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A classical technique to construct polynomial preserving extensions of scalar functions defined on the boundary of an n simplex to the interior is to use so-called rational blending functions. The purpose of this paper is to generalize the construction by blending to the de Rham complex. More precisely, we define polynomial preserving extensions which map traces of k-forms defined on the boundary of the simplex to k-forms defined in the interior. Furthermore, the extensions are cochain maps, i.e., they commute with the exterior derivative. [ABSTRACT FROM AUTHOR]

In one dimension, nodal locations that are distinct are necessary and sufficient to ensure that a unique polynomial interpolant exists for data provided at a set of nodes, i.e. that the set of nodes is unisolvent. In multiple dimensions however, unisolvency for a polynomial interpolant of degree p is not ensured even with nodal locations that are distinct and a set of n nodes, with n equal to the cardinality of a set of polynomial basis functions of at most degree p. In this paper a set of equations is derived for simplices of one to three dimensions with symmetrical nodal distributions to identify a combination of symmetry orbits that can provide a unisolvent set of nodes. The results suggest that there is a unique combination of symmetry orbits that can provide a unisolvent set of nodes for each degree of polynomial interpolant. Consequently, all other combinations of symmetry orbits cannot provide a unisolvent set of nodes for a degree p polynomial interpolant. This is verified numerically up to degree 10 for triangles and degree 7 for tetrahedra. The results suggest that the same is also true for higher-order polynomial interpolants. This significantly reduces the number of combination of symmetry orbits that needs to be considered. For example, for a tetrahedron with a degree seven interpolant, only one combination of symmetry orbits needs to be considered instead of the 161 different combinations of symmetry orbits that provide a set of nodes with n equal to the cardinality of the set of basis functions of at most degree seven. For a symmetrical nodal distribution in a simplex, the conditions presented are necessary but not sufficient to have a unisolvent set of nodes for polynomial interpolation. [ABSTRACT FROM AUTHOR]

A clutter is clean if it has no delta or the blocker of an extended odd hole minor, and it is tangled if its covering number is two and every element appears in a minimum cover. Clean tangled clutters have been instrumental in progress towards several open problems on ideal clutters, including the τ = 2 Conjecture. Let C be a clean tangled clutter. It was recently proved that C has a fractional packing of value two. Collecting the supports of all such fractional packings, we obtain what is called the core of C. The core is a duplication of the cuboid of a set of 0 − 1 points, called the setcore of C. In this paper, we prove three results about the setcore. First, the convex hull of the setcore is a full-dimensional polytope containing the center point of the hypercube in its interior. Secondly, this polytope is a simplex if, and only if, the setcore is the cocycle space of a projective geometry over the two-element field. Finally, if this polytope is a simplex of dimension more than three, then C has the clutter of the lines of the Fano plane as a minor. Our results expose a fascinating interplay between the combinatorics and the geometry of clean tangled clutters. [ABSTRACT FROM AUTHOR]

In this paper, we study dot-product sets and k-simplices in Z n d for odd n, where Z n is the ring of residues modulo n. We show that if E is sufficiently large then the dot-product set of E covers the whole ring. In higher dimensional cases, if E is sufficiently large then the set of simplices and the set of dot-product simplices determined by E, up to congurence, have positive densities. [ABSTRACT FROM AUTHOR]

This work investigates finite differences and the use of (diagonal) quadratic interpolation models to obtain approximations to the first and (non-mixed) second derivatives of a function. Here, it is shown that if a particular set of points is used in the interpolation model, then the solution to the associated linear system (i.e., approximations to the gradient and diagonal of the Hessian) can be obtained in O (n) computations, which is the same cost as finite differences, and is a saving over the O (n 3) cost when solving a general unstructured linear system. Moreover, if the interpolation points are chosen in a particular way, then the gradient approximation is O (h 2) accurate, where h is related to the distance between the interpolation points. Numerical examples confirm the theoretical results. [ABSTRACT FROM AUTHOR]

For each positive integer m and each real finite dimensional Banach space X, we set β (X , m) to be the infimum of δ ∈ (0 , 1 ] such that each set A ⊂ X having diameter 1 can be represented as the union of m subsets of A whose diameters are at most δ . Elementary properties of β (X , m) , including its stability with respect to X in the sense of Banach-Mazur metric, are presented. Two methods for estimating β (X , m) are introduced. The first one estimates β (X , m) using the knowledge of β (Y , m) , where Y is a Banach space sufficiently close to X. The second estimation uses the information about β X (K , m) , the infimum of δ ∈ (0 , 1 ] such that K ⊂ X is the union of m subsets having diameters not greater than δ times the diameter of K, for certain classes of convex bodies K in X. In particular, we show that β (l p 3 , 8) ≤ 0.925 holds for each p ∈ [ 1 , + ∞ ] by applying the first method, and we proved that β (X , 8) < 1 whenever X is a three-dimensional Banach space satisfying β X (B X , 8) < 221 328 , where B X is the unit ball of X, by applying the second method. These results and methods are closely related to the extension of Borsuk's problem in finite dimensional Banach spaces and to C. Zong's computer program for Borsuk's conjecture. [ABSTRACT FROM AUTHOR]

A rule for constructing interpolation nodes for nth degree polynomials on the simplex is presented. These nodes are simple to define recursively from families of 1D node sets, such as the Lobatto{Gauss{Legendre (LGL) nodes. The resulting nodes have attractive properties: they are fully symmetric, they match the 1D family used in construction on the edges of the simplex, and the nodes constructed for the (d1)-simplex are the boundary traces of the nodes constructed for the d-simplex. When compared using the Lebesgue constant to other explicit rules for defining interpolation nodes, the nodes recursively constructed from LGL nodes are nearly as good as the warp & blend nodes of Warburton [J. Engrg. Math., 56 (2006), pp. 247{262] in 2D (which, though defined differently, are very similar) and in 3D are better than other known explicit rules by increasing margins for n > 6. By that same measure, these recursively defined nodes are not as good as implicitly defined nodes found by optimizing the Lebesgue constant or related functions, but such optimal node sets have yet to be computed for the tetrahedron. A reference Python implementation has been distributed as the recursivenodes package, but the simplicity of the recursive construction makes them easy to implement. [ABSTRACT FROM AUTHOR]

Particle tracking is a computationally advantageous and fast scheme to determine travel times and trajectories in subsurface hydrology. Accurate particle tracking requires element-wise mass-conservative, conforming velocity fields. This condition is not fulfilled by the standard linear Galerkin finite element method (FEM). We present a projection, which maps a non-conforming, element-wise given velocity field, computed on triangles and tetrahedra, onto a conforming velocity field in lowest-order Raviart-Thomas-Nédélec ( R T N 0 ) space, which meets the requirements of accurate particle tracking. The projection is based on minimizing the difference in the hydraulic gradients at the element centroids between the standard FEM solution and the hydraulic gradients consistent with the R T N 0 velocity field imposing element-wise mass conservation. Using the conforming velocity field in R T N 0 space on triangles and tetrahedra, we present semi-analytical particle tracking methods for divergent and non-divergent flow. We compare the results with those obtained by a cell-centered finite volume method defined for the same elements, and a test case considering hydraulic anisotropy to an analytical solution. The velocity fields and associated particle trajectories based on the projection of the standard FEM solution are comparable to those resulting from the finite volume method, but the projected fields are smoother within zones of piecewise uniform hydraulic conductivity. While the R T N 0 -projected standard FEM solution is thus more accurate, the computational costs of the cell-centered finite volume approach are considerably smaller. [ABSTRACT FROM AUTHOR]

A lattice d-simplex is the convex hull of d+1 affinely independent integer points in Rd. It is called empty if it contains no lattice point apart from its d+1 vertices. The classification of empty 3-simplices has been known since 1964 (White), based on the fact that they all have width one. But for dimension 4 no complete classification is known. Haase and Ziegler (2000) enumerated all empty 4-simplices up to determinant 1000 and based on their results conjectured that after determinant 179 all empty 4-simplices have width one or two. We prove this conjecture as follows: - We show that no empty 4-simplex of width three or more can have a determinant greater than 5058, by combining the recent classification of hollow 3-polytopes (Averkov, Krumpelmann and Weltge, 2017) with general methods from the geometry of numbers. We continue the computations of Haase and Ziegler up to determinant 7600, and find that no new 4-simplices of width larger than two arise. In particular, we give the whole list of empty 4-simplices of width larger than two, which is as computed by Haase and Ziegler: There is a single empty 4-simplex of width four (of determinant 101), and 178 empty 4-simplices of width three, with determinants ranging from 41 to 179. [ABSTRACT FROM AUTHOR]

Let K⊂Rn be a convex body with barycenter at the origin. We show there is a simplex S⊂K having also barycenter at the origin such that (vol(S)vol(K))1/n≥cn, where c>0 is an absolute constant. This is achieved using stochastic geometric techniques. Precisely, if K is in isotropic position, we present a method to find centered simplices verifying the above bound that works with extremely high probability. By duality, given a convex body K⊂Rn we show there is a simplex S enclosing Kwith the same barycenter such that vol(S)vol(K)1/n≤dn,for some absolute constant d>0. Up to the constant, the estimate cannot be lessened. [ABSTRACT FROM AUTHOR]

For a d-dimensional convex polytope $$P\subset {\mathbb {R}}^d$$ , the outer normal unit vectors of its facets span another d-dimensional polytope, which is called the outer normal transform of P and denoted by $$P^*$$ . It seems that this transform, which clearly differs from the usual polarity transform, was not seriously investigated until now. We derive results about polytopes having natural properties with respect to this transform, like self-duality or equality for the composition of it. Among other things we prove that if P is a convex polytope inscribed in the unit sphere with the property that the circumradii of its facets are all equal, then $$(P^*)^*$$ coincides with P. We also prove that the converse is true when P or $$P^*$$ has at most 2 d vertices. For the three-dimensional case, we characterize the family of those tetrahedra T such that T and $$T^*$$ are congruent. [ABSTRACT FROM AUTHOR]

Explicit generators for high-order ( r>1) scalar and vector finite element spaces generally used in numerical electromagnetism are presented and classical degrees of freedom, the so-called moments, revisited. Properties of these generators on simplicial meshes are investigated, and a general technique to restore duality between moments and generators is proposed. Algebraic and exponential optimal h- and r-error rates are numerically validated for high-order edge elements on the problem of Maxwell's eigenvalues in a square domain. [ABSTRACT FROM AUTHOR]

In this paper we apply a group action approach to the study of Erdős-Falconer-type problems in vector spaces over finite fields and use it to obtain non-trivial exponents for the distribution of simplices. We prove that there exists such that if , , with , then , where denotes the set of congruence classes of k-dimensional simplices determined by -tuples of points from E. Non-trivial exponents were previously obtained by Chapman, Erdogan, Hart, Iosevich and Koh [] for with . A non-trivial result for in the plane was obtained by Bennett, Iosevich and Pakianathan []. These results are significantly generalized and improved in this paper. In particular, we establish the Wolff exponent , previously established in [] for the case to the case , and this results in a new sum-product type inequality. We also obtain non-trivial results for subsets of the sphere in , where previous methods have yielded nothing. The key to our approach is a group action perspective which quickly leads to natural and effective formulae in the style of the classical Mattila integral from geometric measure theory. [ABSTRACT FROM AUTHOR]

We study upper bounds on the number of lattice points for convex bodies having their centroid at the origin. For the family of simplices as well as in the planar case we obtain best possible results. For arbitrary convex bodies we provide an upper bound, which extends the o-symmetric case and which, in particular, shows that the centroid assumption is indeed much more restrictive than an assumption on the number of interior lattice points even for the class of lattice polytopes. [ABSTRACT FROM AUTHOR]

There are three constructions of which I know that yield higher dimensional analogues of Sierpinski's triangle. The most obvious is to remove the open convex hull of the midpoints of the edges of the -simplex. The complement is a union of simplices. Continue the removal recursively in each of the remaining sub-simplices. The result is an uncountably infinite figure in -dimensional space that is Cantor-like in a manner analogous to the Sierpinski triangle. A countable analogue is obtained by means of playing the chaos game in the -simplex. In this 'game' one chooses a random -ary sequence; starting from the initial point (that is identified with a vertex of the simplex), one continues to plot points by moving half-again as much towards the next point in the sequence. The resulting plot converges to the figure described above. Similarly, coloring the multinomial coefficients black or white according to their parity results in a similar figure, when the -dimensional analogue of the Pascal triangle is rescaled and embedded in space. [ABSTRACT FROM AUTHOR]

Let d and t be fixed positive integers, and let K t , … , t d denote the complete d -partite hypergraph with t vertices in each of its parts, whose hyperedges are the d -tuples of the vertex set with precisely one element from each part. According to a fundamental theorem of extremal hypergraph theory, due to Erdős [6] , the number of hyperedges of a d -uniform hypergraph on n vertices that does not contain K t , … , t d as a subhypergraph, is n d − 1 t d − 1 . This bound is not far from being optimal. We address the same problem restricted to intersection hypergraphs of ( d − 1 ) -dimensional simplices in R d . Given an n -element set S of such simplices, let H d ( S ) denote the d -uniform hypergraph whose vertices are the elements of S , and a d -tuple is a hyperedge if and only if the corresponding simplices have a point in common. We prove that if H d ( S ) does not contain K t , … , t d as a subhypergraph, then its number of edges is O ( n ) if d = 2 , and O ( n d − 1 + ϵ ) for any ϵ > 0 if d ≥ 3 . This is almost a factor of n better than Erdős's above bound. Our result is tight, apart from the error term ϵ in the exponent. In particular, for d = 2 , we obtain a theorem of Fox and Pach [7] , which states that every K t , t -free intersection graph of n segments in the plane has O ( n ) edges. The original proof was based on a separator theorem that does not generalize to higher dimensions. The new proof works in any dimension and is simpler: it uses size-sensitive cuttings , a variant of random sampling. [ABSTRACT FROM AUTHOR]

We introduce an axiom system for a collection of matchings that describes the triangulation of product of simplices.

We prove that both, the repeated interior point based subdivision and the LE m -section ( m ⩾ 4 ) of an initial n -simplex in R n ( n ⩾ 2 ) , produce a sequence of simplicial meshes with minimum interior solid angles converging to zero. [ABSTRACT FROM AUTHOR]

Abstract: We provide an analytical description of the Minkowski sum of simplices in . The convex polytope is determined by the coordinates of facets' vertices, may the polytope be degenerate or not. The pre-established description of the polytope allows a rapid implementation using popular applications, here achieved using Matlab®. [Copyright &y& Elsevier]

Let G be any group and A be a non-empty subset of G. The h -fold product set of A is defined as A h : = { a 1 ⋅ a 2 ⋯ a h : a 1 , ... , a h ∈ A }. Nathanson considered the concept of an asymptotic approximate group. Let r , l ∈ Z > 0. The set A is said to be an (r , l) -approximate group in G if there exists a subset X in G such that | X | ⩽ l and A r ⊆ X A. The set A is an asymptotic (r , l) -approximate group if the product set A h is an (r , l) -approximate group for all sufficiently large h. Recently, Nathanson showed that every finite subset A of an abelian group is an asymptotic (r , l ′) -approximate group (with the constant l ′ explicitly depending on r and A). In this article, our motivations are three-fold: (1) We give an alternate proof of Nathanson's result. (2) From the alternate proof we deduce an improvement in the bound on the explicit constant l ′. (3) We generalise the result and show that, in an arbitrary abelian group G , the union of k (unbounded) generalised arithmetic progressions is an asymptotic (r , (4 r k) k) -approximate group. [ABSTRACT FROM AUTHOR]

During the last 40 years, simplicial partitioning has been shown to be highly useful, including in the field of nonlinear optimization, specifically global optimization. In this article, we consider results on the exhaustivity of simplicial partitioning schemes. We consider conjectures on this exhaustivity which seem at first glance to be true (two of which have been stated as true in published articles). However, we will provide counter-examples to these conjectures. We also provide a new simplicial partitioning scheme, which provides a lot of freedom, whilst guaranteeing exhaustivity. [ABSTRACT FROM AUTHOR]

We show that finding the simplices containing a fixed given point among those defined on a set of n points can be done in O(n + k) time for the two-dimensional case, and in O(n2 + k) time for the three-dimensional case, where k is the number of these simplices. As a byproduct, we give an alternative (to the algorithm in Ref. 4) O(n log r) algorithm that finds the red-blue boundary for n bichromatic points on the line, where r is the size of this boundary. Another byproduct is an O(n2 + t) algorithm that finds the intersections of line segments having two red endpoints with those having two blue endpoints defined on a set of n bichromatic points in the plane, where t is the number of these intersections. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we discuss a family of multipoint flux mixed finite element (MFMFE) methods on simplicial, quadrilateral, hexahedral, and triangular-prismatic grids. The MFMFE methods are locally conservative with continuous normal fluxes, since they are developed within a variational framework as mixed finite element methods with special approximating spaces and quadrature rules. The latter allows for local flux elimination giving a cell-centered system for the scalar variable. We study two versions of the method: with a symmetric quadrature rule on smooth grids and a non-symmetric quadrature rule on rough grids. Theoretical and numerical results demonstrate first order convergence for problems with full-tensor coeffcients. Second order superconvergence is observed on smooth grids. [Copyright &y& Elsevier]

We develop explicit, piecewise-linear formulations of functions f( x):ℝ n ↦ℝ, n≤3, that are defined on an orthogonal grid of vertex points. If mixed-integer linear optimization problems (MILPs) involving multidimensional piecewise-linear functions can be easily and efficiently solved to global optimality, then non-analytic functions can be used as an objective or constraint function for large optimization problems. Linear interpolation between fixed gridpoints can also be used to approximate generic, nonlinear functions, allowing us to approximately solve problems using mixed-integer linear optimization methods. Toward this end, we develop two different explicit formulations of piecewise-linear functions and discuss the consequences of integrating the formulations into an optimization problem. [ABSTRACT FROM AUTHOR]

It is shown that repeated subdivision of triangles by their medians into six smaller triangles tends to yield increasingly thin triangles having lognormally distributed thicknesses whose geometric-mean value approaches zero as the subdivision proceeds, a triangle's 'thickness' being defined as the ratio of its shortest altitude to its longest side. [ABSTRACT FROM AUTHOR]

Abstract: Intersecting hypersurfaces in classical Lovelock gravity are studied exploiting the description of the Lovelock Lagrangian as a sum of dimensionally continued Euler densities. We wish to present an interesting geometrical approach to the problem. The analysis allows us to deal most efficiently with the division of spacetime into a honeycomb network of cells produced by an arbitrary arrangement of membranes of matter. We write the gravitational action as bulk terms plus integrals over each lower dimensional intersection. The spin connection is discontinuous at the shared boundaries of the cells, which are spaces of various dimensionalities. That means that at each intersection there are more than one spin connections. We introduce a multi-parameter family of connections which interpolate between the different connections at each intersection. The parameters live naturally on a simplex. We can then write the action including all the intersection terms in a simple way. The Lagrangian of Lovelock gravity is generalized so as to live on the simplices as well. Each intersection term of the action is then obtained as an integral over an appropriate simplex. Lovelock gravity and the associated topological (Euler) density are used as an example of a more general formulation. In this example one finds that singular sources up to a certain co-dimensionality naturally carry matter without introducing conical or other singularities in spacetime geometry. [Copyright &y& Elsevier]

A geometric formulation of the generalized Law of Sines for simplices in constant curvature spaces is presented. It is explained how the Law of Sines can be seen as an instance of the so-called polar duality, which can be formulated as a duality between Gram matrices representing the simplex. [ABSTRACT FROM AUTHOR]

A geometric formulation of the generalized Law of Sines for simplices in constant curvature spaces is presented. It is explained how the Law of Sines can be seen as an instance of the so-called polar duality, which can be formulated as a duality between Gram matrices representing the simplex. [ABSTRACT FROM AUTHOR]

We study small holes through which regular 3-, 4-, and 5-dimensional simplices can pass. [ABSTRACT FROM AUTHOR]

Abstract: Let be a positive integer and be the free abelian lattice-ordered group on generators. We prove that and do not satisfy the same first-order sentences in the language if . We also show that is decidable iff . Finally, we apply a similar analysis and get analogous results for the free finitely generated vector lattices. [Copyright &y& Elsevier]

We establish in this paper some inequalities for vertex distances of two simplices, and give some applications thereof. [Copyright &y& Elsevier]

We establish in this paper some inequalities for medians and volumes of two simplices whose form is analogous to the Neuberg-Pedoe inequality. [Copyright &y& Elsevier]

Column generation is an increasingly popular basic tool for the solution of large-scale mathematical programming problems. As problems being solved grow bigger, column generation may however become less efficient in its present form, where columns typically are not optimizing, and finding an optimal solution instead entails finding an optimal convex combination of a huge number of them. We present a class of column generation algorithms in which the columns defining the restricted master problem may be chosen to be optimizing in the limit, thereby reducing the total number of columns needed. This first article is devoted to the convergence properties of the algorithm class, and includes global (asymptotic) convergence results for differentiable minimization, finite convergence results with respect to the optimal face and the optimal solution, and extensions of these results to variational inequality problems. An illustration of its possibilities is made on a nonlinear network flow model, contrasting its convergence characteristics to that of the restricted simplicial decomposition (RSD) algorithm. [ABSTRACT FROM AUTHOR]

A statistical model for global optimization is constructed generalizing some properties of the Wiener process to the multidimensional case. An approach to the construction of global optimization algorithms is developed using the proposed statistical model. The convergence of an algorithm based on the constructed statistical model and simplicial partitioning is proved. Several versions of the algorithm are implemented and investigated. [Copyright &y& Elsevier]