Simple Explicit Construction Research Articles

Combinatorial Nullstellensatz and Turán numbers of complete r-partite r-uniform hypergraphs

In this note we describe how Lasoń's generalization of Alon's Combinatorial Nullstellensatz gives a framework for constructing lower bounds on the Turán number ex(n,Ks1,…,sr(r)) of the complete r-partite r-uniform hypergraph Ks1,…,sr(r). To illustrate the potential of this method, we give a short and simple explicit construction for the Erdős box problem, showing that ex(n,K2,…,2(r))=Ω(nr−1/r), which asymptotically matches best known bounds when r≤4.

Mirror symmetry for five-parameter Hulek-Verrill manifolds

We study the mirrors of five-parameter Calabi-Yau threefolds first studied by Hulek and Verrill in the context of observed modular behaviour of the zeta functions for Calabi-Yau manifolds. Toric geometry allows for a simple explicit construction of these mirrors, which turn out to be familiar manifolds. These are elliptically fibred in multiple ways. By studying the singular fibres, we are able to identify the rational curves of low degree on the mirror manifolds. This verifies the mirror symmetry prediction obtained by studying the mirror map near large complex structure points. We undertake also an extensive study of the periods of the Hulek-Verrill manifolds and their monodromies. On the mirror, we compute the genus-zero and -one instanton numbers, which are labelled by 5 indices, as h^{1,1}\!=5h1,1=5. There is an obvious permutation symmetry on these indices, but in addition there is a surprising repetition of values. We trace this back to an S_{6}S6 symmetry made manifest by certain constructions of the complex structure moduli space of the Hulek-Verrill manifold. Among other consequences, we see in this way that the moduli space has six large complex structure limits. It is the freedom to expand the prepotential about any one of these points that leads to this symmetry in the instanton numbers. An intriguing fact is that the group that acts on the instanton numbers is larger than S_6S6 and is in fact an infinite hyperbolic Coxeter group, that we study. The group orbits have a “web” structure, and with certain qualifications the instanton numbers are only nonzero if they belong to what we term “positive webs”. This structure has consequences for instanton numbers at all genera.

The space of C1-smooth isogeometric spline functions on trilinearly parameterized volumetric two-patch domains

We study the C1-smooth isogeometric spline space over a specific class of unstructured hexahedral meshes, namely over the class of trilinearly parameterized volumetric two-patch domains. Recently, the structure of this space was experimentally analyzed in Birner et al. (2018) by numerically computing a basis and the dimension of this space. In this work, we develop the theoretical framework to explore the C1-smooth isogeometric space. Amongst others, we use the framework to prove the numerically obtained dimension from Birner et al. (2018) and to describe a simple explicit basis construction which consists of locally supported functions.

Symmetric Pseudo-Random Matrices

We consider the problem of generating symmetric pseudo-random sign (±1) matrices based on the similarity of their spectra to Wigner's semicircular law. Using binary m-sequences (Golomb sequences) of lengths n = 2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</sup> - 1, we give a simple explicit construction of circulant n x n sign matrices and show that their spectra converge to the semicircular law when n grows. The Kolmogorov complexity of the proposed matrices equals to that of Golomb sequences and is at most 2log <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> (n) bits.

Nonvanishing cohomology classes on finite groups of Lie type with Coxeter number at most p

We prove that the degree 2p−3 cohomology of any finite group of Lie type over Fpr, with coefficients in characteristic p, is nonzero as long as its Coxeter number is at most p. We do this by providing a simple explicit construction of a nonzero element.

Quasilocal conservation laws in XXZ spin-1/2 chains: Open, periodic and twisted boundary conditions

A continuous family of quasilocal exact conservation laws is constructed in the anisotropic Heisenberg (XXZ) spin-1/2 chain for periodic (or twisted) boundary conditions and for a set of commensurate anisotropies densely covering the entire easy plane interaction regime. All local conserved operators follow from the standard (Hermitian) transfer operator in fundamental representation (with auxiliary spin s=1/2), and are all even with respect to a spin flip operation. However, the quasilocal family is generated by differentiation of a non-Hermitian highest weight transfer operator with respect to a complex auxiliary spin representation parameter s and includes also operators of odd parity. For a finite chain with open boundaries the time derivatives of quasilocal operators are not strictly vanishing but result in operators localized near the boundaries of the chain. We show that a simple modification of the non-Hermitian transfer operator results in exactly conserved, but still quasilocal operators for periodic or generally twisted boundary conditions. As an application, we demonstrate that implementing the new exactly conserved operator family for estimating the high-temperature spin Drude weight results, in the thermodynamic limit, in exactly the same lower bound as for almost conserved family and open boundaries. Under the assumption that the bound is saturating (suggested by agreement with previous thermodynamic Bethe ansatz calculations) we propose a simple explicit construction of infinite time averages of local operators such as the spin current.

A constructive approach to cubic Hermite Fractal Interpolation Function and its constrained aspects

The theory of splines is a well studied topic, but the kinship of splines with fractals is novel. We introduce a simple explicit construction for a Open image in new window-cubic Hermite Fractal Interpolation Function (FIF). Under some suitable hypotheses on the original function, we establish a priori estimates (with respect to the Lp-norm, 1≤p≤∞) for the interpolation error of the Open image in new window-cubic Hermite FIF and its first derivative. Treating the first derivatives at the knots as free parameters, we derive suitable values for these parameters so that the resulting cubic FIF enjoys Open image in new window global smoothness. Consequently, our method offers an alternative to the standard moment construction of Open image in new window-cubic spline FIFs. Furthermore, we identify appropriate values for the scaling factors in each subinterval and the derivatives at the knots so that the graph of the resulting Open image in new window-cubic FIF lies within a prescribed rectangle. These parameters include, in particular, conditions for the positivity of the cubic FIF. Thus, in the current article, we initiate the study of the shape preserving aspects of fractal interpolation polynomials. We also provide numerical examples to corroborate our results.

A simple construction of Grassmannian polylogarithms

The classical n-logarithm is a multivalued analytic function defined inductively: Lin(z):=∫0zLin−1(t)dlogt,Li1(z)=−log(1−z). In this paper we give a simple explicit construction of the Grassmannian n-logarithm, which is a multivalued analytic function on the quotient of the Grassmannian of n-dimensional subspaces in C2n in generic position to the coordinate hyperplanes by the natural action of the torus (C∗)2n. The classical n-logarithm appears at a certain one dimensional boundary stratum.We study Tate iterated integrals, which are homotopy invariant integrals of 1-forms dlogfi where fi are rational functions. We give a simple explicit formula for the Tate iterated integral which describes the Grassmannian n-logarithm.Another example is the Tate iterated integrals for the multiple polylogarithms on the moduli spaces M0,n, calculated in Section 4.4 of Goncharov (2005) [13] using the combinatorics of plane trivalent trees decorated by the arguments of the multiple polylogarithms.Variations of mixed Hodge–Tate structures on X are described by a Hopf algebra H•(X). We upgrade Tate iterated integrals on a (rational) complex variety X to elements of H•(X). The coproducts of these elements are very interesting invariants of the iterated integrals. In general their calculation is a non-trivial problem. We show however, that working modulo the ideal of H•(X) generated by constant variations, there is a simple way to calculate the coproduct.It is a pleasure to dedicate this paper to Andrey Suslin, whose works Suslin (1984) [21] and Suslin (1991) [22] played an essential role in the development of the story.

Sparsity and spectral properties of dual frames

We study sparsity and spectral properties of dual frames of a given finite frame. We show that any finite frame has a dual with no more than n2 non-vanishing entries, where n denotes the ambient dimension, and that for most frames no sparser dual is possible. Moreover, we derive an expression for the exact sparsity level of the sparsest dual for any given finite frame using a generalized notion of spark. We then study the spectral properties of dual frames in terms of singular values of the synthesis operator. We provide a complete characterization for which spectral patterns of dual frames are possible for a fixed frame. For many cases, we provide simple explicit constructions for dual frames with a given spectrum, in particular, if the constraint on the dual is that it be tight.

Free products, cyclic homology, and the Gauss–Manin connection

We use the techniques of Cuntz and Quillen to present a new approach to periodic cyclic homology. Our construction is based on ((Ω•A)[t],d+t⋅ıΔ), a noncommutative equivariant de Rham complex of an associative algebra A. Here d is the Karoubi–de Rham differential and ıΔ is an operation analogous to contraction with a vector field. As a byproduct, we give a simple explicit construction of the Gauss–Manin connection, introduced earlier by E. Getzler, on the relative periodic cyclic homology of a flat family of associative algebras over a central base ring.We introduce and study free-product deformations of an associative algebra, a new type of deformation over a not necessarily commutative base ring. Natural examples of free-product deformations arise from preprojective algebras and group algebras for compact surface groups.

The convex minorant of a Lévy process

We offer a unified approach to the theory of convex minorants of L\'{e}vy processes with continuous distributions. New results include simple explicit constructions of the convex minorant of a L\'{e}vy process on both finite and infinite time intervals, and of a Poisson point process of excursions above the convex minorant up to an independent exponential time. The Poisson-Dirichlet distribution of parameter 1 is shown to be the universal law of ranked lengths of excursions of a L\'{e}vy process with continuous distributions above its convex minorant on the interval $[0,1]$.

Affine Slice for the Coadjoint Action of a Class of Biparabolic Subalgebras of a Semisimple Lie Algebra

In this article, we give a simple explicit construction of an affine slice for the coadjoint action of a certain class of biparabolic (also called seaweed) subalgebras of a semisimple Lie algebra over an algebraically closed field of characteristic zero. In particular, this class includes all Borel subalgebras.

Bianchi type A hyper-symplectic and hyper-Kähler metrics in 4D

We present a simple explicit construction of hyper-Kähler and hyper-symplectic (also known as neutral hyper-Kähler or hyper-para-Kähler) metrics in 4D using the Bianchi type groups of class A. The construction underlies a correspondence between hyper-Kähler and hyper-symplectic structures of dimension 4.

On the minimum number of completely 3-scrambling permutations

A family P = { π 1 , … , π q } of permutations of [ n ] = { 1 , … , n } is completely k-scrambling [Spencer, Acta Math Hungar 72; Füredi, Random Struct Algor 96] if for any distinct k points x 1 , … , x k ∈ [ n ] , permutations π i 's in P produce all k ! possible orders on π i ( x 1 ) , … , π i ( x k ) . Let N * ( n , k ) be the minimum size of such a family. This paper focuses on the case k = 3 . By a simple explicit construction, we show the following upper bound, which we express together with the lower bound due to Füredi for comparison. 2 log 2 e log 2 n ⩽ N * ( n , 3 ) ⩽ 2 log 2 n + ( 1 + o ( 1 ) ) log 2 log 2 n . We also prove the existence of lim n → ∞ N * ( n , 3 ) / log 2 n = c 3 . Determining the value c 3 and proving the existence of lim n → ∞ N * ( n , k ) / log 2 n = c k for k ⩾ 4 remain open.

Poisson Representation of a Ewens Fragmentation Process

A simple explicit construction is provided of a partition-valued fragmentation process whose distribution on partitions of [n] = 1,. . .,nat time θ ≥ 0 is governed by the Ewens sampling formula with parameter θ. These partition-valued processes are exchangeable and consistent, asnvaries. They can be derived by uniform sampling from a corresponding mass fragmentation process defined by cutting a unit interval at the points of a Poisson process with intensity θx−1dx on/mathbbR+, arranged to beintensifying as θ increases.

Explicit construction of attracting integral manifolds for a dissipative hyperbolic equation

A dissipative sine-Gordon-type equation in a bounded domain is considered. A simple explicit construction of finite-dimensional integral manifolds with exponential tracking is proposed.

Symmetries in Generalized Kähler Geometry

We define the notion of a moment map and reduction in both generalized complex geometry and generalized Kähler geometry. As an application, we give very simple explicit constructions of bi-Hermitian structures on $$\mathbb{C}\mathbb{P}^{N}$$ , Hirzebruch surfaces, the blow up of $$\mathbb{C}\mathbb{P}^{N}$$ at arbitrarily many points, and other toric varieties, as well as complex Grassmannians.

Construction of asymmetric connectors of depth two

An ( n , N ) -connector of depth d is an acyclic digraph with n inputs and N outputs in which for any injective mapping of input vertices into output vertices there exist n vertex disjoint paths of length at most d joining each input to its corresponding output. In this paper we consider the problem of construction of sparse depth two connectors with n ≪ N . We use posets of star products and their matching properties to construct such connectors. In particular, this gives a simple explicit construction for connectors of size O ( N log n / log log n ) . Thus our earlier idea to use other posets than the family of subsets of a finite set was successful.

The holomorphic contractibility of two generalized Teichmuller spaces

We show by simple explicit constructions that the Teichmüller spaces To(Δ) and To(Δ \ {0}) are holomorphically contractible. We also call attention to a useful criterion for a map between domains in complex Banach spaces to be holomorphic.

Entropy Waves, the Zig-Zag Graph Product, and New Constant-Degree Expanders

The main contribution of this work is a new type of graph product, which we call the zig-zag product. Taking a product of a large graph with a small graph, the resulting graph inherits (roughly) its size from the large one, its degree from the small one, and its expansion properties from both! Iteration yields simple explicit constructions of constant-degree expanders of arbitrary size, starting from one constant-size expander. Crucial to our intuition (and simple analysis) of the properties of this graph product is the view of expanders as functions which act as wave propagators -they transform probability distributions in which entropy is concentrated in one area to distributions where that concentration is dissipated. In these terms, the graph product affords the constructive interference of two such waves.