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Several artists, artisans, and mathematicians described fascinating solid bodies in the fifteenth and sixteenth centuries. The knowledge they developed on the subject was so progressive that it is considered a milestone in the history of polyhedra. In the first part of this study we analyze, from a chronological and comparative perspective, the consistent studies developed between 1460 and 1583 on those that came to be recognized as Archimedean Solids. The authors who engaged in such studies were Piero della Francesca, Luca Pacioli, Leonardo da Vinci, Albrecht Dürer, Augustin Hirschvogel, an Anonymous Author who accomplished remarkable studies between 1538 and 1556, Wentzel Jamnitzer, Daniele Barbaro, Lorenz Stöer, Rafael Bombelli, and Simon Stevin. In the second part, we discuss how the revolutionary method of describing solid bodies with planar nets contributed to the rediscovery of the Archimedean Solids. We also present our interpretation of some of the studies by the Anonymous Author and our conclusions on his identity and influence on other authors. [ABSTRACT FROM AUTHOR]

With an educational issue that has caught the attention of many countries in the world (study load), a population of 8th graders from a typical Chinese metropolitan city (40,536 from 118 schools), and an advanced statistical strategy (multilevel piecewise regression), we examined whether there was a turning point in terms of the effects of study load on science achievement. We did identify a turning point for each and every measure of study load. For weekday learning on science achievement, we identified a turning point of 22.50 hr for the effects of in-school learning, 7.50 hr for the effects of homework, and 12 hr for the effects of after-school learning. For weekend learning on science achievement, we identified a turning point of 1.50 hr for the effects of in-school learning, 5 hr for the effects of homework, and 1 hr for the effects of after-school learning. In each case, the difference in effects before and after the turning point was statistically significant, indicating that the effects of study load on science achievement were nonlinear. All of these turning points offered important implications for science education. [ABSTRACT FROM AUTHOR]

This study investigates when and how university students in first-semester introductory calculus interpret multiple representations of the same function. Specifically, it focuses on three tasks. The first task has students give their definitions of 'function sameness', the results of which suggests that many students understand a function as being determined by more than its points. The second has them assess the derivative of a piecewise-defined representation of a typical polynomial function, the results of which suggest that students see a piecewise function as two separate functions with the conditions as instructions. The third involves an instance of the fundamental theorem of calculus, the results of which suggest that students view (what we see as) the same function as two different functions. Taken together, these results suggest that many students see two functions where we as mathematicians see only one. In particular, students see various features of a function's analytic and graphical representations as being essential to the function's identity. [ABSTRACT FROM AUTHOR]

We give a procedure to construct trisections for closed manifolds generated by colored tensor models without restrictions on the number of simplices in the triangulation, therefore generalizing previous works in the context of crystallizations and PL-manifolds. We give a description of how trisection diagrams can arise from colored tensor model graphs for closed 4-manifolds. We further speculate on generalization of similar constructions for a class of singular-manifolds generated by simplicial colored tensor models. [ABSTRACT FROM AUTHOR]

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The paper considers periodic selector-linear difference inclusions. A class of timeperiodic quasi-quadratic Lyapunov functions is distinguished, as well as parametric classes of piecewise-quadratic and piecewise-linear Lyapunov functions. These functions establish necessary and sufficient conditions for asymptotic stability. An example leading to periodic selector-linear differential and difference inclusions is given. The results can find applications in the stability analysis of control systems with periodic parameters, in particular, servomechanisms whose elements operate on alternating current, control systems with amplitude-frequency modulation. [ABSTRACT FROM AUTHOR]

The aim of this article is to obtain convergence in mean in the uniform topology of piecewise linear approximations of stochastic differential equations (SDEs) with C1 drift and C2 diffusion coefficients with uniformly bounded derivatives. Convergence analyses for such Wong-Zakai approximations most often assume that the coefficients of the SDE are uniformly bounded. Almost sure convergence in the unbounded case can be obtained using now standard rough path techniques, although Lq convergence appears yet to be established and is of importance for several applications involving Monte-Carlo approximations. We consider L2 convergence in the unbounded case using a combination of traditional stochastic analysis and rough path techniques. We expect our proof technique extend to more general piecewise smooth approximations. [ABSTRACT FROM AUTHOR]

Generally, there is no analytical method to calculate the chaotic solution of a dynamical system. In this paper, a simple proof of the existence of analytical solution in a discontinuous 2D discrete time piecewise chaotic map is reported, i.e., we show that a map that has a simple analytical solution can also have a chaotic attractor by the use of the classical definition of the Jordan canonical form defined for matrices available in most kinds of literature on linear algebra. [ABSTRACT FROM AUTHOR]

In the previous paper [2], we have classified simple regular polyhedral BP-complexes, which are polyhedral complexes satisfying certain natural conditions (B) and (P) on their vertex structures. A regular polyhedral complex X is defined as satisfying the condition (B) if the diameter of the vertex structure of X is equal to its injectivity radius. On the other hand, we have shown in [3] that 2-skeletons of higher dimensional regular polytopes do not satisfy the condition (B). So it seems very likely that the diameter of the vertex structure and its injective radius are not equal. In this short paper, we consider the geometric implications of the diameter and the injectivity radius of the vertex structure of regular polyhedral complexes. [ABSTRACT FROM AUTHOR]

This paper investigates practical stability for a class of fractional-order impulsive control coupled systems with noninstantaneous impulses on networks. Using graph theory and Lyapunov method, new criteria for practical stability, uniform practical stability as well as practical asymptotic stability are established. In this paper, we extend graph theory to fractional-order system via piecewise Lyapunov-like functions in each vertex system to construct global Lyapunovlike functions. Our results are generalization of some known results of practical stability in the literature and provide a new method of impulsive control law for impulsive control systems with noninstantaneous impulses. Examples are given to illustrate the effectiveness of our results. [ABSTRACT FROM AUTHOR]

We show that the reduced SL2(C)-twisted Burau representation can be obtained from the quantum group Uq(sl2) for q = i a fourth root of unity and that representations of Uq.sl2/satisfy a type of Schur-Weyl duality with the Burau representation. As a consequence, the SL2.C/-twisted Reidemeister torsion of links can be obtained as a quantum invariant. Our construction is closely related to the quantum holonomy invariant of Blanchet-Geer-Patureau-Mirand-Reshetikhin, and we interpret their invariant as a twisted Conway potential. [ABSTRACT FROM AUTHOR]

In this paper the problem of synchronization for delayed chaotic systems is considered based on aperiodic intermittent control. First, delayed chaotic systems are proposed via aperiodic adaptive intermittent control. Next, to cut down the control gain, a new generalized intermittent control and its adaptive strategy is introduced. Then, by constructing a piecewise Lyapunov auxiliary function and making use of piecewise analysis technique, some effective and novel criteria are obtained to ensure the global synchronization of delayed chaotic systems by means of the designed control protocols. At the end, two examples with numerical simulations are provided to verify the effectiveness of the theoretical results proposed scheme. [ABSTRACT FROM AUTHOR]

We propose a new modified Parker's method for efficient gravitational forward modeling and inversion using general polyhedral models. We have made several important modifications to the classical method, including: (a) The new method is now applicable to a general polyhedron represented by triangulated surface or tetrahedral mesh, and with arbitrarily variable 3D density distribution. (b) An optimal Fourier‐domain sampling strategy is used to improve the numerical accuracy of the new algorithm significantly. (c) A simple and effective automatic layering technique is introduced to accelerate the convergence rate of Parker's method. The method is demonstrated using both synthetic and real polyhedral models, including a sphere model approximated by a polyhedron, two asteroids, a digital elevation model in the Himalaya region, and the Yucca Flat basin model in Nevada. The numerical results show that, compared with analytical solutions of polyhedral models in the space domain, the modified Parker's method can improve the computational efficiency by several orders of magnitude while obtaining almost the same simulation results. The difference is well below existing instrumentation error level. By embedding the new forward algorithm into an iterative process, it can be used for fast inversion of density interfaces. Our new method is suitable for the efficient modeling and inversion of gravitational potential, gravitational vector, and gravitational gradient tensor caused by polyhedral models with a large number of faces, representing geological abnormal bodies, asteroids, and single or multilayer density interface models with triangulated surfaces. Plain Language Summary: The original Parker's method for gravitational forward modeling of layered models using regular grid data is modified to include general polyhedron models with irregular surface and either constant or variable density distributions. An optimal Fourier‐domain sampling strategy, together with an effective automatic layering technique, are introduced to improve the numerical accuracy and convergence rate of the new modified Parker's method. A combination of the forward algorithm with an iterative process provides an effective way for the inversion of density interfaces. We have successfully applied the new algorithm to the gravitational forward modeling of a synthetic sphere model, two asteroids, a digital elevation model in the Himalaya region, and to the inversion of basement structure of Yucca Flat basin (Nevada). Key Points: Parker's method for layered models with regular grids is modified to include polyhedrons with irregular surfaces and arbitrary densityOptimal spectral sampling and automatic layering techniques are introduced to improve the accuracy and convergence of the new methodThe new method has been successfully applied to the gravity forward and inverse modeling of synthetic and real‐world polyhedral models [ABSTRACT FROM AUTHOR]

In this paper, we consider a nonautonomous piecewise linear difference equation that describes a discrete version of a single neuron model with a periodic (period two and period three) internal decay rate. We investigated the periodic behavior of solutions relative to the periodic internal decay rate in our previous papers. Our goal is to prove that this model contains a large quantity of initial conditions that generate eventually periodic solutions. We will show that only periodic solutions and eventually periodic solutions exist in several cases. [ABSTRACT FROM AUTHOR]

A classical question in PL topology, asked among others by Hudson, Lickorish, and Kirby, is whether every linear subdivision of the d-simplex is simplicially collapsible. The answer is known to be positive for d ≤ 3 . We solve the problem up to one subdivision, by proving that any linear subdivision of any polytope is simplicially collapsible after at most one barycentric subdivision. Furthermore, we prove that any linear subdivision of any star-shaped polyhedron in R d is simplicially collapsible after d - 2 derived subdivisions at most. This presents progress on an old question by Goodrick. [ABSTRACT FROM AUTHOR]

Trajectory piecewise linear (TPWL) approximation is a well-known model order reduction technique for non-linear systems. It involves weighted summation of the dimensionally reduced linear systems obtained at certain points on the system trajectory. The points at which the linearised systems are produced are called linearization points (LP). The LPs are important as they decide the quality and the complexity of the approximation. The criterion of selecting LPs play a crucial role as the fidelity of the reduced model depends upon it. According to the previous methods, a new linearised model is formed whenever the distance of the linear model being simulated from all the previously selected LPs is greater than some chosen distance. This distance is heuristically chosen in the conventional LP selection schemes. This article proposes a new criterion of selecting the distance and the LPs that not only removes the ambiguity but also enhances the performance of the approximation. With the proposed scheme, much smaller number of LPs are needed and the distances are computed automatically, instead of being arbitrarily selected. A number of relevant examples are presented, showing the advantages of the method, both in the number of LPs and in the error of the TPWL approximation. [ABSTRACT FROM AUTHOR]

In the last few years, the interest for studying the piecewise linear differential systems has increased strongly, mainly due to their applications to many physical phenomena. In the study of these differential systems, the limit cycles play a main role. Up to now, the major part of papers which study the limit cycles of the piecewise linear differential systems consider only two pieces. Here, we consider piecewise linear differential systems with three pieces. In this paper, we study the limit cycles of the discontinuous piecewise linear differential systems in the plane R 2 formed by three arbitrary linear centers separated by the set Σ = { (x , y) ∈ R 2 : y = 0 {or} x = 0 {and} y ≥ 0 }. We prove that such discontinuous piecewise linear differential systems can have 1 , 2 , or 3 limit cycles, with 3 the maximum number of limit cycles that such systems can have. Moreover, the limit cycles are nested and must intersect Σ in three or four points. The limit cycles having three intersection points with Σ can reach the maximum number 3. The limit cycles having four intersection points with Σ are at most 1 , and if it exists, the systems could simultaneously have 1 or 2 limit cycles intersecting Σ in three points. [ABSTRACT FROM AUTHOR]

We prove that the Scott-Vogelius finite elements are inf-sup stable on shape-regular meshes for piecewise quartic velocity fields and higher (k ≥ 4). [ABSTRACT FROM AUTHOR]

Hypernym-hyponym ("is-a") relations are key components in taxonomies, object hierarchies and knowledge graphs. Robustly harvesting of such relations requires the analysis of the linguistic characteristics of is-a word pairs in the target language. While there is abundant research on is-a relation extraction in English, it still remains a challenge to accurately identify such relations from Chinese knowledge sources due to the flexibility of language expression and the significant differences between the two language families. In this paper, we introduce a weakly supervised framework to extract Chinese is-a relations from user-generated categories. It employs piecewise linear projection models trained on an existing Chinese taxonomy built from Wikipedia and an iterative learning algorithm to update model parameters incrementally. A pattern-based relation selection method is proposed to prevent "semantic drift" in the learning process using bi-criteria optimization. Experimental results on the publicly available test set illustrate that the proposed approach outperforms state-of-the-art methods. [ABSTRACT FROM AUTHOR]

We establish the regularity results for solutions of nonlocal Venttsel' problems in polygonal and piecewise smooth two-dimensional domains. [ABSTRACT FROM AUTHOR]

A piecewise linear weighting function for Black-Scholes implied volatilities is used in conjunction with the Black-Scholes call pricing model to approximate stochastic volatility European call prices. A sensitivity analysis is conducted to compare simulated stochastic volatility call prices to the Black-Scholes prices calculated with the weighted implied volatilities. The analysis indicates that a simple model can provide close approximations to the simulated prices with far less computational effort. [ABSTRACT FROM AUTHOR]

Abstract The cell-wall thickness induced and cell-wall angle induced piecewise linear graded honeycombs (PLGHs) are presented and investigated under in-plane compression. Effects of cell-wall angle on the strength of honeycombs are discussed. Results reveal that the normalized crushing strength displays similar variety to that of the shape ratio within a given range. Based on this, traditional theoretical models of plateau stresses are improved. Comparative studies on different kinds of PLGHs are conducted. Quasi-static and dynamic deformation mode, strength and energy absorption capacity are intensively discussed. Both weak-to-strong and top-to-bottom layer-by-layer deformation mode are observed. Comparison of strain-stress curves shows that cell-wall angle induced PLGHs generally express lower level of stress than that of cell-wall thickness induced PLGHs under quasi-static compression, but the difference is insignificant as the crushing velocity increases. Theoretical prediction is also conducted to estimate the stress level under quasi-static and high crushing velocities. Both the constant velocity and initial velocity loadings are applied to studying the energy absorption capacity. It is interesting to note that cell-wall angle induced PLGHs shows a time saving characteristic as the normalized initial kinetic energy increases from 0.75 to 1, which has not been discovered in other types of honeycombs. [ABSTRACT FROM AUTHOR]

A novel chaotic image encryption scheme based on hash function and cyclic shift is designed. The key streams are dependent on the hash value of the plain text and changed in every encryption process. The one-time keys are generated by the hash values of the plain images to get six initial values of the piecewise linear chaotic map (PWLCM). Then, the PWLCM system will produce three pseudo-random sequences to diffusion and confusion image. The mask operation and new fast ranks transformation method are used to encrypt image based on pixel level. Cycle shift is applied to the encryption system to change the position and the value of the pixels based on bit level. Many experiments are done and security analyses are discussed, which show that the novel algorithm has good effect and it can resist common attacks. [ABSTRACT FROM AUTHOR]

Highlights • Piecewise model predicting sphere vertical distance and velocity time evolution upon vertical impact on liquid surface. • Model compared to experimental data for varying sphere mass ratios and impact velocities. • Additional model of cavity pinch-off confirmed to match experimental results well for full range of operating conditions. • Relatively simple model performs as well as complicated models; experimentally verified to deeper depths past pinch-off. • Light spheres affected by trailing cavity pinch-off and vortex ring formation for which the model does not account. Abstract This study presents a piecewise model for determining the vertical distance and velocity evolution with time for a sphere impacting a water surface and submerging to depths beyond deep-seal cavity pinch-off. Experimental data taken with a high-speed camera are presented for varying sphere mass ratios and impact velocities. The semi-empirical model incorporates results from previously published research and is shown to be in good agreement with experiments for heavier spheres but deviates when the sphere is only slightly denser than water. Two causes for the deviation are presented which relate to the dynamics of the cavity pinch-off event and the inception of a trailing vortex ring after the trailing cavity sloughs from the sphere. A model for predicting cavity pinch-off time and sphere position and velocity at the moment of cavity pinch-off is shown to agree well with experimental results for varying sphere mass ratios and impact velocities. The key experimental values are provided for comparison with current and future modeling efforts. [ABSTRACT FROM AUTHOR]

In this work, we study the nonlinear travelling waves in density stratified fluids with piecewise-linear shear currents. Beginning with the formulation of the water-wave problem due to Ablowitz et al. (J. Fluid Mech., vol. 562, 2006, pp. 313-343), we extend the work of Ashton & Fokas (J. Fluid Mech., vol. 689, 2011, pp. 129-148) and Haut & Ablowitz (J. Fluid Mech., vol. 631, 2009, pp. 375-396) to examine the interface between two fluids of differing densities and varying linear shear. We derive a systems of equations depending only on variables at the interface, and numerically solve for periodic travelling wave solutions using numerical continuation. Here, we consider only branches which bifurcate from solutions where there is no slip in the tangential velocity at the interface for the trivial flow. The spectral stability of these solutions is then determined using a numerical Fourier-Floquet technique. We find that the strength of the linear shear in each fluid impacts the stability of the corresponding travelling wave solutions. Specifically, opposing shears may amplify or suppress instabilities. [ABSTRACT FROM AUTHOR]

Highlights • Land cover proportion can almost substitute for area-related landscape configuration indicators. • Landscape composition has relatively stable correlation values at different scales. • Vegetated land surfaces composition is more indicative than unvegetated land surfaces. • Landscape metrics are better correlated with LST in high temperature than in low temperature. • Unvegetated land surfaces proportion weakly influences the correlations (R2 < 0.2). Abstract Identification of the relationship between landscape metrics and urban land surface temperature (LST) provides a basic understanding of the interaction of landscape pattern and ecological process. However, the evidenced relationships between landscape metrics and LST are still uncertain, and cannot provide fundamental support to landscape management. Other than a test of statistical significance, four judgments (median correlation value; temporal variation and statistical scale; threshold; and effects of key variables) are considered "efficient" criteria in this study. Partial correlation and piecewise linear regression are used to focus on the indicators of land cover proportion, biophysical proportion, and mainly area- or shape-related landscape configuration indicators. The results show: (1) land cover proportion can almost substitute for area-related landscape configuration indicators; (2) landscape composition is more efficient than configuration as an indicator because of its relatively stable temporal correlation values at different statistical scales; (3) the evident landscape composition threshold of vegetated land surfaces (50–70% for land cover proportion and 0.2–0.3 for biophysical proportion) is more indicative in application than the linear relationship for unvegetated land surfaces; and (4) landscape metrics are better correlated with LST in high temperature than in low temperature, and urban area weakly influences this correlation (R2 < 0.2). Additionally, the configuration metrics at a landscape level are not recommended for characterizing LST. Depended on the efficient indicators, such as the application of thresholds, landscape planning can be linked to the quantitative observations and statistical evidence in landscape metrics. [ABSTRACT FROM AUTHOR]

Dubey et al. [40] have shown that solving an Atanassov's I-fuzzy Linear Programming Problem represented by Atanassov's I-fuzzy sets with linear membership and non-membership functions is equivalent to solving an appropriate fuzzy optimisation problem with piecewise linear S-shaped membership functions. The equivalence is established using Hurwicz optimism-pessimism criterion [38] and indeterminacy resolution in Atanassov's I-fuzzy sets. Moreover, in case of convex break points in the piecewise linear membership function, the crisp counterpart of the equivalent optimisation problem involves binary variables. Here, in this paper we first convert the resulting fuzzy optimisation problem having convex break points into an equivalent fuzzy optimisation problem having concave break points on the lines of Inuiguchi et al. [34], before formulating its crisp equivalent. The advantage of this strategy is that the resulting crisp equivalent problem has no binary variables. Further, we also make use of the indeterminacy factor resolution principle to establish a duality relation which can be interpreted as a Atanassov's I-fuzzy variant of the (crisp) weak duality theorem. [ABSTRACT FROM AUTHOR]

We reconsider a regime-switching model of credit frictions which has been proposed in a general framework by Matsuyama for the case of Cobb-Douglas production functions. This results in a piecewise linear map with two discontinuity points and all three branches having the same slope. We offer a complete characterization of the bifurcation structure in the parameter space, as well as of the attracting sets and related basins of attraction in the phase space. We also discuss parameter regions associated with overshooting, leapfrogging, poverty traps, reversal of fortune and growth miracle, as well as cycles with any kind of switching between the expansionary and contractionary phases. [ABSTRACT FROM AUTHOR]

Simon’s congruence, denoted by ∼k, relates the words having the same subwords of length at most k. In this paper a normal form is presented for the equivalence classes of ∼k. The length of this normal form is the shortest possible. Moreover, a canonical solution of the equation pwq∼kr is also shown (the words p, q, r are parameters), which can be viewed as a generalization of giving a normal form for ∼k. In this paper, there can be found an algorithm with which the canonical solution can be determined in O((L+n)nk) time, where L denotes the length of the word pqr and n is the size of the alphabet. [ABSTRACT FROM AUTHOR]

Narendra-Shapiro (NS) algorithms are bandit-type algorithms developed in the 1960s. NS-algorithms have been deeply studied in infinite horizon but little non-asymptotic results exist for this type of bandit algorithms. In this paper, we focus on a non-asymptotic study of the regret and address the following question: are Narendra-Shapiro bandit algorithms competitive from this point of view? In our main result, we obtain some uniform explicit bounds for the regret of (over)-penalized-NS algorithms. We also extend to the multi-armed case some convergence properties of penalized-NS algorithms towards a stationary Piecewise Deterministic Markov Process (PDMP). Finally, we establish some new sharp mixing bounds for these processes. [ABSTRACT FROM AUTHOR]

In this paper we study the limit cycles of some classes of piecewise smooth vector fields defined in the two dimensional torus. The piecewise smooth vector fields that we consider are composed by linear, Ricatti with constant coefficients and perturbations of these one, which are given in (3). Considering these piecewise smooth vector fields we characterize the global dynamics, studying the upper bound of number of limit cycles, the existence of non-trivial recurrence and a continuum of periodic orbits. We also present a family of piecewise smooth vector fields that posses a finite number of fold points and, for this family we prove that for any 2k number of limit cycles there exists a piecewise smooth vector fields in this family that presents k number of limit cycles and prove that some classes of piecewise smooth vector fields presents a non-trivial recurrence or a continuum of periodic orbits. [ABSTRACT FROM AUTHOR]

We study the limit cycles of two families of piecewise-linear differential systems in R 3 with two pieces separated by a plane Σ . In one family the differential systems are continuous on the plane Σ , and in the other family they are discontinuous on the plane Σ . The usual tool for studying these limit cycles is the Poincaré map, but here we shall use recent results which extend the averaging theory to continuous and discontinuous differential systems. All the computations have been done with the algebraic manipulator Mathematica. [ABSTRACT FROM AUTHOR]

We consider in this article the properties of topological conjugacy of the tent-like maps f v : [ 0 , 1 ] → [ 0 , 1 ] , which are piecewise linear and whose graph consists of straight line segments, extending from (0 , 0) to (v , 1) to (1 , 0) , where v ∈ (0 , 1) is a parameter. For any point x ∈ [ 0 , 1 ] , we reduce the calculation of the derivative of the conjugacy h of functions f v 1 and f v 2 to the limit of a recurrently defined sequence, which is defined by x. In the case v 1 = 1 / 2 , this result is reduced to a study of some properties of the binary expansion of x. [ABSTRACT FROM AUTHOR]

In this paper, the robust peak-to-peak filtering problem for periodic piecewise systems with polytopic uncertainties is investigated. Attention is focused on designing of robust peak-to-peak filters that guarantee the robust asymptotic stability of periodic piecewise filtering error system and satisfy a prescribed peak-to-peak disturbance attenuation level for all admissible uncertainties. A sufficient condition is proposed for robust stability and peak-to-peak performance by employing the constructed time-varying Lyapunov function. Two design approaches based on the Lyapunov function with parameter-dependent matrices and parameter-independent matrices are utilised to solve this problem. An algorithm is proposed to compute the periodic filter parameters governed by non-convex feasibility conditions. Two numerical examples are presented to demonstrate the effectiveness of the proposed techniques. [ABSTRACT FROM AUTHOR]

In this paper, we propose an optimization-based slotting approach to automated ICs design for generating a power transistor. The original slotting problem is formulated as a mixed-integer linear program. It is solved through parametric optimization with advantage that usage of any commercial optimization solvers on user’s side is avoided with short evaluation time and simple implementation. The approach is applied in specific very large scale integration production technology and demonstrated on an example. [ABSTRACT FROM AUTHOR]

In 1938, Tarski proved that a formula is not intuitionistically valid if, and only if, it has a counter-model in the Heyting algebra of open sets of some topological space. In fact, Tarski showed that any Euclidean space R n with n ⩾ 1 suffices, as does e.g. the Cantor space. In particular, intuitionistic logic cannot detect topological dimension in the Heyting algebra of all open sets of a Euclidean space. By contrast, we consider the lattice of open subpolyhedra of a given compact polyhedron P ⊆ R n , prove that it is a locally finite Heyting subalgebra of the (non-locally-finite) algebra of all open sets of P , and show that intuitionistic logic is able to capture the topological dimension of P through the bounded-depth axiom schemata. Further, we show that intuitionistic logic is precisely the logic of formulæ valid in all Heyting algebras arising from polyhedra in this manner. Thus, our main theorem reconciles through polyhedral geometry two classical results: topological completeness in the style of Tarski, and Jaśkowski's theorem that intuitionistic logic enjoys the finite model property. Several questions of interest remain open. E.g., what is the intermediate logic of all closed triangulable manifolds? [ABSTRACT FROM AUTHOR]

We consider the wide class of all piecewise Chebyshevian splines with connection matrices at the knots. We prove that a spline space of this class is “good for interpolation” if and only if the spline space obtained by integration is “good for design”. As a consequence, this provides us with a simple practical description of all such spline spaces which can be used for solving Hermite interpolation problems. These results strongly rely on the properties of blossoms. [ABSTRACT FROM AUTHOR]

Consider a mechanical linkages whose underlying graph is a polygon with a diagonal constraint, or more general, a partial two-tree. We show that (with an appropriate definition) the oriented area is a Bott–Morse function on the configuration space of the linkage. Its critical points are described and Bott–Morse indices are computed. This paper is a generalization of analogous results for polygonal linkages (obtained earlier by G. Khimshiashvili, G. Panina, and A. Zhukova). [ABSTRACT FROM AUTHOR]

We develop a coupled volume of fluid and level set (VOSET) method for unstructured quadrilateral grids to simulate incompressible two-phase interfacial flows in irregular domains. In the method, an analytic piecewise linear interface calculation (PLIC) is first proposed and its solution speed is much faster (about five times) than that of Brent's iterative method; then an iterative geometric operation by which the level set function near interfaces can be calculated, is extended to unstructured quadrilateral grids; moreover, the volume fraction advection is solved by a Lagrangian-Eulerian advection scheme. Finally, the present method is validated by the single vortex flow problem, bubble rising problem and falling droplet problem. Our simulation results show good agreement with those in previous studies. [ABSTRACT FROM AUTHOR]

The paper presents an improved plane layout for stabilizing piles based on a proposed piecewise function expression for the irregular driving force. Based on the specific morphological characteristics of a highway landslide, the piecewise function is used to calculate the irregular driving force by dividing the landslide into several sub-areas. Furthermore, the reasonable layout range and pile spacing can be obtained based on the piecewise function expression of the irregular driving force and on relevant research results of the plane layout for stabilizing piles. Therefore, an improved plane layout of stabilizing piles is presented in consideration of a piecewise function expression of the irregular driving force. A highway landslide located in eastern Guizhou Province, China, is analyzed as a case study using the proposed method. The results demonstrate that the theory presented in this paper provides improved economic benefits and can reduce the required number of stabilizing piles by 28.6% compared with the conventional plane layout scheme. [ABSTRACT FROM AUTHOR]

Given α ∈ (0, 1) and c = h + iβ, h, β ∈ R, the function fα,c: R → C defined as follows is considered: (1) fα,c is Hermitian, i.e., fα,c(−x)fα,c(x)¯,x∈ℝ;, x ∈ R; (2) fα,c(x) = 0 for x > 1; moreover, on each of the closed intervals [0, α] and [α, 1], the function fα,c is linear and satisfies the conditions fα,c(0) = 1, fα,c(α) = c, and fα,c(1) = 0. It is proved that the complex piecewise linear function fα,c is positive definite on R if and only if m(α) ≤ h ≤ 1 − α and |β| ≤ γ(α, h), m(α)={0if1/α∉ℕ,−αif1/α∈ℕ. If m(α) ≤ h ≤ 1 − α and α ∈ Q, then γ(α, h) > 0; otherwise, γ(α, h) = 0. This result is used to obtain a criterion for the complete monotonicity of functions of a special form and prove a sharp inequality for trigonometric polynomials. [ABSTRACT FROM AUTHOR]

Piecewise potential vorticity inversion (PPVI) seeks to determine the impact of observed potential vorticity (PV) anomalies on the surrounding flow. This widely used technique is based on dividing a flow domain D into subdomains D1 and D2 = D − D1. The influence of PV in D1 on the flow in D2 is assessed by removing all PV anomalies in D2 and then inverting the modified PV in D. The resulting flow with streamfunction ψ1 is attributed to the PV anomalies in D1. The relation of PV in D1 to ψ1 in D2 is not unique, because there are many PV distributions in D1 that induce the same ψ1. There is, however, a unique solution if the ageostrophic circulation is included in the inversion procedure. The superposition principle requires that the sum of inverted flows with PV = 0 in D2 and the complementary ones with PV = 0 in D1 equal the inverted flow for the complete observed PV in D. It is demonstrated, using two isolated PV balls as a paradigmatic example, that the superposition principle is violated if the ageostrophic circulation is included in PPVI, because the ageostrophic circulation cannot be associated with only one of the anomalies. Inversions of Ertel's PV are carried out using Charney's balance condition. PPVI is not unique in that case, because many different PV fields can be specified in D1, which all lead to the same inverted flow in D2. The balance condition assumes vanishing vertical velocity w so that uniqueness cannot be established by including w in the inversion, as was possible in the quasigeostrophic case. [ABSTRACT FROM AUTHOR]