Unit Cube Research Articles

Recycling phosphogypsum in subbase of pavement: Treatment, testing, and application

This study aims to recycle the phosphogypsum (PG) as pavement subbase material. First, the physical and chemical properties of the PG before and after calcination treatment were characterized. Then the mix design, as well as the performance evaluation of the PG mixture, were detailed. Finally, the engineering application and economic analysis of the PG mixture for paving subbase pavement were presented. It is found that incorporation of the calcined PG (CPG) can help form gel structure to stabilize the PG. Meanwhile, addition of lime can neutralize the water-soluble phosphorus contained in the PG to improve the early strength of the CPG-stabilized PG. The optimal mix proportion of the CPG-lime stabilized PG by mass is determined as PG: CPG: lime = 89: 9: 2. In addition, as compared to the PG stabilized with the traditional inorganic binder materials, the CPG-lime stabilized PG has higher unconfined compressive strength (UCS) and better resistance to the water damage and fatigue damage caused by the water content variation. The field cores extracted from the paved CPG-lime stabilized PG subbase were experimentally observed to have a 7 d UCS as high as 3.69 MPa. Moreover, reuse of the CPG-lime stabilized PG is proven to be a highly economic and environment-friendly pathway for subbase paving, of which total cost per unit cube meter is 504.5 % lower than that of the commonly-used lime-fly ash stabilized soil. These findings identify the feasibility of utilizing the PG for pavement subbase construction.

Iterative square roots of functions

AbstractAn iterative square root of a self-map f is a self-map g such that $g(g(\cdot ))=f(\cdot )$ . We obtain new characterizations for detecting the non-existence of such square roots for self-maps on arbitrary sets. They are used to prove that continuous self-maps with no square roots are dense in the space of all continuous self-maps for various topological spaces. The spaces studied include those that are homeomorphic to the unit cube in ${\mathbb R}^{m}$ and to the whole of $\mathbb {R}^{m}$ for every positive integer $m.$ However, we also prove that every continuous self-map on a space homeomorphic to the unit cube in $\mathbb {R}^{m}$ with a fixed point on the boundary can be approximated by iterative squares of continuous self-maps.

Analysis of the economic effects on biopsychosocial homeostasis from a three-dimensional probability measurement system.

Describes the economic effects on bio-psychosocial homeostasis from a three-dimensional analysis of the joint interaction of biological, psychological and economic-social comorbidities, in order to know the probabilities of occurrence of these conditions simultaneously. It is a cross-sectional study and random quota sampling. 353 men and women over 20 years of age participated, and answered a 27 questions application (bio-psycho-economic-social), the data was analyzed three-dimensionally in order to know the probability of occurrence of the variables studied and their possible location in the probabilistic unit cube. Based on the unit cube proposal, it can be observed that in the block of people with incomes from 2200 to 6600 pesos in the group of men there is a probability of 12.9 per 1000 inhabitants of presenting debt problems, overweight and severe depression, while in the case of women, the probability is 2.08 per 1000 inhabitants. There is a priority axis that determines the change of the remaining variables, in this case it is the economic axis the one that determines the biological and psychological conditions.

The Resolution of Keller’s Conjecture

We consider three graphs, $$G_{7,3}$$ , $$G_{7,4}$$ , and $$G_{7,6}$$ , related to Keller’s conjecture in dimension 7. The conjecture is false for this dimension if and only if at least one of the graphs contains a clique of size $$2^7 = 128$$ . We present an automated method to solve this conjecture by encoding the existence of such a clique as a propositional formula. We apply satisfiability solving combined with symmetry-breaking techniques to determine that no such clique exists. This result implies that every unit cube tiling of $$\mathbb {R}^7$$ contains a facesharing pair of cubes. Since a faceshare-free unit cube tiling of $$\mathbb {R}^8$$ exists (which we also verify), this completely resolves Keller’s conjecture.

Robust numerical integration on curved polyhedra based on folded decompositions

We present a novel method to perform numerical integration over curved polyhedra enclosed by high-order parametric surfaces. Such a polyhedron is first decomposed into a set of triangular and/or rectangular pyramids, whose certain faces correspond to the given parametric surfaces. Each pyramid serves as an integration cell with a geometric mapping from a standard parent domain (e.g., a unit cube), where the tensor-product Gauss quadrature is adopted. As no constraint is imposed on the decomposition, certain resulting pyramids may intersect with themselves, and thus their geometric mappings may present negative Jacobian values. We call such cells folded cells and refer to the corresponding decomposition as a folded decomposition. We show that folded cells do not cause any issues in practice as they are only used to numerically compute certain integrals of interest. The same idea can be applied to planar curved polygons as well. We demonstrate both theoretically and numerically that folded cells can retain the same accuracy as the cells with strictly positive Jacobians. On the other hand, folded cells allow for a much easier and much more flexible decomposition for general curved polyhedra, on which one can robustly compute integrals. In the end, we show that folded cells can flexibly and robustly accommodate real-world complex geometries by presenting several examples in the context of immersed isogeometric analysis, where involved sharp features can be well respected in generating integration cells.

A symplectic embedding of the cube with minimal sections and a question by Schlenk

I prove that the open unit cube can be symplectically embedded into a longer polydisc in such a way that the area of each section satisfies a sharp bound and the complement of each section is path-connected. This answers a variant of a question by F. Schlenk.

Tensor rank bounds for point singularities in ℝ3

We analyze rates of approximation by quantized, tensor-structured representations of functions with isolated point singularities in ℝ3. We consider functions in countably normed Sobolev spaces with radial weights and analytic- or Gevrey-type control of weighted semi-norms. Several classes of boundary value and eigenvalue problems from science and engineering are discussed whose solutions belong to the countably normed spaces. It is shown that quantized, tensor-structured approximations of functions in these classes exhibit tensor ranks bounded polylogarithmically with respect to the accuracy ε ∈ (0,1) in the Sobolev space H1. We prove exponential convergence rates of three specific types of quantized tensor decompositions: quantized tensor train (QTT), transposed QTT and Tucker QTT. In addition, the bounds for the patchwise decompositions are uniform with respect to the position of the point singularity. An auxiliary result of independent interest is the proof of exponential convergence of hp-finite element approximations for Gevrey-regular functions with point singularities in the unit cube Q = (0,1)3. Numerical examples of function approximations and of Schrödinger-type eigenvalue problems illustrate the theoretical results.

A Szemerédi-type theorem for subsets of the unit cube

We investigate gaps of $n$-term arithmetic progressions $x, x+y, \ldots, x+(n-1)y$ inside a positive measure subset $A$ of the unit cube $[0,1]^d$. If lengths of their gaps $y$ are evaluated in the $\ell^p$-norm for any $p$ other than $1, 2, \ldots, n-1$, and $\infty$, and if the dimension $d$ is large enough, then we show that the numbers $\|y\|_{\ell^p}$ attain all values from an interval, the length of which depends only on $n$, $p$, $d$, and the measure of $A$. Known counterexamples prevent generalizations of this result to the remaining values of the exponent $p$. We also give an explicit bound for the length of the aforementioned interval. The proof makes the bound depend on the currently available bounds in Szemer\'{e}di's theorem on the integers, which are used as a black box. A key ingredient of the proof are power-type cancellation estimates for operators resembling the multilinear Hilbert transforms. As a byproduct of the approach we obtain a quantitative improvement of the corresponding (previously known) result for side lengths of $n$-dimensional cubes with vertices lying in a positive measure subset of $([0,1]^2)^n$.

Stabilization of a class of switched systems with state constraints via time-varying Lyapunov functions

This paper studies the stabilization problem of a class of continuous-time switched linear systems with state constraints under pre-specified dwell-time switchings. Such systems are defined on a closed hypercube as all state variables are constrained to the unit hypercube. The dwell time in this paper is an arbitrarily pre-specified rather than a calculated constant, which is independent of any parameters. First, a class of multiple time-varying Lyapunov functions is introduced to study the stability analysis, and sufficient conditions on stability of the studied switched systems without control input are derived in the framework of pre-specified dwell-time switchings. The distinguishing feature of the proposed Lyapunov functions is that this type of delicately constructed Lyapunov functions can efficiently eliminate the “jump” phenomena of adjacent Lyapunov functions at switching times. Second in the same framework of the dwell time, sufficient conditions for stabilization are proposed for that of the switched systems with state constraints by further designing state feedback controllers. Finally, two examples are provided to demonstrate the effectiveness of the proposed results. The results of this paper do not require to calculate the total time of each subsystem during which the state is saturated or non-saturated separately, which makes the pre-specified dwell-time switchings easy to apply.

Параллельное решение краевых задач с помощью технологии OpenMP

The paper introduces the results of the theoretical and experimental analysis of the Robust Multigrid Technique (RMT), a computational algorithm designed for the numerical solution of (initial-)boundary value problems for the equations of mathematical physics in black-box software. The purpose of the study was to develop robust, efficient and large-scale granulated algorithm for solving a wide class of nonlinear applied problems. The paper describes the algebraic and geometric parallelisms of the RMT and the multigrid cycle for solving nonlinear problems. The OpenMP technology was used to implement the parallel RMT. Computational experiments related to the solution of the Dirichlet problem for the Poisson equation in the unit cube were performed on a personal computer using 3, 9 and 27 threads (p = 3, 9, 27) and on a multiprocessor computer system with shared memory using 27 threads (p = 27). The highest achieved efficiency of the parallel RMT is E ≈ 0.95 at N > 106 and p = 3 and E ≈ 0.80 at N > 107 and p = 27. Findings of the research reveal that the determining factor affecting the efficiency of the parallel RMT is the limited memory performance of multicore computing systems. The complexity of the sequential iteration of the V-cycle and the parallel iteration of the RMT was theoretically analyzed. The study shows that the parallel iteration of the RMT, implemented on 27 threads, will be executed several times faster than the sequential iteration of the V-loop

A sharp discrepancy bound for jittered sampling

For m , d ∈ N m, d \in {\mathbb N} , a jittered (or stratified) sampling point set P P having N = m d N = m^d points in [ 0 , 1 ) d [0,1)^d is constructed by partitioning the unit cube [ 0 , 1 ) d [0,1)^d into m d m^d axis-aligned cubes of equal size and then placing one point independently and uniformly at random in each cube. We show that there are constants c > 0 c > 0 and C C such that for all d d and all m ≥ d m \ge d the expected non-normalized star discrepancy of a jittered sampling point set satisfies \[ c d m d − 1 2 1 + log ⁡ ( m d ) ≤ E D ∗ ( P ) ≤ C d m d − 1 2 1 + log ⁡ ( m d ) . c \,dm^{\frac {d-1}{2}} \sqrt {1 + \log (\tfrac md)} \le {\mathbb E} D^*(P) \le C\, dm^{\frac {d-1}{2}} \sqrt {1 + \log (\tfrac md)}. \] This discrepancy is thus smaller by a factor of Θ ( ( 1 + log ⁡ ( m / d ) m / d ) 1 / 2 ) \Theta \big (\big (\frac {1+\log (m/d)}{m/d}\big )^{1/2}\big ) than the one of a uniformly distributed random point set (Monte Carlo point set) of cardinality m d m^d . This result improves both the upper and the lower bound for the discrepancy of jittered sampling given by Pausinger and Steinerberger [J. Complexity 33 (2016), pp. 199–216]. It also removes the asymptotic requirement that m m is sufficiently large compared to d d .

RETRACTED - THE KRONECKER–WEYL EQUIDISTRIBUTION THEOREM AND GEODESICS IN 3-MANIFOLDS

Abstract Given any rectangular polyhedron $3$ -manifold $\mathcal {P}$ tiled with unit cubes, we find infinitely many explicit directions related to cubic algebraic numbers such that all half-infinite geodesics in these directions are uniformly distributed in $\mathcal {P}$ .

Periodic heat transfer characteristics of additively manufactured lattices

In the present study, the flow and heat transfer characteristics of lattices with four different unit cell topologies, namely, (a) Octet, (b) FD-Cube, (c) Tetrakaidecahedron (TKD), and (d) Cube were investigated with air as the working fluid. All the samples had the designed porosity of ∼ 0.88 and unit cell size of 10 mm. The distribution of local heat transfer coefficient revealed that depending on the unit cell topology, the flow became thermally developed after travelling over five or six rows of unit cells in streamwise direction. The heat transfer coefficient values in thermally developed zones were similar for Octet, FD-Cube and TKD. Overall heat transfer coefficient provided by Octet, FD-Cube, and TKD was in the range of ∼216 W/m2K to 415W/m2K and Cube yielded the lowest heat transfer coefficient values between 167W/m2K and 290 W/m2K.Three different characteristic length scales, viz. (a) channel hydraulic diameter, (b) pore-diameter, and (c) fiber-diameter were used to define the Nusselt number. The usage of pore-scale dimensions such as pore-diameter and fiber-diameter provided a very coherent data set for Nusselt number of all the topologies which helped in deriving a single correlation for predicting the thermal performance. The lattices provided heat transfer augmentation of 10-20 times higher than the smooth channel counterparts. Amongst all the investigated topologies, FD-Cube incurred the highest and Cube yielded the lowest pressure drop. It was also observed that convective heat transfer coefficients obtained for different lattices can be conveniently reported in Nusselt number forms with pore diameter and/or fiber diameters as reference length scales, where such Nusselt numbers for different lattice configurations converges to a single trendline when plotted against Reynolds numbers (also based on same reference length scales). We have provided a single correlation for Nusselt numbers as a function of Reynolds numbers which fits all the lattice configurations studied here. Finally, the thermal-hydraulic performance of lattices is reported which had similar magnitude levels for all the investigated topologies and a single correlation has been proposed for THP covering all the configurations of present study. The investigated topologies provided THP of ∼ 3 at the lowest investigated Reynolds number.

Minimal dispersion of large volume boxes in the cube

In this note we present a construction which improves the best known bound on the minimal dispersion of large volume boxes in the unit cube. Let d>1. The dispersion of T⊂[0,1]d is defined as the supremum of the volume taken over all axis parallel boxes in the cube which do not intersect T. The minimal dispersion of n points in the cube is defined as the infimum of the dispersion taken over all T such that |T|=n. Define the “large volume” regime as the class of all volumes 14<r≤12. The inverse of the minimal dispersion is denoted as N(r,d). When the volume is large, the best known upper bound on N(r,d) is of the order (r−14)−1. The construction presented in this note yields an upper bound given byN(r,d)≤⌊πr−14⌋−3. Some of our intermediate estimates are sharp given the condition that d≥Cr, where Cr is a positive constant which depends only on the volume r.

Development of climatic zoning and energy demand prediction for Ethiopian cities in degree days

The energy need for comfort rises as the living standard and urbanization escalate. This study intends to regionalize the climatic zones of Ethiopia and estimate the energy needs of respective regions based on the prevalent method of degree days. For this temperature data for 952 National Center for Environmental Prediction (NCEP) stations from 1979 to 2013 were used to compute the heating degree days (HDD) and cooling degree days (CDD) with base temperature of 18.3 °C and to regionalize the country into 5 ASHRAE standard zones. Within each zone, five reference cities were identified for their population size and urbanization level, namely, Goba Robe, Robe, Addis Ababa, Dire Dawa and Gode. The result has shown that the eastern and western peripheries of the country (402,552 km2) attained high CDD (2659 DD) and low HDD. The central spine of the country attained high HDD (29,981 km2) and low CDD. Between the two there are intermediate zones (zone 3 and 4) of 706,178 km2 coverage. The yearly total energy demand for comfort in Ethiopia is estimated to be 0.068 to 0.168 kJ (kJ)/°C in a unit cube of room space. That means, if all households are furnished with AC and standard of living raised, with the population size of 143,028,465 in 2030, Ethiopia demands to produce 4,689,205.7 kJ of energy.

Exact order of extreme L_p discrepancy of infinite sequences in arbitrary dimension

We study the extreme L_p discrepancy of infinite sequences in the d-dimensional unit cube, which uses arbitrary sub-intervals of the unit cube as test sets. This is in contrast to the classical star L_p discrepancy, which uses exclusively intervals that are anchored in the origin as test sets. We show that for any dimension d and any p>1, the extreme L_p discrepancy of every infinite sequence in [0,1)^d is at least of order of magnitude (log N)^{d/2}, where N is the number of considered initial terms of the sequence. For p in (1,infty ), this order of magnitude is best possible.

New bounds on the minimal dispersion

We provide a new construction for a set of boxes approximating axis-parallel boxes of fixed volume in [0,1]d. This improves upper bounds for the minimal dispersion of a point set in the unit cube and its inverse in both the periodic and non-periodic settings in certain regimes. In the case of random choice of points our bounds are sharp up to double logarithmic factor. We also apply our construction to k-dispersion.

A note on concatenation of quasi-Monte Carlo and plain Monte Carlo rules in high dimensions

In this note, we study a concatenation of quasi-Monte Carlo and plain Monte Carlo rules for high-dimensional numerical integration in weighted function spaces. In particular, we consider approximating the integral of periodic functions defined over the s-dimensional unit cube by using rank-1 lattice point sets only for the first d(<s) coordinates and random points for the remaining s−d coordinates. We prove that, by exploiting a decay of the weights of function spaces, almost the optimal order of the mean squared worst-case error is achieved by such a concatenated quadrature rule as long as d scales at most linearly with the number of points. This result might be useful for numerical integration in extremely high dimensions, such as partial differential equations with random coefficients for which even the standard fast component-by-component algorithm is considered computationally expensive.

Approximate Bayesian conditional copulas

Copula models are flexible tools to represent complex structures of dependence for multivariate random variables. According to Sklar's theorem, any multidimensional absolutely continuous distribution function can be uniquely represented as a copula, i.e. a joint cumulative distribution function on the unit hypercube with uniform marginals, which captures the dependence structure among the vector components. In real data applications, the interest of the analyses often lies on specific functionals of the dependence, which quantify aspects of it in a few numerical values. A broad literature exists on such functionals, however extensions to include covariates are still limited. This is mainly due to the lack of unbiased estimators of the conditional copula, especially when one does not have enough information to select the copula model. Several Bayesian methods to approximate the posterior distribution of functionals of the dependence varying according covariates are presented and compared; the main advantage of the investigated methods is that they use nonparametric models, avoiding the selection of the copula, which is usually a delicate aspect of copula modelling. These methods are compared in simulation studies and in two realistic applications, from civil engineering and astrophysics.

On some similarity of finite sets (and what we can say today about certain old problem)

The paper presents a new way of classifying numerical sets (division into abstraction classes, called cuts hereafter in this paper) after imposing a constraint on the sum of elements of their subsets. A simple and not too expensive operation of sorting a set (with maximum nlog(n), sorting by merge sort) allowed to determine pairs of subsets (which shall be described in the paper as pairs in a solid compound) representing a given set with the accepted constraint, which in turn allowed to prove various properties and limit the number of necessary comparisons. Then, the properties of the ratio of the sum of elements to the constraint within its cut were demonstrated, and to this cut (the abstraction class) an interval on the real number axis was assigned. To illustrate the obtained results, a mutation tree is used as an image of a unit hypercube. For introduced concepts, for example, their application for the classical number partitioning problem is shown, which is NP-complete according to the classification (which was proved by Richard Karp in 1972 in his work [1] and classified it as a subproblem of the knapsack problem).