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In this paper we define the concept of a near-infinity concentrated norm on a Banach space X with a boundedly complete Schauder basis. When ‖ ⋅ ‖ is such a norm, we prove that ( X , ‖ ⋅ ‖ ) has the fixed point property (FPP); that is, every nonexpansive self-mapping defined on a closed, bounded, convex subset has a fixed point. In particular, P.K. Lin's norm in ℓ 1 [14] and the norm ν p ( ⋅ ) (with p = ( p n ) and lim n ⁡ p n = 1 ) introduced in [3] are examples of near-infinity concentrated norms. When ν p ( ⋅ ) is equivalent to the ℓ 1 -norm, it was an open problem as to whether ( ℓ 1 , ν p ( ⋅ ) ) had the FPP. We prove that the norm ν p ( ⋅ ) always generates a nonreflexive Banach space X = R ⊕ p 1 ( R ⊕ p 2 ( R ⊕ p 3 … ) ) satisfying the FPP, regardless of whether ν p ( ⋅ ) is equivalent to the ℓ 1 -norm. We also obtain some stability results. [ABSTRACT FROM AUTHOR]

Perfect dyadic operators were first introduced in [1] , where a local T ( b ) theorem was proved for such operators. In [3] it was shown that for every singular integral operator T with locally bounded kernel on R n × R n there exists a perfect dyadic operator T such that T − T is bounded on L p ( d x ) for all 1 < p < ∞ . In this paper we show a decomposition of perfect dyadic operators on real line into four well known operators: two selfadjoint operators, paraproduct and its adjoint. Based on this decomposition we prove a sharp weighted version of the T ( 1 ) theorem for such operators, which implies A 2 conjecture for such operators with constant which only depends on ‖ T ( 1 ) ‖ BMO d , ‖ T ⁎ ( 1 ) ‖ BMO d and the constant in testing conditions for T . Moreover, the constant depends on these parameters at most linearly. In this paper we also obtain sufficient conditions for the two weight boundedness for a perfect dyadic operator and simplify these conditions under additional assumptions that weights are in the Muckenhoupt class A ∞ d . [ABSTRACT FROM AUTHOR]

In an earlier paper, [1] , the authors treated a family of Allen–Cahn model problems for which 0 and 1 are solutions and further solutions were found that are near 1 on a prescribed set, T + Ω , where T ⊂ Z n , and near 0 on ( Z n ∖ T ) + Ω . Here Ω ⊂ ( 0 , 1 ) n . In this paper, a more general class of potentials is treated for which the pair, { 0 , 1 } , is replaced by Z and the existence of a far richer structure of shadowing solutions, including unbounded ones, is established. [ABSTRACT FROM AUTHOR]

Martin-Löf defined algorithmic randomness, the use of martingales in this definition and several variants were explored by Schnorr, and Lutz introduced resource-bounded randomness. Ambos-Spies et al. have studied resource-bounded randomness under time-bounds and have shown that resource-bounded randomness implies resource-bounded genericity. While the genericity of Ambos-Spies is based on time-bound on finite-extension strategy, the genericity of Dowd, the main topic of this paper, is based on an analogy of the forcing theorem. For a positive integer r , an oracle D is r -generic in the sense of Dowd ( r -Dowd) if the following holds: If a certain formula F on an exponential-sized portion of D is a tautology then a polynomial-sized subfunction of D forces F to be a tautology. Here, r is the number of occurrences of query symbols in F . The relationship between resource-bounded randomness and Dowd-type genericity has been not clear so far. We show that there exists a primitive recursive function t ( n ) with the following property: Every t ( n ) -random set is r -Dowd for each positive integer r . A proof is done by means of constructing resource-bounded martingales. In our former paper, we left an open problem whether there exists a primitive recursive set which is r -Dowd for every positive integer r . In our recent work, we give an affirmative answer to the problem. The main theorem of the present paper gives an alternative proof of the affirmative answer. [ABSTRACT FROM AUTHOR]

We say that a pair ( A , B ) is a recovering pair if A and B are set systems on an n -element ground set, such that for every A , A ′ ∈ A and B , B ′ ∈ B we have ( A ∖ B = A ′ ∖ B ′ implies A = A ′ ) and symmetrically ( B ∖ A = B ′ ∖ A ′ implies B = B ′ ). G. Simonyi conjectured that if ( A , B ) is a recovering pair, then | A | | B | ≤ 2 n . For the quantity | A | | B | the best known upper bound is 2 . 326 4 n due to Holzman and Körner. In this paper we improve this upper bound to 2 . 281 4 n . [ABSTRACT FROM AUTHOR]

The typical extremal problem asks how large a structure can be without containing a forbidden substructure. The Erdős–Rothschild problem, introduced in 1974 by Erdős and Rothschild in the context of extremal graph theory, is a coloured extension, asking for the maximum number of colourings a structure can have that avoid monochromatic copies of the forbidden substructure. The celebrated Erdős–Ko–Rado theorem is a fundamental result in extremal set theory, bounding the size of set families without a pair of disjoint sets, and has since been extended to several other discrete settings. The Erdős–Rothschild extensions of these theorems have also been studied in recent years, most notably by Hoppen, Koyakayawa and Lefmann for set families, and Hoppen, Lefmann and Odermann for vector spaces. In this paper we present a unified approach to the Erdős–Rothschild problem for intersecting structures, which allows us to extend the previous results, often with sharp bounds on the size of the ground set in terms of the other parameters. In many cases we also characterise which families of vector spaces asymptotically maximise the number of Erdős–Rothschild colourings, thus addressing a conjecture of Hoppen, Lefmann and Odermann. [ABSTRACT FROM AUTHOR]

In an earlier work, the first author and Petsche solved an energy minimization problem for local fields and used the result to obtain lower bounds on the height of algebraic numbers all whose conjugates lie in various local fields, such as totally real and totally p -adic numbers. In this paper, we extend these techniques and solve the corresponding minimization programs for real intervals and p -adic discs, obtaining several new lower bounds for the height of algebraic numbers all of whose conjugates lie in such sets. [ABSTRACT FROM AUTHOR]

The main result of this paper is to show that all binomial identities are orderable. This is a natural statement in the combinatorial theory of finite sets, which can also be applied in distributed computing to derive new strong bounds on the round complexity of the weak symmetry breaking task. Furthermore, we introduce the notion of a fundamental binomial identity and find an infinite family of values, other than the prime powers, for which no fundamental binomial identity can exist. [ABSTRACT FROM AUTHOR]

In this paper, we use methods from spectral graph theory to obtain some results on the sum–product problem over finite valuation rings R of order q r which generalize recent results given by Hegyvári and Hennecart (2013). More precisely, we prove that, for related pairs of two-variable functions f ( x , y ) and g ( x , y ) , if A and B are two sets in R ∗ with | A | = | B | = q α , then max { | f ( A , B ) | , | g ( A , B ) | } ≫ | A | 1 + Δ ( α ) , for some Δ ( α ) > 0 . [ABSTRACT FROM AUTHOR]

The Hunter–Saxton equation determines a flow of conservative solutions taking values in the space H 1 ( R + ) . However, the solution typically includes finite time gradient blowups, which make the solution flow not continuous w.r.t. the natural H 1 distance. The aim of this paper is to first study the generic properties of conservative solutions of some initial boundary value problems to the Hunter–Saxton type equations. Then using these properties, we give a new way to construct a Finsler type metric which renders the flow uniformly Lipschitz continuous on bounded subsets of H 1 ( R + ) . [ABSTRACT FROM AUTHOR]

Let n , s be positive integers, n ≥ s + 2 . In 1964 Katona [5] established the maximum possible size of a family of subsets of { 1 , 2 , … , n } such that the union of any two members of the family has size of at most s . Katona also proved that the optimal families are unique up to isomorphism. In the present paper we sharpen this result by showing that excluding those optimal families one can get better bounds. These new bounds are best possible. [ABSTRACT FROM AUTHOR]

In this paper, we first give a corollary to Snevily’s Theorem on L -intersecting families, which implies a result that cuts by almost half the bound given by Grolmusz and Sudakov (2002), and provide a k -wise extension to the theorem by Babai et al. (2001) on set systems with L -intersections modulo prime powers which implies polynomial bounds for such families. We then extend Alon–Babai–Suzuki type inequalities on set systems to k -wise L -intersecting families and derive a result which improves the existing bound substantially for the non-modular case. We also provide the first known polynomial bounds for codes with restricted Hamming distances for all prime powers moduli p t , in contrast with Grolmusz’s result from Grolmusz (2006) that for non-prime power composite moduli, no polynomial bound exists for such codes. [ABSTRACT FROM AUTHOR]

In this paper, we find a lower bound for the order of the coset x + b in the Artin–Schreier extension F q [ x ] / ( x p − x − a ) , where b ∈ F q satisfies a generic special condition. [ABSTRACT FROM AUTHOR]

This paper studies by means of standard analytic tools the small time behavior of the heat content over a bounded Lebesgue measurable set of finite perimeter by working with the set covariance function and by imposing conditions on the heat kernels. Applications concerning the heat kernels of rotational invariant α -stable processes are given. [ABSTRACT FROM AUTHOR]

In a recent paper Kwon and Oum (2014), Kwon and Oum claim that every graph of bounded rank-width is a pivot-minor of a graph of bounded tree-width (while the converse has been known true already before). We study the analogous questions for “depth” parameters of graphs, namely for the tree-depth and related new shrub-depth. We show how a suitable adaptation of known results implies that shrub-depth is monotone under taking vertex-minors, and we prove that every graph class of bounded shrub-depth can be obtained via vertex-minors of graphs of bounded tree-depth. While we exhibit an example that pivot-minors are generally not sufficient (unlike Kwon and Oum (2014)) in the latter statement, we then prove that the bipartite graphs in every class of bounded shrub-depth can be obtained as pivot-minors of graphs of bounded tree-depth. [ABSTRACT FROM AUTHOR]

In this paper, by analyzing the solutions of certain equations over F 3 m , we present four classes of optimal ternary cyclic codes with parameters [ 3 m − 1 , 3 m − 1 − 2 m , 4 ] . It is shown that some recent work on this class of optimal ternary cyclic codes are special cases of our results. [ABSTRACT FROM AUTHOR]

In this paper we study several properties of Chebyshev sets in geodesic spaces. We focus on analyzing if some well-known results that characterize convexity of such sets in Hilbert spaces are also valid in the setting of geodesic spaces with bounded curvature. [ABSTRACT FROM AUTHOR]

We introduce a stratified version of Combinatory Logic 1 in which there are two classes of terms called player and opponent such that the class of player terms is strictly contained in the class of opponent terms. We show that the system characterizes Polynomial Time in a similar way to Soft Linear Logic. With the addition of explicit “lazy” products to the player terms and various notions of distributivity, we obtain a characterization of Polynomial Space. This paper is an expanded version of the abstract presented at DICE 2013. [ABSTRACT FROM AUTHOR]

In this paper we will give some characterizations for the boundedness of the maximal operators, potential operators and singular integral operators on the vanishing generalized Morrey spaces V L Π p ( ⋅ ) , φ ( Ω ) with variable exponent p ( ⋅ ) and bounded set Ω. Moreover, we also obtain some corresponding results for unbounded set Ω on such these spaces. [ABSTRACT FROM AUTHOR]

Over finite fields, if the image of a polynomial map is not the entire field, then its cardinality can be bounded above by a significantly smaller value. Earlier results bound the cardinality of the value set using the degree of the polynomial, but more recent results make use of the powers of all monomials. In this paper, we explore the geometric properties of the Newton polytope and show how they allow for tighter upper bounds on the cardinality of the multivariate value set. We then explore a method which allows for even stronger upper bounds, regardless of whether one uses the multivariate degree or the Newton polytope to bound the value set. Effectively, this provides improvement of a degree matrix-based result given by Zan and Cao, making our new bound the strongest upper bound thus far. [ABSTRACT FROM AUTHOR]

A well-known result of W. Ray asserts that if C is an unbounded convex subset of a Hilbert space, then there is a nonexpansive mapping T : C → C that has no fixed point. In this paper we establish some common fixed point properties for a semitopological semigroup S of nonexpansive mappings acting on a closed convex subset C of a Hilbert space, assuming that there is a point c ∈ C with a bounded orbit and assuming that certain subspace of C b ( S ) has a left invariant mean. Left invariant mean (or amenability) is an important notion in harmonic analysis of semigroups and groups introduced by von Neumann in 1929 [28] and formalized by Day in 1957 [5] . In our investigation we use the notion of common attractive points introduced recently by S. Atsushiba and W. Takahashi. [ABSTRACT FROM AUTHOR]

Set agreement and renaming are two tasks that allow processes to coordinate, even when agreement is impossible. In k -set agreement , n processes must decide on at most k of their input values. While n -set agreement is trivially wait-free solvable by each process deciding on its input, ( n − 1 ) -set agreement is not wait-free solvable. In M -renaming , processes must decide on distinct names in a range of size M . For any number n of processes, ( 2 n − 1 ) -renaming is wait-free solvable, but surprisingly, ( 2 n − 2 ) -renaming is wait-free solvable if and only if n is not a prime power; the only previous lower bound on the number of names necessary for renaming, when n is not a prime power, is n + 1 . In adaptive renaming, M decreases when the number p of participants in the execution decreases. It is known that ( 2 p − 1 ) -adaptive renaming is wait-free solvable, while ( 2 p − ⌈ p n − 1 ⌉ ) -adaptive renaming is not. This paper presents counting-based proofs for the above mentioned impossibility results: n processes can wait-free solve neither ( n − 1 ) -set agreement nor ( 2 p − ⌈ p n − 1 ⌉ ) -adaptive renaming; if n is a prime power, n processes cannot wait-free solve ( 2 n − 2 ) -renaming. For an arbitrary number of processes, we give a lower bound for renaming, by reduction from renaming for a different number of processes, and relying on the distribution of prime numbers. Our proofs combine simple operational properties of a restricted set of executions with elementary counting arguments to show the existence of an execution violating the task’s conditions. This makes the proofs easier to understand, verify, and, we hope, extend. [ABSTRACT FROM AUTHOR]

The paper deals with extremal problems concerning colorings of hypergraphs. By using a random recoloring algorithm we show that any n -uniform simple (i.e. every two distinct edges share at most one vertex) hypergraph H with maximum edge degree at most Δ ( H ) ⩽ c ⋅ n r n − 1 is r -colorable, where c > 0 is an absolute constant. As an application of our proof technique we establish a new lower bound for Van der Waerden number W ( n , r ) , the minimum N such that in any r -coloring of the set { 1 , … , N } there exists a monochromatic arithmetic progression of length n . We show that W ( n , r ) > c ⋅ r n − 1 , for some absolute constant c > 0 . [ABSTRACT FROM AUTHOR]

In this paper, we investigate the complexity of deciding determinism of unary languages. First, we give a method to derive a set of arithmetic progressions from a regular expression E over a unary alphabet, and establish relations between numbers represented by these arithmetic progressions and words in L ( E ) . Next, we define a problem relating to arithmetic progressions and investigate the complexity of this problem. Then by a reduction from this problem we show that deciding determinism of unary languages is coNP -complete. Finally, we extend our derivation method to expressions with counting, and prove that deciding whether an expression over a unary alphabet with counting defines a deterministic language is in Π 2 p . We also establish a tight upper bound for the size of the minimal DFA for expressions with counting. [ABSTRACT FROM AUTHOR]

Text A sum-dominant set is a finite set A of integers such that | A + A | > | A − A | . As a typical pair of elements contributes one sum and two differences, we expect sum-dominant sets to be rare in some sense. In 2006, however, Martin and O'Bryant showed that the proportion of sum-dominant subsets of { 0 , … , n } is bounded below by a positive constant as n → ∞ . Hegarty then extended their work and showed that for any prescribed s , d ∈ N 0 , the proportion ρ n s , d of subsets of { 0 , … , n } that are missing exactly s sums in { 0 , … , 2 n } and exactly 2 d differences in { − n , … , n } also remains positive in the limit. We consider the following question: are such sets, characterized by their sums and differences, similarly ubiquitous in higher dimensional spaces? We generalize the integers in a growing interval to the lattice points in a dilating polytope. Specifically, let P be a polytope in R D with vertices in Z D , and let ρ n s , d now denote the proportion of subsets of L ( n P ) that are missing exactly s sums in L ( n P ) + L ( n P ) and exactly 2 d differences in L ( n P ) − L ( n P ) . As it turns out, the geometry of P has a significant effect on the limiting behavior of ρ n s , d . We define a geometric characteristic of polytopes called local point symmetry, and show that ρ n s , d is bounded below by a positive constant as n → ∞ if and only if P is locally point symmetric. We further show that the proportion of subsets in L ( n P ) that are missing exactly s sums and at least 2 d differences remains positive in the limit, independent of the geometry of P . A direct corollary of these results is that if P is additionally point symmetric, the proportion of sum-dominant subsets of L ( n P ) also remains positive in the limit. Video For a video summary of this paper, please visit http://youtu.be/2M8Qg0E0RAc . [ABSTRACT FROM AUTHOR]

This paper concerns μ -limit sets of cellular automata: sets of configurations made of words whose probability to appear does not vanish with time, starting from an initial μ -random configuration. More precisely, we investigate the computational complexity of these sets and of related decision problems. Main results: first, μ -limit sets can have a Σ 3 0 -hard language, second, they can contain only α -complex configurations, third, any non-trivial property concerning them is at least Π 3 0 -hard. We prove complexity upper bounds, study restrictions of these questions to particular classes of CA, and different types of (non-)convergence of the measure of a word during the evolution. [ABSTRACT FROM AUTHOR]

In this paper, we prove explicit lower bounds for the cost of fast boundary controls for a class of linear equations of parabolic or dispersive type involving the spectral fractional Laplace operator. We notably deduce the following striking result: in the case of the heat equation controlled on the boundary, Miller's conjecture formulated in Miller (2004) [16] is not verified. Moreover, we also give a new lower bound for the minimal time needed to ensure the uniform controllability of the one-dimensional convection–diffusion equation with negative speed controlled on the left boundary, proving that the conjecture formulated in Coron and Guerrero (2005) [2] concerning this problem is also not verified at least for negative speeds. The proof is based on complex analysis, and more precisely on a representation formula for entire functions of exponential type, and is quite related to the moment method . [ABSTRACT FROM AUTHOR]

In this paper, by using the optimization idea, a new ultimate bound for the complex Lorenz system is derived. It is shown that a hyperelliptic estimate of the ultimate bound set can be analytically calculated based on the optimization method and the Lagrange multiplier method. And based on the ellipsoidal bound set and set operations, one further obtains a more conservative boundary for each variable in the complex system, which only relies on the system parameters. Afterwards, the estimated results are applied to the exponential finite-time synchronization of the complex Lorenz system. Especially, the designed control depends on the parameters of the exponential convergence rate, the finite-time convergence rate, the bound of the initial states of the master system, and the system parameter. Finally, numerical simulations are given to verify the effectiveness and correctness of the obtained results. [ABSTRACT FROM AUTHOR]

Spherical t -design is a finite subset on sphere such that, for any polynomial of degree at most t , the average value of the integral on sphere can be replaced by the average value at the finite subset. It is well-known that an equivalent condition of spherical design is given in terms of harmonic polynomials. In this paper, we define a spherical design of harmonic index t from the viewpoint of this equivalent condition, and we give its construction and a Fisher type lower bound on the cardinality. Also we investigate whether there is a spherical design of harmonic index attaining the bound. [ABSTRACT FROM AUTHOR]

A theorem due to Cartan asserts that every origin-preserving automorphism of bounded circular domains with respect to the origin is linear. In the present paper, by employing the theory of Bergman's representative domain, we prove that under certain circumstances Cartan's assertion remains true for quasi-circular domains in C n . Our main result is applied to obtain some simple criterions for the case n = 3 and to prove that Braun–Kaup–Upmeier's theorem remains true for our class of quasi-circular domains. [ABSTRACT FROM AUTHOR]

A stability theorem says that a nearly extremal object can be obtained from an extremal one by “small changes”. In this paper, we prove a sharp stability theorem of sets of even type in PG ( 2 , q ) , q even. As a consequence, we improve Blokhuis and Bruen's stability theorem on hyperovals and also on the minimum number of lines intersecting a point set of size at most q + 2 ⌊ q ⌋ − 2 ; furthermore we improve on the lower bound for untouchable sets. [ABSTRACT FROM AUTHOR]

Suppose that f is a Lipschitz function on R with ‖ f ‖ Lip ≤ 1 . Let A be a bounded self-adjoint operator on a Hilbert space H . Let p ∈ ( 1 , ∞ ) and suppose that x ∈ B ( H ) is an operator such that the commutator [ A , x ] is contained in the Schatten class S p . It is proved by the last two authors, that then also [ f ( A ) , x ] ∈ S p and there exists a constant C p independent of x and f such that ‖ [ f ( A ) , x ] ‖ p ≤ C p ‖ [ A , x ] ‖ p . The main result of this paper is to give a sharp estimate for C p in terms of p . Namely, we show that C p ∼ p 2 p − 1 . In particular, this gives the best estimates for operator Lipschitz inequalities. We treat this result in a more general setting. This involves commutators of n self-adjoint operators A 1 , … , A n , for which we prove the analogous result. The case described here in the abstract follows as a special case. [ABSTRACT FROM AUTHOR]

In this paper we introduce and study a new property of infinite words: An infinite word x ∈ A N , with values in a finite set A , is said to be k - self-shuffling ( k ≥ 2 ) if x admits factorizations: x = ∏ i = 0 ∞ U i ( 1 ) ⋯ U i ( k ) = ∏ i = 0 ∞ U i ( 1 ) = ⋯ = ∏ i = 0 ∞ U i ( k ) . In other words, there exists a shuffle of k -copies of x which produces x . We are particularly interested in the case k = 2 , in which case we say x is self-shuffling. This property of infinite words is shown to be independent of the complexity of the word as measured by the number of distinct factors of each length. Examples exist from bounded to full complexity. It is also an intrinsic property of the word and not of its language (set of factors). For instance, every aperiodic word contains a non-self-shuffling word in its shift orbit closure. While the property of being self-shuffling is a relatively strong condition, many important words arising in the area of symbolic dynamics are verified to be self-shuffling. They include for instance the Thue–Morse word fixed by the morphism 0 ↦ 01 , 1 ↦ 10 . As another example we show that all Sturmian words of slope α ∈ R ∖ Q and intercept 0 < ρ < 1 are self-shuffling (while those of intercept ρ = 0 are not). Our characterization of self-shuffling Sturmian words can be interpreted arithmetically in terms of a dynamical embedding and defines an arithmetic process we call the stepping stone model . One important feature of self-shuffling words stems from their morphic invariance: The morphic image of a self-shuffling word is self-shuffling. This provides a useful tool for showing that one word is not the morphic image of another. In addition to its morphic invariance, this new notion has other unexpected applications particularly in the area of substitutive dynamical systems. For example, as a consequence of our characterization of self-shuffling Sturmian words, we recover a number theoretic result, originally due to Yasutomi, on a classification of pure morphic Sturmian words in the orbit of the characteristic. [ABSTRACT FROM AUTHOR]

The term low-discrepancy sequences is widely used to refer to s-dimensional sequences X for which the bound D*(N,X)≤cs(logN)s+O((logN)s-1) is satisfied, where D* denotes the usual star discrepancy. In this paper, we study such bounds for (t,s)-sequences and a newer class of low-discrepancy sequences called (t,e,s)-sequences, introduced recently by Tezuka (2013). In the first case, by using a combinatorial argument coupled with a careful worst-case analysis, we are able to improve the discrepancy bounds from Faure and Lemieux (2012) for (t,s)-sequences. In the second case, an adaptation of the same pair of arguments allows us to improve the asymptotic behavior of the bounds from Tezuka (2013) in the case of even bases. [ABSTRACT FROM AUTHOR]

Abstract: We consider nondeterministic concurrent games played on event structures and study their determinacy problem—the existence of winning strategies. It is known that when the winning conditions of the games are characterised by a collection of finite winning sets/plays, a restriction (called race-freedom) on the boards where the games are played guarantees determinacy. However the games may no longer be determined when the winning sets are infinite. This paper provides a study of concurrent games and nondeterministic winning strategies by analysing conditions that ensure determinacy when infinitely many events are played, that is, when the winning sets are infinite. The main result is a determinacy theorem for a class of games with a bounded concurrency property and infinite winning sets shown to be finitely decidable. [Copyright &y& Elsevier]

Abstract: This paper is concerned with the existence of segregated vector solutions for where Ω is a bounded or unbounded domain in with , is a small parameter, is an integer, ( ) is a potential function, ( ) is constant and ( ) is coupling constant. This system describes some physical phenomena such as the propagation in birefringent optical fibers, Kerr-like photorefractive in optics and Bose–Einstein condensates. For , which corresponds to the synchronization case for the above system with constant potentials, we prove that the system has multiple positive vector solutions, whose components may have spikes clustering at the same point as , but the distance between them divided by ε will go to infinity. [Copyright &y& Elsevier]

Abstract: In this paper, we are concerned with the partial regularity of the stationary solution to semilinear elliptic equations. For almost all supercritical exponents, we improve the upper bounds of the Hausdorff dimension of the singular set, which are derived by Pacard, to the optimal ones. [Copyright &y& Elsevier]

Abstract: In this paper we consider the following Schrödinger–Poisson system where potentials , ( , ), and is asymptotically linear at infinity. Working in a variational setting, we prove the existence and multiplicity of solutions for the system when λ is small and belong to different ranges. We also study the boundedness of the set of the solutions found, as , in the spirit of Ruiz (2006) [21]. [Copyright &y& Elsevier]

Abstract: A popular approach for analyzing high-dimensional datasets is to perform dimensionality reduction by applying non-parametric affinity kernels. Usually, it is assumed that the represented affinities are related to an underlying low-dimensional manifold from which the data is sampled. This approach works under the assumption that, due to the low-dimensionality of the underlying manifold, the kernel has a low numerical rank. Essentially, this means that the kernel can be represented by a small set of numerically-significant eigenvalues and their corresponding eigenvectors. We present an upper bound for the numerical rank of Gaussian convolution operators, which are commonly used as kernels by spectral manifold-learning methods. The achieved bound is based on the underlying geometry that is provided by the manifold from which the dataset is assumed to be sampled. The bound can be used to determine the number of significant eigenvalues/eigenvectors that are needed for spectral analysis purposes. Furthermore, the results in this paper provide a relation between the underlying geometry of the manifold (or dataset) and the numerical rank of its Gaussian affinities. The term cover-based bound is used because the computations of this bound are done by using a finite set of small constant-volume boxes that cover the underlying manifold (or the dataset). We present bounds for finite Gaussian kernel matrices as well as for the continuous Gaussian convolution operator. We explore and demonstrate the relations between the bounds that are achieved for finite and continuous cases. The cover-oriented methodology is also used to provide a relation between the geodesic length of a curve and the numerical rank of Gaussian kernel of datasets that are sampled from it. [Copyright &y& Elsevier]

Abstract: Consider a graph and, for each vertex , a subset of neighbors of . A -coloring is a coloring of the elements of so that vertices appearing together in some receive pairwise distinct colors. An obvious lower bound for the minimum number of colors in such a coloring is the maximum size of a set , denoted by . In this paper we study graph classes for which there is a function , such that for any graph and any , there is a -coloring using at most colors. It is proved that if such a function exists for a class , then can be taken to be a linear function. It is also shown that such classes are precisely the classes having bounded star chromatic number. We also investigate the list version and the clique version of this problem, and relate the existence of functions bounding those parameters to the recently introduced concepts of classes of bounded expansion and nowhere-dense classes. [Copyright &y& Elsevier]

Abstract: Let be a graph. A clique-transversal set is a subset of vertices of such that meets all cliques of , where a clique is defined as a complete subgraph maximal under inclusion and having at least two vertices. The clique-transversal number, denoted by , of is the cardinality of a minimum clique-transversal set in . A -clique-coloring of is a -coloring of its vertices such that no clique is monochromatic. All planar graphs have been proved to be 3-clique-colorable by Mohar and Škrekovski [B. Mohar, R. Škrekovski, The Grötzsch theorem for the hypergraph of maximal cliques, Electron. J. Combin. 6 (1999) #R26]. Erdős et al. [P. Erdős, T. Gallai, Zs. Tuza, Covering the cliques of a graph with vertices, Discrete Math. 108 (1992) 279–289] proposed to find sharp estimates on for planar graphs. In this paper we first show that every outerplanar graph of order has and the bound is tight. Secondly, we prove that every claw-free planar graph different from an odd cycle is -clique-colorable and we present a polynomial-time algorithm to find the -clique-coloring. As a by-product of the result, we obtain a tight upper bound on the clique-transversal number for claw-free planar graphs. [Copyright &y& Elsevier]

Abstract: A dominating set of graph is called metric-locating–dominating if it is also locating, that is, if every vertex is uniquely determined by its vector of distances to the vertices in . If moreover, every vertex not in is also uniquely determined by the set of neighbors of belonging to , then it is said to be locating–dominating. Locating, metric-locating–dominating and locating–dominating sets of minimum cardinality are called -codes, -codes and -codes, respectively. A Nordhaus–Gaddum bound is a tight lower or upper bound on the sum or product of a parameter of a graph and its complement . In this paper, we present some Nordhaus–Gaddum bounds for the location number , the metric-location–domination number and the location–domination number . Moreover, in each case, the graph family attaining the corresponding bound is fully characterized. [Copyright &y& Elsevier]

Abstract: Digital Kronecker-sequences are a non-archimedean analog of classical Kronecker-sequences whose construction is based on Laurent series over a finite field. In this paper it is shown that for almost all digital Kronecker-sequences the star discrepancy satisfies for infinitely many , where only depends on the dimension s and on the order q of the underlying finite field, but not on N. This result shows that a corresponding metrical upper bound due to Larcher is up to some term best possible. [Copyright &y& Elsevier]

Abstract: In this paper it is shown that there exists a constant such that there do not exist Cameron–Liebler line classes in with parameter x satisfying . This improves all known bounds except for small values of q. [Copyright &y& Elsevier]

Abstract: This paper is dedicated to inverse quasi-variational inequalities which correspond to a mixture of quasi-variational inequalities and inverse variational inequalities. Our aim is to obtain local/global error bounds for inverse quasi-variational inequality problems in terms of different gap functions/merit functions i.e. the residual gap function, the regularized gap function and the -gap function. These bounds provide effective estimated distances between a specific point and the exact solution of the inverse quasi-variational inequality problem. Since the class of inverse quasi-variational inequalities includes the classes of general quasi-variational inequalities and variational inequalities, our results cover and extend similar results for these problems. [Copyright &y& Elsevier]

Abstract: The sum–product conjecture of Erdős and Szemerédi states that, given a finite set of positive numbers, one can find asymptotic lower bounds for of the order of for every . In this paper we consider the set of all spectral radii of matrices with entries in , and find lower bounds for the cardinality of this set. In the case , this cardinality is necessarily larger than . [Copyright &y& Elsevier]

A hypergraph is -colorable if there is a -coloring of the vertices with no monochromatic hyperedge. Let denote the class of all -uniform -regular hypergraphs. The Lovász Local Lemma, devised by Erdös and Lovász in 1975 to tackle the problem of hypergraph 2-colorings, implies that every hypergraph is -colorable, provided . Alon and Bregman [N. Alon, Z. Bregman, Every 8-uniform 8-regular hypergraph is 2-colorable, Graphs Combin. 4 (1988) 303–306] proved the slightly stronger result that every hypergraph is -colorable, provided . It is implicitly known in the literature that the Alon–Bregman result is true for all , as remarked by Vishwanathan [S. Vishwanathan, On 2-coloring certain -uniform hypergraphs, J. Combin. Theory Ser. A 101 (2003) 168–172] even though we have not seen it explicitly proved. For completeness, we provide a short proof of this result. As remarked by Alon and Bregman the result is not true when , as may be seen by considering the Fano plane. Our main result in this paper is a strengthening of the above results. For this purpose, we define a set of vertices in a hypergraph to be a free set in if we can -color such that every edge in receives at least one vertex of each color. Equivalently, is a free set in if it is the complement of two disjoint transversals in . For every , we prove that every hypergraph of order has a free set of size at least . For any where and for sufficiently large , we prove that every hypergraph of order has a free set of size at least , where , and so as . As an application, we show that the total restrained domination number of a graph on vertices with sufficiently large minimum degree is at most , which significantly improves the best known bound of . [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we give sharp estimates of the smallest principal curvature of level sets of n-dimensional p-harmonic functions which extends the result of 2-dimensional minimal surface case due to Longinetti [Longinetti, On minimal surfaces bounded by two convex curves in parallel planes, J. Differential Equations 67 (3) (1987) 344–358]. More precisely, we prove that the function is a convex function with respect to the layer parameter of the level sets for all and . [Copyright &y& Elsevier]

Abstract: In this paper, we give an example of a Fourier integral operator with a symbol belonging to that cannot be extended as a bounded operator on . [Copyright &y& Elsevier]

Abstract: Let be the triangle in bounded by the lines (the standard simplex in ). Denote by the set of all real bivariate algebraic polynomials of total degree at most two. Let be the unit ball of the space endowed with the supremum norm on . We give a full description of the semidefinite extreme points of . The present paper completes the description of the set of all extreme points of started in Milev and Naidenov (2008) [13] and Milev and Naidenov (2011) [14]. We study, as an application, the path-connectedness of . The conclusion is that consists of two path-connected components. [Copyright &y& Elsevier]