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Abstract: When a v-set can be equipped with a set of k-subsets so that every 2-subset of the v-set appears in exactly (or at most, or at least) one of the chosen k-subsets, the result is a balanced incomplete block design (or packing, or covering, respectively). For each k, balanced incomplete block designs are known to exist for all sufficiently large values of v that meet certain divisibility conditions. When these conditions are not met, one can ask for the packing with the most blocks and/or the covering with the fewest blocks. Elementary necessary conditions furnish an upper bound on the number of blocks in a packing and a lower bound on the number of blocks in a covering. In this paper it is shown that for all sufficiently large values of v, a packing and a covering on v elements exist whose numbers of blocks differ from the basic bounds by no more than an additive constant depending only on k. [Copyright &y& Elsevier]

Abstract: We consider nonlinear elliptic partial differential equations for quasilinear operators of the form subject to fully nonlinear boundary conditions involving boundary operators of the form, for each , The main goal of this paper is to give, under suitable assumptions on and , an explicit estimate for bounded solutions of these elliptic boundary value problems. Then, we establish the existence of at least one solution to such problems extending the authorʼs previous work. Our methods rely on the definition of approximate problems, deducing a priori estimates for their solutions and compactness arguments in order to pass to the limit. These methods can be applied to a large class of equations involving operators of Leray–Lions type (on suitable Banach spaces) for a general class of boundary operators which are, possibly, of the same order as . As examples, these results are shown to apply to a class of uniformly elliptic equations that occur in the theory of phase transitions, and certain elliptic systems associated with climate problems which describe the evolution of atmospheric sea-level temperatures for relatively long time scales. [Copyright &y& Elsevier]

Abstract: Okounkov bodies, which are closed convex sets defined for big line bundles, have rich information on the line bundles. On the other hand, Seshadri constants are invariants which measure the positivity of line bundles. In this paper, we prove that Okounkov bodies give lower bounds of Seshadri constants. [Copyright &y& Elsevier]

Abstract: When constructing finite frames for a given application, the most important consideration is the spectrum of the frame operator. Indeed, the minimum and maximum eigenvalues of the frame operator are the optimal frame bounds, and the frame is tight precisely when this spectrum is constant. Often, the second-most important design consideration is the lengths of frame vectors: Gabor, wavelet, equiangular and Grassmannian frames are all special cases of equal norm frames, and unit norm tight frame-based encoding is known to be optimally robust against additive noise and erasures. We consider the problem of constructing frames whose frame operator has a given spectrum and whose vectors have prescribed lengths. For a given spectrum and set of lengths, the existence of such frames is characterized by the Schur–Horn Theorem—they exist if and only if the spectrum majorizes the squared lengths—the classical proof of which is nonconstructive. Certain construction methods, such as harmonic frames and spectral tetris, are known in the special case of unit norm tight frames, but even these provide but a few examples from the manifold of all such frames, the dimension of which is known and nontrivial. In this paper, we provide a new method for explicitly constructing any and all frames whose frame operator has a prescribed spectrum and whose vectors have prescribed lengths. The method itself has two parts. In the first part, one chooses eigensteps—a sequence of interlacing spectra—that transform the trivial spectrum into the desired one. The second part is to explicitly compute the frame vectors in terms of these eigensteps; though nontrivial, this process is nevertheless straightforward enough to be implemented by hand, involving only arithmetic, square roots and matrix multiplication. [Copyright &y& Elsevier]

Abstract: A hypergraph is called an grid if it is isomorphic to a pattern of horizontal and vertical lines, i.e., a family of sets such that for and for . Three sets form a triangle if they pairwise intersect in three distinct singletons, , . A hypergraph is linear, if holds for every pair of edges . In this paper we construct large linear -hypergraphs which contain no grids. Moreover, a similar construction gives large linear -hypergraphs which contain neither grids nor triangles. For our constructions are almost optimal. These investigations are motivated by coding theory: we get new bounds for optimal superimposed codes and designs. [Copyright &y& Elsevier]

Abstract: We discuss parametric polynomial systems, with algorithms for real root classification and triangular decomposition of semi-algebraic systems as our main applications. We exhibit new results in the theory of border polynomials of parametric semi-algebraic systems: in particular a geometric characterization of its “true boundary” (Definition 1). In order to optimize the corresponding decomposition algorithms, we also propose a technique, that we call relaxation, which can simplify the decomposition process and reduce the number of components in the output. This paper extends our earlier works (Chen et al., 2010, 2011). [Copyright &y& Elsevier]

Abstract: Let be an integral self-affine set (not necessarily a self-similar set) satisfying , where is an integer expanding matrix and is a finite set of integer vectors. For “totally disconnected ”, in 1992, Falconer obtained formulas for lower and upper bounds for the Hausdorff dimension of . In order to have such bounds for arbitrary , we consider an extension of Falconer’s formulas to certain graph directed sets and define new bounds. For a very few classes of self-affine sets, the Hausdorff dimension and Falconer’s upper bound are known to be different. In this paper, we present a new such class by using the new upper bound, and show that our upper bound is the box dimension for that class. We also study the computation of those bounds. [Copyright &y& Elsevier]

Abstract: For in , a subset of is called a Furstenberg set of type or -set if for each direction in the unit circle there is a line segment in the direction of such that the Hausdorff dimension of the set is greater than or equal to . In this paper we use generalized Hausdorff measures to give estimates on the size of these sets. Our main result is to obtain a sharp dimension estimate for a whole class of zero-dimensional Furstenberg type sets. Namely, for , , we construct a set of Hausdorff dimension not greater than . Since in a previous work we showed that is a lower bound for the Hausdorff dimension of any , with the present construction, the value is sharp for the whole class of Furstenberg sets associated to the zero dimensional functions . [Copyright &y& Elsevier]

Abstract: In 2001 Heinrich, Novak, Wasilkowski and Woźniakowski proved the upper bound for the inverse of the star discrepancy . This is equivalent to the fact that for any and there exists a set of points in the -dimensional unit cube whose star-discrepancy is bounded by . Dick showed that there exists a double infinite matrix of elements of such that for any and the star discrepancy of the -dimensional -element sequence is bounded by In the present paper we show that this upper bound can be reduced to , which is (up to the value of the constant) the same upper bound as the one obtained by Heinrich et al. in the case of fixed and . [Copyright &y& Elsevier]

Abstract: This article concerns nonconvolutional type operators (also known as Journéʼs type operators) associated with a multiparameter family of dilations given by where and . We are especially interested in the boundedness of such operators on the multiparameter Hardy spaces. This work is motivated by Pipherʼs result on the boundedness of these operators from the multiparameter spaces to spaces for , and Journéʼs counter-example which shows that the number of parameter plays a crucial role in boundedness of singular integral operators on multiparameter Hardy spaces. Journéʼs work shows that there is a sharp difference between the situations for two and three or more parameters. We establish in this paper the necessary and sufficient conditions under which the singular integral operators in Journéʼs class are bounded on multiparameter spaces () with arbitrary number of parameters. [Copyright &y& Elsevier]

Abstract: In this paper, we give existence, uniqueness and regularity of the solutions of problems whose prototype is where is a bounded open set of , , and . Here is a norm on , is its polar and is the anisotropic Laplacian. [Copyright &y& Elsevier]

Abstract: In Duursma and Park (2010) , the authors formulate new coset bounds for algebraic geometric codes. The bounds give improved lower bounds for the minimum distance of algebraic geometric codes as well as improved thresholds for algebraic geometric linear secret sharing schemes. The bounds depend on the delta set of a coset and on the choice of a sequence of divisors inside the delta set. In this paper we give general properties of delta sets and we analyze sequences of divisors supported in two points on Hermitian and Suzuki curves. [Copyright &y& Elsevier]

Abstract: Text: A More Sums Than Differences (or sum-dominant) set is a finite set with . Though it was believed that the percentage of subsets of that are sum-dominant tends to zero, Martin and OʼBryant proved a positive percentage is sum-dominant. We generalize their result to other sums and differences. We prove that a positive percent of the time for all nontrivial choices of , and give explicit constructions. We construct sets exhibiting different behavior as more sums/differences are taken. We prove that for any m, a positive percentage of the time. We find the limiting behavior of for an arbitrary set A as and an upper bound on k for such behavior to settle down. Finally, we say A is k-generational sum-dominant if are all sum-dominant. We prove that for any k a positive percentage of sets is k-generational, and no set is k-generational for all k. Video: For a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=ERSMlrEAijY;list=UUfJicAn0WSCOS0IZWMy7HsA;index=1;feature=plcp. [Copyright &y& Elsevier]

Abstract: We show that one proposition in our original paper allows for a much simpler proof resulting in certain improvements of the bounds in the main theorems. [Copyright &y& Elsevier]