity of the centered fractional maximal function on radial functions abstract we study the regularity properties of the centered fractional maximal function m more precisely we prove that the map f m f is bounded and continuous from w 1 1 r d to l q r d in the endpoint case q d d if f is a radial function for d 1 the radiality assumption can be removed this corresponds to the counterparts of known results for the non centered fractional maximal function the main new idea consists in relating the centered and non centered fractional maximal function at the derivative level

convergent sequences of dense graphs i subgraph frequencies metric properties and testing we consider sequences of graphs gn and define various notions of convergence related to these sequences left convergence defined in terms of the densities of homomorphisms from small graphs into gn right convergence defined in terms of the densities of homomorphisms from gn into small graphs and convergence in a suitably defined metric in part i of this series we show that left convergence is equivalent to convergence in metric both for simple graphs gn and for graphs gn with nodeweights and edgeweights one of the main steps here is the introduction of a cut distance comparing graphs not necessarily of the same size we also show how these notions of convergence provide natural

endpoint sobolev and bv continuity for maximal operators abstract in this paper we investigate some questions related to the continuity of maximal operators in w 1 1 and bv spaces complementing some well known boundedness results letting m be the one dimensional uncentered hardy littlewood maximal operator we prove that the map f m f is continuous from w 1 1 r to l 1 r in the discrete setting we prove that m b v z b v z is also continuous for the one dimensional fractional hardy littlewood maximal operator we prove by means of counterexamples that the corresponding continuity statements do not hold both in the continuous and discrete settings and for the centered and uncentered versions

robust convex optimization we study convex optimization problems for which the data is not specified exactly and it is only known to belong to a given uncertainty set u yet the constraints must hold for all possible values of the data from u the ensuing optimization problem is called robust optimization in this paper we lay the foundation of robust convex optimization in the main part of the paper we show that if u is an ellipsoidal uncertainty set then for some of the most important generic convex optimization problems linear programming quadratically constrained programming semidefinite programming and others the corresponding robust convex program is either exactly or approximately a tractable problem which lends itself to efficientalgorithms such as polynomial time interior point methods

matching measure benjamini schramm convergence and the monomer dimer free energy we define the matching measure of a lattice l as the spectral measure of the tree of self avoiding walks in l we connect this invariant to the monomer dimer partition function of a sequence of finite graphs converging to l this allows us to express the monomer dimer free energy of l in terms of the matching measure exploiting an analytic advantage of the matching measure over the mayer series then leads to new rigorous bounds on the monomer dimer free energies of various euclidean lattices while our estimates use only the computational data given in previous papers they improve the known bounds significantly

canonical measures on metric graphs and a kazhdan s theorem we extend the notion of canonical measures to all possibly non compact metric graphs this will allow us to introduce a notion of hyperbolic measures on universal covers of metric graphs kazhdan s theorem for riemann surfaces describes the limiting behavior of canonical arakelov measures on finite covers in relation to the hyperbolic measure we will prove a generalized version of this theorem for metric graphs allowing any infinite galois cover to replace the universal cover we will show all such limiting measures satisfy a version of gauss bonnet formula which using the theory of von neumann dimensions can be interpreted as a trace formula in the special case where the infinite cover is the universal cover we will provide explicit methods to compute the corresponding limiting hyperbolic measure our ideas are motivated by non archimedean analytic and tropical geometry

robust counterparts of inequalities containing sums of maxima of linear functions abstract this paper addresses the robust counterparts of optimization problems containing sums of maxima of linear functions these problems include many practical problems e g problems with sums of absolute values and arise when taking the robust counterpart of a linear inequality that is affine in the decision variables affine in a parameter with box uncertainty and affine in a parameter with general uncertainty in the literature often the reformulation is used that is exact when there is no uncertainty however in robust optimization this reformulation gives an inferior solution and provides a pessimistic view we observe that in many papers this conservatism is not mentioned some papers have recognized this problem but existing solutions are either conservative or their performance for different uncertainty regions is not known a comparison between them is not available and they are restricted to specific problems we describe techniques for general problems and compare them with numerical examples in inventory management regression and brachytherapy based on these examples we give recommendations for reducing the conservatism

on the variation of maximal operators of convolution type abstract in this paper we study the regularity properties of two maximal operators of convolution type the heat flow maximal operator associated to the gauss kernel and the poisson maximal operator associated to the poisson kernel in dimension d 1 we prove that these maximal operators do not increase the l p variation of a function for any p 1 while in dimensions d 1 we obtain the corresponding results for the l 2 variation similar results are proved for the discrete versions of these operators

percolation on dense graph sequences in this paper we determine the percolation threshold for an arbitrary sequence of dense graphs g n let n be the largest eigenvalue of the adjacency matrix of g n and let g n p n be the random subgraph of g n obtained by keeping each edge independently with probability p n we show that the appearance of a giant component in g n p n has a sharp threshold at p n 1 n in fact we prove much more if g n converges to an irreducible limit then the density of the largest component of g n c n tends to the survival probability of a multi type branching process defined in terms of this limit here the notions of convergence and limit are those of borgs chayes lovasz s6s and vesztergombi in addition to using basic properties of convergence we make heavy use of the methods of bollobas janson and riordan who used multi type branching processes to study the emergence of a giant component in a very broad family of sparse inhomogeneous random graphs

convergent sequences of dense graphs ii multiway cuts and statistical physics we consider sequences of graphs gn and dene various notions of convergence related to these sequences including left convergence dened in terms of the densities of homomorphisms from small graphs into gn and right convergence dened in terms of the densities of homomorphisms from gn into small graphs we show that right convergence is equivalent to left convergence both for simple graphs gn and for graphs gn with nontrivial nodeweights and edgeweights other equivalent conditions for convergence are given in terms of fundamental notions from combinatorics such as maximum cuts and szemer edi partitions and fundamental notions from statistical physics like energies and free energies we thereby relate local and global properties of graph sequences quantitative forms of these results express the relationships among dierent measures of similarity of large graphs

boundedness and unboundedness results for some maximal operators on functions of bounded variation abstract we characterize the space bv i of functions of bounded variation on an arbitrary interval i r in terms of a uniform boundedness condition satisfied by the local uncentered maximal operator m r from bv i into the sobolev space w 1 1 i by restriction the corresponding characterization holds for w 1 1 i we also show that if u is open in r d d 1 then boundedness from bv u into w 1 1 u fails for the local directional maximal operator m t v the local strong maximal operator m t s and the iterated local directional maximal operator m t d m t 1 nevertheless if u satisfies a cone condition then m t s bv u l 1 u boundedly and the same happens with m t v m t d m t 1 and m r

interpolation between holder and lebesgue spaces with applications abstract classical interpolation inequality of the type u x c u y u z 1 is well known in the case when x y z are lebesgue spaces in this paper we show that this result may be extended by replacing norms y or x by suitable holder semi norm we shall even prove sharper version involving weak lorentz norm we apply this result to prove the gagliardo nirenberg inequality for a wider scale of parameters

sobolev regularity of polar fractional maximal functions abstract we study the sobolev regularity on the sphere s d of the uncentered fractional hardy littlewood maximal operator m at the endpoint p 1 when acting on polar data we first prove that if q d d 0 d and f is a polar w 1 1 s d function we have m f q d f 1 we then prove that the map f m f is continuous from w 1 1 s d to l q s d when restricted to polar data our methods allow us to give a new proof of the continuity of the map f m f from w rad 1 1 r d to l q r d moreover we prove that a conjectural local boundedness for the centered fractional hardy littlewood maximal operator m implies the continuity of the map f m f from w 1 1 to l q in the context of polar functions on s d and radial functions on r d

a combinatorial strongly polynomial algorithm for minimizing submodular functions this paper presents a combinatorial polynomial time algorithm for minimizing submodular functions answering an open question posed in 1981 by grotschel lovasz and schrijver the algorithm employs a scaling scheme that uses a flow in the complete directed graph on the underlying set with each arc capacity equal to the scaled parameter the resulting algorithm runs in time bounded by a polynomial in the size of the underlying set and the length of the largest absolute function value the paper also presents a strongly polynomial version in which the number of steps is bounded by a polynomial in the size of the underlying set independent of the function values

mathematica implementation of output feedback pole assignment for uncertain systems via symbolic algebra this paper presents the application of symbolic algebra techniques to the mathematica implementation of a set of output feedback pole assignment algorithms for systems characterized by parametric uncertainty for multivariable systems there may be more than one feedback matrix solution leading to the same closed loop poles based on the same algorithm used thus over parameterized solutions are sought by generalizing the existing algorithms with extra degrees of freedom retained in the symbolic variables the general parametric form of output feedback compensators is developed in terms of the uncertain parameters and symbols representing the extra degrees of freedom the implementation of three output feedback pole assignment techniques is presented with the theory briefly introduced and examples illustrating the effectiveness of the algorithms described

functions of bounded variation the derivative of the one dimensional maximal function and applications to inequalities we prove that if f i r r is of bounded variation then the uncentered maximal function mf is absolutely continuous and its derivative satisfies the sharp inequality dmf l 1 i df i this allows us to obtain under less regularity versions of classical inequalities involving derivatives

large deviations of subgraph counts for sparse erdős renyi graphs abstract for any fixed simple graph h v e and any fixed u 0 we establish the leading order of the exponential rate function for the probability that the number of copies of h in the erd s renyi graph g n p exceeds its expectation by a factor 1 u assuming n h p 1 with h 1 2 where 1 is the maximum degree of h this improves on a previous result of chatterjee and the second author who obtained h c e for a constant c 0 moreover for the case of cycle counts we can take as large as 1 2 we additionally obtain the sharp upper tail for schatten norms of the adjacency matrix as well as the sharp lower tail for counts of graphs for which sidorenko s conjecture holds as a key step we establish quantitative versions of szemeredi s regularity lemma and the counting lemma suitable for the analysis of random graphs in the large deviations regime

local convergence for permutations and local limits for uniform rho avoiding permutations with rho 3 we set up a new notion of local convergence for permutations and we prove a characterization in terms of proportions of consecutive pattern occurrences we also characterize random limiting objects for this new topology introducing a notion of shift invariant property corresponding to the notion of unimodularity for random graphs we then study two models in the framework of random pattern avoiding permutations we compute the local limits of uniform rho avoiding permutations for rho 3 when the size of the permutations tends to infinity the core part of the argument is the description of the asymptotics of the number of consecutive occurrences of any given pattern for this result we use bijections between rho avoiding permutations and rooted ordered trees local limit results for galton watson trees the second moment method and singularity analysis

on the continuous and discontinuous maximal operators abstract in the first part of this paper we study the regularity properties of a wide class of maximal operators these results are used to show that the spherical maximal operator is continuous w 1 p r n w 1 p r n when p n n 1 other given applications include fractional maximal operators and maximal singular integrals on the other hand we show that the restricted hardy littlewood maximal operator m where the supremum is taken over the cubes with radii greater than 0 is bounded from l p r n to w 1 p r n but discontinuous

a parametric solution to the pole assignment problem using dynamic output feedback a technique is presented for pole placement of linear time invariant systems using dynamic feedback a previously developed method for partial pole assignment using constant feedback is generalized to the dynamic output feedback case subject to a mild assumption on the number of complex conjugate poles to be assigned it is almost always possible to arbitrarily assign all the closed loop system poles using a compensator of order n spl phi max m l using this new method here n m and l are the order of the system and the number of inputs and outputs respectively and spl phi spl delta max m l max m l 2 max m l min m l where x denotes the nearest integer lower than or equal to x i e floor x and x denotes the nearest integer greater than or equal to x i e ceiling x an equivalent result is that using a compensator of order q it is almost always possible to arbitrarily assign min n q max m l 1 q spl phi closed loop system poles only the normal procedures of linear algebra are required to implement the technique note that spl phi spl ges l m 1 and therefore the result is stronger than previous exact pole assignment results since it does not involve iteration or any other numerical techniques it is possible to implement the method symbolically and therefore to obtain general parametric solutions to the pole assignment problem the freedom in this design approach can also often be used to guarantee the internal stability and or robustness of the resulting closed loop system

semiconductor lot allocation using robust optimization abstract in this work the problem of allocating a set of production lots to satisfy customer orders is considered this research is of relevance to lot to order matching problems in semiconductor supply chain settings we consider that lot splitting is not allowed during the allocation process due to standard practices furthermore lot sizes are regarded as uncertain planning data when making the allocation decisions due to potential yield loss in order to minimize the total penalties of demand un fulfillment and over fulfillment a robust mixed integer optimization approach is adopted to model is proposed the problem of allocating a set of work in process lots to customer orders where lot sizes are modeled using ellipsoidal uncertainty sets to solve the optimization problem efficiently we apply the techniques of branch and price and benders decomposition the advantages of our model are that it can represent uncertainty in a straightforward manner with little distributional assumptions and it can produce solutions that effectively hedge against the uncertainty in the lot sizes using very reasonable amounts of computational effort

on the growth of l 2 invariants for sequences of lattices in lie groups we study the asymptotic behaviour of betti numbers twisted torsion and other spectral invariants of sequences of locally symmetric spaces our main results are uniform versions of the degeorge wallach theorem of a theorem of delorme and various other limit multiplicity theorems a basic idea is to adapt the notion of benjamini schramm convergence bs convergence originally introduced for sequences of finite graphs of bounded degree to sequences of riemannian manifolds and analyze the possible limits we show that bs convergence of locally symmetric spaces g k implies convergence in an appropriate sense of the normalized relative plancherel measures associated to l2 g this then yields convergence of normalized multiplicities of unitary representations betti numbers and other spectral invariants on the other hand when the corresponding lie group g is simple and of real rank at least two we prove that there is only one possible bs limit i e when the volume tends to infinity locally symmetric spaces always bs converge to their universal cover g k this leads to various general uniform results when restricting to arbitrary sequences of congruence covers of a fixed arithmetic manifold we prove a strong quantitative version of bs convergence which in turn implies upper estimates on the rate of convergence of normalized betti numbers in the spirit of sarnak xue an important role in our approach is played by the notion of invariant random subgroups for higher rank simple lie groups g we exploit rigidity theory and in particular the nevo stuck zimmer theorem and kazhdan s property t to obtain a complete understanding of the space of irs s of g 2017 department of mathematics princeton university

sparse regular random graphs spectral density and eigenvectors we examine the empirical distribution of the eigenvalues and the eigenvectors of adjacency matrices of sparse regular random graphs we find that when the degree sequence of the graph slowly increases to infinity with the number of vertices the empirical spectral distribution converges to the semicircle law moreover we prove concentration estimates on the number of eigenvalues over progressively smaller intervals we also show that with high probability all the eigenvectors are delocalized

an l p theory of sparse graph convergence i limits sparse random graph models and power law distributions we introduce and develop a theory of limits for sequences of sparse graphs based on lp graphons which generalizes both the existing l1 theory of dense graph limits and its extension by bollob as and riordan to sparse graphs without dense spots in doing so we replace the no dense spots hypothesis with weaker assumptions which allow us to analyze graphs with power law degree distributions this gives the rst broadly applicable limit theory for sparse graphs with unbounded average degrees in this paper we lay the foundations of the lp theory of graphons characterize convergence and develop corresponding random graph models while we prove the equivalence of several alternative metrics in a companion paper