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A signed graph is a graph with edges marked positive and negative; it is unbalanced if some cycle has negative sign product. We introduce the concept of vector valued switching function in signed graphs, which extends the concept of switching to higher dimensions. Using this concept, we define balancing dimension and strong balancing dimension for a signed graph, which can be used for a new classification of degree of imbalance of unbalanced signed graphs. We provide bounds for the balancing and strong balancing dimensions, and calculate these dimensions for some classes of signed graphs. [ABSTRACT FROM AUTHOR]

This comprehensive investigation explores the application and significance of q-differential calculus in the realm of vector functions of several variables, addressing critical aspects such as q-Rolle’s theorem, the q-Mean-value theorem, and q-chain rule for vector functions. Additionally, we investigate the q-gradient, q-Jacobian, and q-Hessian operators, elucidating their roles in quantifying rates of change, determining directional derivatives, and characterizing critical points of multivariate functions. Furthermore, this research provides a rigorous treatment of Multivariate and Bivariate Taylor theorems in the context of q-differential calculus, presenting analytical expansions of functions around specific points and showcasing their utility in approximating functions in higher dimensions. The q-Maximum and Minimum are demonstrated and discussed as well. [ABSTRACT FROM AUTHOR]

The accurate representation of objects holds pivotal significance in the realm of panoptic segmentation. Presently, prevalent object representation methodologies, including box-based, keypoint-based and query-based techniques, encounter a challenge known as the 'representation confusion' issue in specific scenarios, often resulting in the mislabeling of instances. In response, this paper introduces Convex Object Representation (COR), a straightforward yet highly effective approach to address this problem. COR leverages a CNN-based Euclidean Distance Transform to convert the target instance into a convex heatmap. Simultaneously, it offers a parallel embedding method for encoding the object. Subsequently, COR characterizes objects based on the distinctive embedding vectors of their convex vertices. This paper seamlessly integrates COR into a state-of-the-art query-based panoptic segmentation framework. Experimental findings validate that COR successfully mitigates the representation confusion predicament, enhancing segmentation accuracy. The COR-augmented methods exhibit notable improvements of +1.3 and +0.7 points in PQ on the Cityscapes validation and MS COCO panoptic 2017 validation datasets, respectively. [ABSTRACT FROM AUTHOR]

The construction of explicit structures of conserved vectors plays diverse crucial roles in the study of nonlinear science inclusive of the fact that they are invoked in developing appropriate numerical schemes and for other mathematical analyses. Therefore, in this paper, we examine the conserved quantities of a nonlinear wave equation, existing in three dimensions, and highlight their applications in physical sciences. The robust technique of the Lie group theory of differential equations (DEs) is invoked to achieve analytic solutions to the equation. This technique is used in a systematic way to generate the Lie point symmetries of the equation under study. Consequently, an optimal system of one-dimensional (1-D) Lie subalgebras related to the equation is obtained. Thereafter, we engage the formal Lagrangian of the nonlinear wave equation in conjunction with various gained subalgebras to construct conservation laws of the equation under study using Ibragimov's theorem for conserved vectors. [ABSTRACT FROM AUTHOR]

Data normalization is a crucial step in multi-criteria decision-making (MCDM) processes. The choice of data normalization method significantly influences the ranking of alternatives. The available data normalization methods in the MARA (Magnitude of the Area for the Ranking of Alternatives) method may not be applicable in certain cases. This study aims to broaden the application scope of the MARA method. Therefore, an investigation into the compatibility of data normalization methods when combined with the MARA method has been conducted. Ten data normalization methods were utilized, including Linear normalization, Weitendorf normalization, Sum linear normalization, Vector normalization, Logarithmic normalization, Max linear normalization, Min linear normalization, Jüttler-Körth normalization, Peldschus normalization, and Stop normalization. The compatibility between the MARA method and these ten data normalization methods was tested in five different scenarios. In the initial four scenarios, variations in the number of alternatives, criteria, and criterion types were introduced. Eight out of the ten data normalization methods were proven to be suitable for integration with the MARA method. In the fifth scenario, a hypothetical situation was presented where the data normalization methods available in the MARA method could not be used. Alternative data normalization methods were employed, and their combination with the MARA method was compared to using other multi-criteria decision-making methods. The results affirmed the accuracy of these combinations. This exploration has expanded the application scope of the MARA method compared to its original version. [ABSTRACT FROM AUTHOR]

Fractal geometry plays an important role in the description of the characteristics of nature. Local fractional calculus, a new branch of mathematics, is used to handle the non-differentiable problems in mathematical physics and engineering sciences. The local fractional inequalities, local fractional ODEs and local fractional PDEs via local fractional calculus are studied. Fractional calculus is also considered to express the fractal behaviors of the functions, which have fractal dimensions. The interesting problems from fractional calculus and fractals are reported. With the scaling law, the scaling-law vector calculus via scaling-law calculus is suggested in detail. Some special functions related to the classical, fractional, and power-law calculus are also presented to express the Kohlrausch–Williams–Watts function, Mittag-Leffler function and Weierstrass–Mandelbrot function. They have a relation to the ODEs, PDEs, fractional ODEs and fractional PDEs in real-world problems. Theory of the scaling-law series via Kohlrausch–Williams–Watts function is suggested to handle real-world problems. The hypothesis for the tempered Xi function is proposed as the Fractals Challenge, which is a new challenge in the field of mathematics. The typical applications of fractal geometry are proposed in real-world problems. [ABSTRACT FROM AUTHOR]

In this paper, the scaling-law vector calculus, which is connected between the vector calculus and the scaling law in fractal geometry, is addressed based on the Leibniz derivative and Stieltjes integral for the first time. The scaling-law Gauss–Ostrogradsky-like, Stokes-like and Green-like theorems, and Green-like identities are considered in sense of the scaling-law vector calculus. The strong and weak conjectures for the scaling-law flows are obtained in detail. The obtained result is a potentially mathematical tool proposed to develop an important way of approaching this challenge for the scaling-law flows. [ABSTRACT FROM AUTHOR]

In this paper, we use differential forms to prove a number of theorems of integral vector calculus that are rarely found in textbooks. Two of them, as far as the author knows, have not been published before. Some possible applications to problems in physics are shared including a general approach for computing net forces and torques on current-carrying loops that yields insights that are not evident from the standard approach. Readers who are familiar with differential forms will enjoy seeing how they can be employed to prove several new vector calculus identities. But even readers who do not follow those derivations will benefit from seeing how these identities can be employed to find forces and torques on current-carrying loops. [ABSTRACT FROM AUTHOR]

We derive the optimal rate of convergence for the mean squared error at the terminal point for anticipating linear stochastic differential equations, where the integral is interpreted in Skorohod sense. Although alternative proof techniques are needed, our results can be seen as generalizations of the corresponding results for Itō SDEs. As a key tool we extend optimal approximation results for vectors of correlated Wiener integrals to general random vectors, which contain the solutions of our Skorohod SDEs. [ABSTRACT FROM AUTHOR]

Researchers are showing great interest in Generative Adversarial Networks (GANs), which use deep learning techniques to mimic the content of datasets and are particularly adept at data generation. Despite their impressive performance, there is uncertainty about how GANs precisely map latent space vectors to realistic images and how the chosen dimensionality of the latent space affects the quality of the generated images. In this paper, we explored the potential of generative models in generating animal face images. For this purpose, we used the Deep Convolutional Generative Adversarial Network (DCGAN) model as a reference. To analyze the impact of selected latent space vectors, we synthesized animal face images by training data representations in the DCGAN model with the well-known AFHQ dataset from the literature. We compared the quantitative evaluation of the produced images using Fréchet Inception Distance (FID) and Inception Score (IS). As a result, we demonstrated that generative models can produce images with latent sizes significantly smaller and larger than the standard size of 100. [ABSTRACT FROM AUTHOR]

Over the past decade, the techniques of topological data analysis (TDA) have grown into prominence to describe the shape of data. In recent years, there has been increasing interest in developing statistical methods and in particular hypothesis testing procedures for TDA. Under the statistical perspective, persistence diagrams - the central multi-scale topological descriptors of data provided by TDA - are viewed as random observations sampled from some population or process. In this context, one of the earliest works on hypothesis testing focuses on the two-group permutation-based approach where the associated loss function is defined in terms of within-group pairwise bottleneck or Wasserstein distances between persistence diagrams [42]. However, in situations where persistence diagrams are large in size or number, the permutation test in question gets computationally more costly to apply. To address this limitation, we instead consider pairwise distances between vectorized functional summaries of persistence diagrams for the loss function. In the present work, we explore the utility of the Betti function in this regard, which is one of the simplest function summaries of persistence diagrams. We introduce an alternative vectorization method for the Betti function based on integration and prove stability results with respect to the Wasserstein distance. Moreover, we propose a new shuffling technique of group labels to increase the power of the test. Through several experimental studies, on both synthetic and real data, we show that the vectorized Betti function leads to competitive results compared to the baseline method involving the Wasserstein distances for the permutation test. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new look at Finsler surfaces. Landsberg surfaces are Finsler surfaces that are solutions of a system of non-linear partial differential equations. Considering the unicorn's Landsberg problem, we reduce this system to a single non-linear PDE which we call the Landsberg's PDE. By making use of the new look of Finsler surfaces, we solve the Landsberg's PDE and get a class of solutions. Moreover, we show that these solutions and their conformal transformations are Berwaldain. [ABSTRACT FROM AUTHOR]

In this study, we investigate the relations between a multiple partitioned linear model and its correctly-reduced models. We consider the comparison problem of covariance matrices of the best linear unbiased predictors (BLUPs) of all unknown vectors including partial parameters under these models by using the block matrix rank and inertia formulas. We derive various inequalities and equalities for covariance matrices of BLUPs under some general assumptions. Also, results for special cases are given. [ABSTRACT FROM AUTHOR]

The calculation of the view factor is an important consideration in radiative heat transfer analysis. Thermal engineering requires the view factor calculations between complex geometries containing parallel and inclined surfaces. There is currently no analytic formula available for this, so the summation rule (SR) must be used. However, when complex geometries are analyzed, processing SR becomes a very intensive task. Therefore, the present study aims to exploit vector calculus to develop a formula for calculating the view factors for radiative heat transfer from any two rectangular oblique or parallel surfaces. The exact solution obtained by means of the contour integration was compared with the numerical integration of the quadruple integral using the multiple Simpson's rule 1/3 with five intervals (MSR) and the summation rule via the Hamilton–Morgan solution (HMS). The lack of similar precedents in the literature reinforces the scientific and practical nature of the proposed method. [ABSTRACT FROM AUTHOR]

Fractional vector calculus is the building block of the fractional partial differential equations that model non-local or long-range phenomena, e.g., anomalous diffusion, fractional electromagnetism, and fractional advection-dispersion. In this work, we reformulate a type of fractional vector calculus that uses Caputo fractional partial derivatives and discretize this reformulation using discrete exterior calculus on a cubical complex in the structure-preserving way, meaning that the continuous-level properties curl α grad α = 0 and div α curl α = 0 hold exactly on the discrete level. We discuss important properties of our fractional discrete exterior derivatives and verify their second-order convergence in the root mean square error numerically. Our proposed discretization has the potential to provide accurate and stable numerical solutions to fractional partial differential equations and exactly preserve fundamental physics laws on the discrete level regardless of the mesh size. [ABSTRACT FROM AUTHOR]

We prove that the realization space of the d-dimensional cube is contractible. For this we first show that any two realizations are connected by a finite sequence of projective transformations and normal transformations. As an application we use this fact to define an analog of the connected sum construction for cubical d-polytopes, and apply this construction to certain cubical d-polytopes to conclude that the rays spanned by f -vectors of cubical d-polytopes are dense in Adin’s cone. The connectivity result on cubes extends to any product of simplices, and further it shows that the respective realization spaces are contractible. [ABSTRACT FROM AUTHOR]

Inverse data envelopment analysis (InvDEA) is a remarkable and popular management tool. This paper deals with one application of this tool. In fact, the problem of the merging of units is investigated in the presence of negative data. The problem of merging units refers to the fact that a set of units create a new unit based on synergy to improve their performance. We use multiple objective programming for this purpose and suggest new models based on predetermined conditions for new units. The proposed models estimate inputs and outputs simul- taneously. Importance advantages of the proposed models are: i) We can follow multiple goals in the problem of merging units because mul- tiple objective programming is applied. ii) Models can simultaneously estimate the inputs and outputs of the combined unit. iii) Unlike the existing methods in the InvDEA-based merging literature, the negative data do not need to be transferred to positive data. Finally, a numerical example is used to explain and validate the model proposed in this paper. [ABSTRACT FROM AUTHOR]

The Inverse Gaussian (IG) distribution is widely used to model positively skewed data. In this article, we examine goodness of fit tests for the Inverse Gaussian distribution based on the empirical distribution function. In order to compute the test statistics, parameters of the Inverse Gaussian distribution are estimated by maximum likelihood estimators (MLEs), which are simple explicit estimators. Critical points and the actual sizes of the tests are obtained by Monte Carlo simulation. Through a simulation study, power values of the tests are compared with each other. Finally, an illustrative example is presented and analyzed. [ABSTRACT FROM AUTHOR]

In this paper the meanings of neutrosophic special n-dominating set, neutrosophic special n-domination number, inverse neutrosophic special domination set (number) and inverse neutrosophic special n-domination number are introduced and some of related results are investigated. Finally, an application of inverse neutrosophic special dominating set in decision making under ashy clauses between certainty and uncertainty is provided. In fact, we present a decision-making problem in real-world applied example which discusses the factors influencing a companys efficiency. The presented model is, in fact, a factor-based model wherein the impact score of each factor is divided into two types of direct and indirect influences. [ABSTRACT FROM AUTHOR]

In this paper, an attempt has been taken to develop a method to solve the neutrosophic multiobjective linear fractional programming (NMOLFPP) problem. In the first step of our method, the problem is linearized based on some transformations. Then, the linearized model is reduced to a crisp multi-objective programming problem with the help of the accuracy function for each objective. In the following, we extend Zimmerman’s approach to maximize the truth membership and minimize the indeterminacy and falsity membership functions in the solution pro- cedure. Finally, to illustrate the proposed approach, a numerical example is included. [ABSTRACT FROM AUTHOR]

Here, we investigate the existence of solutions for the initial value problem of fractional-order differential inclusion containing a nonlocal infinite-point or Riemann–Stieltjes integral boundary conditions. A sufficient condition for the uniqueness of the solution is given. The continuous dependence of the solution on the set of selections and on some data is studied. At last, examples are designed to illustrate the applicability of the theoretical results. [ABSTRACT FROM AUTHOR]

By considering a fixed point in the punctured unit disk and using the q–derivative, a new subfamily of meromorphic and univalent functions is defined. Also, the first and second order q–derivative of meromorphic functions are introduced. Coefficient bounds, extreme points, radii of starlikeness and convexity are obtained. Furthermore, the convexity and preserving under convolution with some restrictions on parameters are investigated. [ABSTRACT FROM AUTHOR]

Today, the great performance of Deep Neural Networks(DNN) has been proven in various fields. One of its most attractive applications is to produce artistic designs. A carpet that is known as a piece of art is one of the most important items in a house, which has many enthusiasts all over the world. The first stage of producing a carpet is to prepare its map, which is a difficult, time-consuming, and expensive task. In this research work, our purpose is to use DNN for generating a Modern Persian Carpet Map. To reach this aim, three different DNN style transfer methods are proposed and compared against each other. In the proposed methods, the Style-Swap method is utilized to create the initial carpet map, and in the following, to generate more diverse designs, methods Clip-Styler, Gatys, and Style-Swap are used separately. In addition, some methods are examined and introduced for coloring the produced carpet maps. The designed maps are evaluated via the results of filled questionnaires where the outcomes of user evaluations confirm the popularity of generated car- pet maps. Eventually, for the first time, intelligent methods are used in producing carpet maps, and it reduces human intervention. The proposed methods can successfully produce diverse carpet designs, and at a higher speed than traditional ways. [ABSTRACT FROM AUTHOR]

In the context of observers, any mathematical model accord- ing to the viewpoint of an observer Θ is called a relative model. The purpose of the present paper is to study the relative model of logical en- tropy. Given an observer Θ, we define the relative logical entropy and relative conditional logical entropy of a sub-σΘ-algebra having finitely many atoms on the relative probability Θ−measure space and prove the ergodic properties of these measures. Finally, it is shown that the relative logical entropy is invariant under the relation of equivalence modulo zero. [ABSTRACT FROM AUTHOR]

We consider a planar symmetric vector field that undergoes a homoclinic bifurcation‎. ‎In order to study the existence of exterior periodic solutions of the vector field around broken symmetric homoclinic orbits‎, ‎we investigate the existence of fixed points of the exterior Poincaré map around these orbits‎. This Poincaré map is the result of the combination of flows inside and outside the homoclinic orbits. It shows how ‎a big periodic orbit converts to two small periodic orbits by passing through a double homoclinic structure‎. Finally‎, ‎we use the results to investigate the existence of periodic solutions of the perturbed Duffing equation. [ABSTRACT FROM AUTHOR]

We relate the notion of dimension expanders to quiver representations and their general subrepresentations and use this relation to establish sharp existence results. [ABSTRACT FROM AUTHOR]

We investigate a class of Nash equilibrium problems on networks where both the utility function and the strategy space of each player are parametrized by means of continuous functions of a vector parameter. We provide a variational inequality reformulation of the Nash equilibrium problem which turns out to be much simpler than the original problem. Moreover, under additional assumptions, the continuity of the solution with respect to the parameter is proved. [ABSTRACT FROM AUTHOR]

This paper exploits non-autonomous Caputo-type systems associated with unbounded delayed arguments. We formulate a new theorem and provide proof that gives limiting behaviour of nontrivial solutions to zero vector. Two examples are provided to illustrate the proposed theorem. [ABSTRACT FROM AUTHOR]

In this paper, we consider invariant (α, β)- metrics which are induced by invariant Riemannian metrics and invariant vector fields on homogeneous spaces. We first study geodesic vectors and investigates the set of all homogeneous geodesics of (α, β)- metrics. Then we study the geometry of simply connected two-step nilpotent Lie groups of dimension five equipped with a left invariant (α, β)- metrics and we examine Lie algebras with 1-dimensional center, 2-dimensional center and 3-dimensional center. [ABSTRACT FROM AUTHOR]

This work contributes to nonlocal vector calculus as an indispensable mathematical tool for the study of nonlocal models that arises in a variety of applications. We define the nonlocal half-ball gradient, divergence and curl operators with general kernel functions (integrable or fractional type with finite or infinite supports) and study the associated nonlocal vector identities. We study the nonlocal function space on bounded domains associated with zero Dirichlet boundary conditions and the half-ball gradient operator and show it is a separable Hilbert space with smooth functions dense in it. A major result is the nonlocal Poincaré inequality, based on which a few applications are discussed, and these include applications to nonlocal convection–diffusion, nonlocal correspondence model of linear elasticity and nonlocal Helmholtz decomposition on bounded domains. [ABSTRACT FROM AUTHOR]

In an online asynchronous vector calculus course, we observed exam answers solved with a formula from online homework instead of the formula from lecture. Our exploratory study investigated (1) why students learned from homework instead of lecture for this topic and (2) their epistemological frames (e-frames) for lecture and homework. Per (1), reasons included studying homework as more efficient than reviewing lectures, difficulty understanding the lecture, and incomplete lecture notes. Per (2), some students see lecture as explaining mathematical meaning and giving them tools to do homework. Students primarily see homework as a space to learn procedures. The results are significant for several reasons. First, they connect homework and lecture learning, rather than treating them separately (as in prior research). A key contribution of the work is both the empirical documentation that students' lecture and homework learning influence one another, and the implication from this that researchers should attend to learning at the intersections of milieu. A second significance of the work is the preliminary documentation of students' e-frames for lecture and homework. Characterizing e-frames is important because improvements in student learning may come in part from helping students shift their beliefs about learning and about mathematics as a discipline. [ABSTRACT FROM AUTHOR]

We provide an elementary derivation of an orthogonal coordinate system for boundary layers around evolving smooth surfaces and curves based on the signed-distance function. We go beyond previous works on the signed-distance function and collate useful vector calculus identities for these coordinates. These results and provided code enable consistent accounting of geometric effects to arbitrary order for boundary layer asymptotics in a wide range of physical systems. [ABSTRACT FROM AUTHOR]

In multi-objective (MO) heuristic search, solution costs, as well as heuristic values, are sets of multi-dimensional cost vectors, representing possible non-dominated trade-offs between objectives. The maximum of two or more such vector sets, which is an important operation in creating informative admissible MO heuristics, can be defined in several ways: Geißer et al. recently proposed two MO maximum operators, the component-wise maximum (comax) and the anti-dominance maximum (admax), which represent different trade-offs between informativeness and computational cost. We show that the anti-dominance maximum is not admissibility-preserving, and propose an alternative, the "select one" maximum (somax). We also show that the comax operator is the greatest admissibility-preserving MO maximum, and briefly investigate its efficient implementation. The conclusion of our experimental results is that somax achieves a trade-off similar to that intended with admax - cheaper to compute but less informed - also when compared to an improved comax implementation. [ABSTRACT FROM AUTHOR]

Copyright of Journal of Southeast University (English Edition) is the property of Journal of Southeast University Editorial Office and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

The study assesses science and engineering students' difficulties in understanding vector concepts. Test of understanding of vectors was administered to 101 university students who were completing their second introductory physics course. Rasch measurement software (Winsteps) was used to analyze the raw data. The concepts that the students found most difficult were identified; these included the graphical representation of a unit vector, the graphical addition and subtraction of vectors, and the interpretation of the dot and cross products, among others. On the other hand, some concepts were easier for students, such as the magnitude of a vector presented in unit-vector notation, and determination of the components of vectors. No significant differences in understanding of vector concepts could be attributed to gender; however, engineering students outperformed science students in understanding vector concepts. This study offers recommendations that physics instructors and researchers can use to improve the teaching of vector concepts. [ABSTRACT FROM AUTHOR]

A novel nonlocal thermo‐hydro‐mechanical coupling model is proposed to investigate cracking behaviors of quasi‐brittle materials in the framework of advanced general particle dynamics (GPD). The nonlocal thermal conduction equation and the nonlocal flow diffusion equation are derived based on nonlocal vector calculus and the microscopic constitutive equation. Moreover, the coupling influences are considered based on the nonlocal equation and the coupling principle. The governing equation of the coupling system is established in the framework of GPD. The correctness of the proposed model is verified through comparing with the corresponding experimental data, and the numerical results are in good agreement with experimental results. Finally, the proposed model is applied to analyze the deformation and cracking behaviors of rock mass around tunnels, and the stress redistribution is revealed under thermo‐hydro‐mechanical coupling condition. Highlights: A novel nonlocal thermo‐hydro‐mechanical coupling model is proposed to investigate the cracking behaviors of rocks.The nonlocal thermal conduction equation and the nonlocal flow diffusion equation are derived.The governing equation of the coupling system is established in the framework of advanced general particle dynamics. [ABSTRACT FROM AUTHOR]

Audio spoof detection gained the attention of the researchers recently, as it is vital to detect spoofed speech for automatic speaker recognition systems. Publicly available datasets also accelerated the studies in this area. Many different features and classifiers have been proposed to overcome the spoofed speech detection problem, and some of them achieved considerably high performances. However, under additive noise, the spoof detection performance drops rapidly. On the other hand, the number of studies about robust spoofed speech detection is very limited. The problem becomes more interesting as the conventional speech enhancement methods reportedly performed worse than no enhancement. In this work, i-vectors are used for spoof detection, and discriminative denoising autoencoder (DAE) network is used to obtain enhanced (clean) i-vectors from their noisy counterparts. Once the enhanced i-vectors are obtained, they can be treated as normal i-vectors and can be scored/classified without any modifications in the classifier part. Data from ASVspoof 2015 challenge is used with five different additive noise types, following a similar configuration of previous studies. The DAE is trained in a multicondition manner, using both clean and corrupted i-vectors. Three different noise types at various signal-to-noise ratios are used to create corrupted i-vectors, and two different noise types are used only in the test stage to simulate unknown noise conditions. Experimental results showed that the proposed DAE approach is more effective than the conventional speech enhancement methods. [ABSTRACT FROM AUTHOR]

The geometry and interactions between the constituents of a liquid crystal, which are responsible for inducing the partial order in the fluid, may locally favor an attempted phase that could not be realized in R 3 . While states that are incompatible with the geometry of R 3 were identified more than 50 years ago, the collection of compatible states remained poorly understood and not well characterized. Recently, the compatibility conditions for three-dimensional director fields were derived using the method of moving frames. These compatibility conditions take the form of six differential relations in five scalar fields locally characterizing the director field. In this work, we rederive these equations using a more transparent approach employing vector calculus. We then use these equations to characterize a wide collection of compatible phases. [ABSTRACT FROM AUTHOR]

In practical computational fluid dynamics simulations around industrial products with complex surface shapes, the robustness of preprocessing to "dirty" geometry is an important issue. The dirty STL (Standard Triangle Language) data contains errors such as gaps between facets, overlapping of facets, and flipping of normal vectors. These errors in the STL data are difficult to avoid in 3D modeling of complex geometry. Using a Cartesian grid is advantageous to the boundary-fitted grid in the aspect of preprocessing for dirty STL files. In this study, a robust and automatic generation method of a signed distance function for the preprocessing of Cartesian grid solvers is proposed. To ensure robustness to the complex and dirty STL data, the proposed method uses information of all STL facets to determine each grid point. The proposed preprocessing method is verified by numerical simulation of the flow around the NASA common research model. [ABSTRACT FROM AUTHOR]

In this article we introduce a type of teaching scholarship called Lesson Analysis (LA) that helps build instructional knowledge, and an associated genre of writing called Lesson Analysis Manuscript (LAM) that shares this instructional knowledge. We briefly describe the problem LA is intended to solve, as well as key features of a LAM. In LA, instructors develop lessons that overcome specific instructional challenges. LAMs are lesson-specific manuscripts that help others understand details of the lesson that addressed the instructional challenge, and include such information as the instructional context, the rationale behind the instructional decisions, typical student thinking, and guidance on carrying out discussions and implementing tasks. We have illustrated this genre by including an entire example LAM as supplemental material located online. The example LAM focuses on helping students understand the connection between dot products and the connection of dot products to measures of similarity between vectors. [ABSTRACT FROM AUTHOR]

Let Ω be a bounded domain in Rn with C0;1 boundary, and let dΩ: Ω ! R be the distance function dΩ (x):= dist (x; Ω): Our aim in this paper is to study the existence and properties of principal eigenvalues of selfadjoint elliptic operators with weight function and singular potential, whose model problem is u + bu = mu in Ω, u = 0 on Ω; u > 0 in Ω where b Ω R is a nonnegative function such that d2Ω b 2 L1 () ; m: Ω R is a nonidentically zero function in L1 () that may change sign, and the solutions are understood in weak sense. [ABSTRACT FROM AUTHOR]

We consider real univariate polynomials with all roots real. Such a polynomial with c sign changes and p sign preservations in the sequence of its coefficients has c positive and p negative roots counted with multiplicity. Suppose that all moduli of roots are distinct; we consider them as ordered on the positive half-axis. We ask the question: If the positions of the sign changes are known, what can the positions of the moduli of negative roots be? We prove several new results which show how far from trivial the answer to this question is. [ABSTRACT FROM AUTHOR]

The purpose is to provide a generalization of Carleson's Theorem on interpolating sequences when dealing with a sequence in the open unit ball of a Hilbert space. Precisely, we interpolate a sequence by a function belonging to a weighted Bergman space of infinite order on a unit Hilbert ball and we furnish explicitly the upper bound corresponding to the interpolation constant. [ABSTRACT FROM AUTHOR]

We prove the unique existence of the functions rn (n = 1; 2;:::) on [0; 1] such that the corresponding sequence of King operators approximates each continuous function on [0; 1] and preserves the functions e0(x) = 1 and ej (x) = xj, where j 2 f2; 3;:::g is fixed. We establish the essential properties of rn, and the rate of convergence of the new sequence of King operators will be estimated by the usual modulus of continuity. Finally, we show that the introduced operators are not polynomial and we obtain quantitative Voronovskaja type theorems for these operators. [ABSTRACT FROM AUTHOR]

In this paper, we pose the1-Laplace equation as a Dirichlet Problem in a class of Grushin-type spaces whose vector fields are of the form... and k is not a polynomial for indices m + 1 k n. Solutions to the1-Laplacian in the viscosity sense have been shown to exist and be unique in [3], when k is a polynomial; we extend these results by exploiting the relationship between Grushin-type and Euclidean second-order jets and utilizing estimates on the viscosity derivatives of sub-and supersolutions in order to produce a comparison principle for semicontinuous functions. [ABSTRACT FROM AUTHOR]

In this study, we examine some important subspaces by showing that the set of n-dimensional interval vectors is a quasilinear space. By defining the concept of dimensions in these spaces, we show that the set of n-dimensional interval vectors is actually a (nr, ns)-dimensional quasilinear space and any quasilinear space is (nr, 0s)-dimensional if and only if it is n-dimensional linear space. We also give examples of (2r, 0s) and (0r, 2s)-dimensional subspaces. We define the concept of dimension in a quasilinear space with natural number pairs. Further, we define an inner product on some spaces and talk about them as inner product quasilinear spaces. Further, we show that some of them have Hilbert quasilinear space structure. [ABSTRACT FROM AUTHOR]

Let K be a field. In this paper, we construct a sequence of Cohen–Macaulay standard graded K -domains whose h-vectors are non-flawless and have exponentially deep flaws. [ABSTRACT FROM AUTHOR]

Let a,b ∈ Rn be column vectors, and (u, v) be the inner product of vectors u and v on Rn. Let G ⊂ GL(n,R) be a compact matrix group. For A ∈ G and a continue function f on G, the integral ∫G f (A)dA is the invariant integral of the compact group G. In this paper, we study the inequality ∀x ∈ Rn ∫G e(Aa, x)dA ≥ ∫G e(Ab, x)dA. We prove that the above inequality holds if and only if b ∈Conv(Ga) . This work follows a series of results, that is, Muirhead (1903), Hardy, Littlewood and P`olya (1932), Rado (1952), Daykin (1971), Kimelfeld (1995) and Schulman (2009). Furthermore, We construct an determining algorithm when G is finite. Compared with other effective algorithms, this one is symbolic and easy to implement on computer. [ABSTRACT FROM AUTHOR]

In this work, the complete moment convergence for weighted sums of negatively associated random variables is discussed without assumptions of identical distribution. Under the moment condition E|X|α /(log (1+|X|))α/γ−1 < ∞ for the case 0 < γ < α with 1 < α ≤ 2, the complete moment convergence theorem for weighted sums of negatively associated setup is presented. [ABSTRACT FROM AUTHOR]

We show that the Zariski canonical stratification of complex hypersurfaces is locally bi-Lipschitz trivial along the strata of codimension two. More precisely, we study the Zariski equisingular families of surface, not necessarily isolated, singularities in C³. We show that a natural stratification of such a family, given by the singular set and the generic family of polar curves, provides a Lipschitz stratification in the sense of Mostowski. In particular such families are biLipschitz trivial, with trivializations obtained by integrating Lipschitz vector fields. [ABSTRACT FROM AUTHOR]