Mathematics

We consider properties of residuated lattices with universal quantifier and show that, for a residuated lattice X, (X, ∀) is a residuated lattice with a quantifier if and only if there is an m-relatively complete substructure of X. We also show that, for a strong residuated lattice X, ∩ {P λ | P λ is an m-filter} = {1} and hence that any strong residuated lattice is a subdirect product of a strong residuated lattice with a universal quantifier {X/P λ }, where P λ is a prime m-filter. As a corollary of this result, we prove that every strong monadic MTL-algebra (BL-and MValgebra) is a subdirect product of linearly ordered strong monadic MTLalgebras (BL-and MV-algebras, respectively).

We consider fuzzy rough sets defined on De Morgan Heyting algebras. We present a theorem that can be used to obtain several correspondence results between fuzzy rough sets and fuzzy relations defining them. We characterize fuzzy rough approximation operators corresponding to compositions of reflexive, transitive, mediate, Euclidean and adjoint fuzzy relations defined on De Morgan Heyting algebras by using only a single axiom.

In this paper we shall first show that for every weak DeMorgan algebra L(n) of order n (WDM-n algebra), there is a quotient weak DeMorgan algebra L(n)/∼ which is embeddable in the finite WDM-n algebra (n). We then demonstrate that the finite WDM-n algebra (n) is functionally free for the class CL(n) of WDM-n algebras. That is, we show that any formulas f and g are identically equal in each algebra in CL(n) if and only if they are identically equal in (n). Finally we establish that there is no weak DeMorgan algebra whose quotient algebra by a maximal filter has exactly seven elements. 1. Are there 5-valued (or 6-valued, 7-valued, ...etc.) algebras satisfying DeMorgan's law? 2. What algebras are embeddable in those finite algebras if they exist? In this paper we will answer these questions. The three algebras (2), (3), and (4) satisfy the condition N 2 x = x as well as DML. In general, DML and the condition that x ≤ y implies Ny ≤ Nx are equivalent to each other under the condition N 2 x = x. However the question arises as to whether the converse holds, that is, as to whether the equivalency of these conditions yields N 2 x = x.

The notion of n-normal residuated lattice, as a class of residuated lattices in which every prime filter contains at most n minimal prime filters, is introduced and studied. Before that, the notion of ω-filter is introduced and it is observed that the set of ω-filters in a residuated lattice forms a distributive lattice on its own, which includes the set of coannulets as a sublattice. The class of n-normal residuated lattices is characterized in terms of their prime filters, minimal prime filters, coannulets and ωfilters. 1

In this paper we give an axiom system of a non-linear 4-valued logic which we call a de Morgan logic (ML), whose Lindenbaum algebra is the de Morgan algebra with implication (MI-algebra), and show that (1) For every MI-algebra L, there is a quotient MI-algebra L such that it is embeddable to the simplest 4-valued MI-algebra M = {0, a, b, 1}; (2) The Lindenbaum algebra of ML is the MI-algebra; (3) The completeness theorem of ML is established; (4) ML is decidable.

We prove some fundamental properties of monotone modal operators on bounded commutative integral residuated lattices (CRL). Moreover we give a positive answer to the problem left open in [RACH ŮNEK, J.-ŠALOU-NOV Á, D.: Modal operators on bounded commutative residuated R -monoids, Math. Slovaca 57 (2007), 321-332].

In this paper we consider fundamental properties of some types of filters (implicative, positive implicative and fantastic filters) of non-commutative residuated lattices and prove that every implicative filter and positive implicative filter are normal. Moreover, we characterize some types of filters by the properties of corresponding quotient algebras, that is, for any non-commutative residuated lattice X and a filter F of X, (a) F is an implicative filter if and only if X/F is a Heyting algebra; (b) F is a positive implicative filter if and only if it is a Boolean filter if and only if X/F is a Boolean algebra; (c) If F is normal, then it is a fantastic filter if and only if X/F satisfies the condition (a → b) b = (b → a) a and (a b) → b = (b a) → a for all a, b ∈ X/F , that is, pseudo-MV algebra.

We consider properties of residuated lattices with universal quantifier and show that, for a residuated lattice $X$, $(X, forall)$ is a residuated lattice with a quantifier if and only if there is an $m$-relatively complete substructure of $X$. We also show that, for a strong residuated lattice $X$, $bigcap {P_{lambda} ,|,P_{lambda} {rm is an} m{rm -filter} } = {1}$ and hence that any strong residuated lattice is a subdirect product of a strong residuated lattice with a universal quantifier ${ X/P_{lambda} }$, where $P_{lambda}$ is a prime $m$-filter. As a corollary of this result, we prove that every strong monadic MTL-algebra (BL- and MV-algebra) is a subdirect product of linearly ordered strong monadic MTL-algebras (BL- and MV-algebras, respectively).

The notion of n-normal residuated lattice, as a class of residuated lattices in which every prime filter contains at most n minimal prime filters, is introduced and studied. Before that, the notion of ω-filter is introduced and it is observed that the set of ω-filters in a residuated lattice forms a distributive lattice on its own, which includes the set of coannulets as a sublattice. The class of n-normal residuated lattices is characterized in terms of their prime filters, minimal prime filters, coannulets and ωfilters. 1

In this paper we give an axiom system of a non-linear 4-valued logic which we call a de Morgan logic (ML), whose Lindenbaum algebra is the de Morgan algebra with implication (MI-algebra), and show that (1) For every MI-algebra L, there is a quotient MI-algebra L such that it is embeddable to the simplest 4-valued MI-algebra M = {0, a, b, 1}; (2) The Lindenbaum algebra of ML is the MI-algebra; (3) The completeness theorem of ML is established; (4) ML is decidable.

Earlier, the authors introduced the logic IntGC, which is an extension of intuitionistic propositional logic by two rules of inference mimicking the performance of Galois connections (Logic J. of the IGPL, 18:837-858, 2010). In this paper, the extensions Int2GC and Int2GC+FS of IntGC are studied. Int2GC can be seen as a fusion of two IntGC logics, and Int2GC+FS is obtained from Int2GC by adding instances of duality-like connections $\Diamond(A \to\ B) \to (\Box A \to \Diamond B)$ and $(\Diamond A \to \Box B) \to \Box(A \to B)$, introduced by G. Fischer Servi (Rend. Sem. Mat. Univers. Politecn. Torino, 42:179-194, 1984), for interlinking the two Galois connections of Int2GC. Both Kripke-style and algebraic semantics are presented for Int2GC and Int2GC+FS, and the logics are proved to be complete with respect to both of these semantics. We show that rough lattice-valued fuzzy sets defined on complete Heyting algebras are proper algebraic models for Int2GC+FS. We also prove that Int2GC...

Lukasiewicz 3-valued logic may be seen as a logic with hidden truthfunctional modalities defined by ♦A := ¬A → A and A := ¬(A → ¬A). It is known that axioms (K), (T), (B), (D), (S4), (S5) are provable for these modalities, and rule (RN) is admissible. We show that, if analogously defined modalities are adopted in Lukasiewicz 4-valued logic, then (K), (T), (D), (B) are provable, and (RN) is admissible. In addition, we show that in the canonical n-valued Lukasiewicz-Moisil algebras Ln, identities corresponding to (K), (T), and (D) hold for all n ≥ 3 and 1 = 1. We define analogous operations in residuated lattices and show that residuated lattices determine modal systems in which axioms (K) and (D) are provable and 1 = 1 holds. Involutive residuated lattices satisfy also the identity corresponding to (T). We also show that involutive residuated lattices do not satisfy identities corresponding to (S4) nor (S5). Finally, we show that in Heyting algebras, and thus in intuitionistic logic, ♦ and are equal, and they correspond to the double negation ¬¬.

We prove some fundamental properties of monotone modal operators on bounded commutative integral residuated lattices (CRL). Moreover we give a positive answer to the problem left open in [RACHŮNEK, J.—ŠALOUNOV Á, D.: Modal operators on bounded commutative residuated Rℓ-monoids, Math. Slovaca 57 (2007), 321–332].

We define states on bounded commutative residuated lattices and consider their property. We show that, for a bounded commutative residuated lattice X, (1)If s is a state, then X/ker(s) is an MV-algebra.(2)If s is a state-morphism, then X/ker(s) is a linearly ordered locally finite MV-algebra.Moreover we show that for a state s on X, the following statements are equivalent: (i)s is a state-morphism on X.(ii)ker(s) is a maximal filter of X.(iii)s is extremal on X.

În articolul dat, conform unei abordări cibernetice, se analizează sistemele de producție care pun accentul pe următorii indicatori economici: rentabilitatea optimă (profitul maximal); costul minim la achiziția și transportarea resurselor; costul minim pentru distribuția și transportul bunurilor spre piețele de consum; respectarea strictă a tuturor restricțiilor de producție, transport și a normelor de poluare a mediului ambiant. Toate aceste patru momente enumerate sunt redate, în mod sistemic, prin limbajul economico-matematic. Anterior, doar în unele forme simplificate, au fost cercetate câteva variante de modele de acest gen. Modelul elaborat poate fi utilizat, de exemplu, la evaluarea anumitor soluții propuse în dezbateri, la analiza unui oarecare eșantion de scenarii simulate, iar, în simbioză cu aplicarea metodelor de optimizare, poate fi eficient și în obținerea unor decizii eficiente.

The so called simplicial algorithms are put into use to compute the stationary probability distributions of stochastic matrices. This is a typical exampie of application of sinlplicia~ algorithms to compute the fixed points of a continuous map (Brouwer's theorem). We further demonstrate the variable grid refinement approach involved in the simpljcia~ algorithms. The variable grid refinement scheme gives good accuracy and acceptable average con~putati~~n~~i complexity. 0 199X Elsevier Science Inc. All rights reserved. uctin roblems such as: ~o~~putatiou of zero points [I]. j'(x) = 0; computation of x; com~utat;on of equiti riui~ prices in a free ex-somput~!tion of neutral stability region in uncertain ave received a computational solution firmly rooted gy and usually referred to as .si~t~~li~i(t~ ~~g~trit~tttt.s.

The purpose of the study was to determine the effect of 8 week plyometric training programme on “Horizontal jump”. Total forty (n= 40) students, age ranging from 20 to 25 years, in which two groups were formed randomly (N=20) for Experimental and (N=20) for Control Group for this study. The experimental group performed 8 week plyometric training program of 50 minute a day, 3 times in a week in the morning sessions and the control group did not perform. Standing Broad Jump (SBJ) was used to analyze the performance. The data was analyse by using paired and grouped t-test at 0.05 level of significance. On the basis of statistical findings, the paired statistic shows the difference within the experimental and for control group; t (19) = 5.93 p < 0.001, t (19) = 3.17 p < 0.001 respectively. The grouped statistic shows the difference between experimental and control group; t (38) = 6.64 p < 0.001.So, the null hypothesis was rejected in the both cases, ultimately the experimental ...

The purpose of this study was to find out kinanthropometric profile of 20 Athletes of Middle distance 800 meters, & Long distance runners 5000 meters of Track Event of age 17 years were assessed for the present study. The data of athletes was collected at Athletics Summer Camp 2015 in Kashmir region. The Athletes having participation of at least two years were selected for the current study. All subjects were assessed for height, weight, girths, diameters, and skin fold thickness. The data was analyzed by applying descriptive statistic i.e., mean, standard deviation & t-test to find out the significant differences of/in middle distance and Long distance runners. The long distance runners had taller than long distance runners for standing height, upper leg length, and lower leg length. The middle distance runners 800 significantly greater in weight, skin fold measurements (p 0.05). It is concluded that in some of the parameters there were significant differences between middle distan...

Semi-supervised learning (SSL) is fundamentally a geometric task: in order to classify high-dimensional point sets when only a small fraction of data points are labeled, the geometry of the unlabeled data points is exploited to gain better classifying accuracy. A number of state-of-the-art SSL techniques rely on label propagation through graphbased diffusion, with edge weights that are evaluated either analytically from the data or through compute-intensive training based on nonlinear and nonconvex optimization. In this paper, we bring discrete differential geometry to bear on this problem by introducing a graph-based SSL approach where label diffusion uses a Laplacian operator learned from the geometry of the input data. From a data-dependent graph of the input, we formulate a biconvex loss function in terms of graph edge weights and inferred labels. Its minimization is achieved through alternating rounds of optimization of the Laplacian and diffusion-based inference of labels. The resulting optimized Laplacian diffusion directionally adapts to the intrinsic geometric structure of the data which often concentrates in clusters or around low-dimensional manifolds within the high-dimensional representation space. We show on a range of classical datasets that our variational classification is more accurate than current graph-based SSL techniques. The algorithmic simplicity and efficiency of our discrete differential geometric approach (limited to basic linear algebra operations) also make it attractive, despite the seemingly complex task of optimizing all the edge weights of a graph.

In this paper, we propose a controllable embedding method for high-and low-dimensional geometry processing through sparse matrix eigenanalysis. Our approach is equally suitable to perform non-linear dimensionality reduction on big data, or to offer non-linear shape editing of 3D meshes and pointsets. At the core of our approach is the construction of a multi-Laplacian quadratic form that is assembled from local operators whose kernels only contain locally-affine functions. Minimizing this quadratic form provides an embedding that best preserves all relative coordinates of points within their local neighborhoods. We demonstrate the improvements that our approach brings over existing nonlinear dimensionality reduction methods on a number of datasets, and formulate the first eigen-based as-rigid-as-possible shape deformation technique by applying our affine-kernel embedding approach to 3D data augmented with user-imposed constraints on select vertices.

Visual quality, low computational cost, and numerical stability are foremost goals in computer animation. An important ingredient in achieving these goals is the conservation of fundamental motion invariants. For example, rigid and deformable body simulation benefits greatly from conservation of linear and angular momenta. In the case of fluids, however, none of the current techniques focuses on conserving invariants, and consequently, they often introduce a visually disturbing numerical diffusion of vorticity. Visually just as important is the resolution of complex simulation domains. Doing so with regular (even if adaptive) grid techniques can be computationally delicate. In this paper, we propose a novel technique for the simulation of fluid flows. It is designed to respect the defining differential properties, i.e., the conservation of circulation along arbitrary loops as they are transported by the flow. Consequently, our method offers several new and desirable properties: arbitrary simplicial meshes (triangles in 2D, tetrahedra in 3D) can be used to define the fluid ___domain; the computations involved in the update procedure are efficient due to discrete operators with small support; and it preserves discrete circulation avoiding numerical diffusion of vorticity. Categories and Subject Descriptors: I.3.6 [Computer Graphics]: Methodology and Techniques Fig. : Discrete Fluids: we present a novel geometric integration scheme for fluids applicable to tetrahedral meshes of arbitrary domains. Aside from resolving the boundaries precisely, our approach also provides an accurate treatment of vorticity through a discrete preservation of Kelvin's circulation theorem. Here, a hot smoke cloud rises inside a bunny shaped ___domain of 7K vertices (32K tetrahedra-equivalent complexity of a 19 3 regular grid), significantly reducing the computational cost of the simulation for such an intricate boundary compared to regular grid-based techniques (0.47s/frame on a Pentium 4, 3GHz).

The analysis of electromagnetic scattering has long been performed on a discrete representation of the geometry. This representation is typically continuous but not differentiable. The need to define physical quantities on this geometric representation has led to development of sets of basis functions that need to satisfy constraints at the boundaries of the elements/tesselations (viz., continuity of normal or tangential components across element boundaries). For electromagnetics, these result in either curl/div-conforming basis sets. The geometric representation used for analysis is in stark contrast with that used for design, wherein the surface representation is higher order differentiable. Using this representation for both geometry and physics on geometry has several advantages, and is eludicated in Hughes et al., Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement, Computer Methods in Applied Mechanics and Engineering 194 (39-41) (2005). Until now, a bulk of the literature on isogeometric methods have been limited to solid mechanics, with some effort to create NURBS based basis functions for electromagnetic analysis. In this paper, we present the first complete isogeometry solution methodology for the electric field integral equation as applied to simply connected structures. This paper systematically proceeds through surface representation using subdivision, definition of vector basis functions on this surface, to fidelity in the solution of integral equations. We also present techniques to stabilize the solution at low frequencies, and impose a Calderón preconditioner. Several results presented serve to validate the proposed approach as well as demonstrate some of its capabilities.

Proteins and many other biologically relevant molecules are flexible, and the flexibility of a given molecule is one of its important characteristics. In particular, the degree of global and local flexibility of proteins is an important characteristic of protein small-molecule binding sites. In this article, the binding site comparison problem is presented as a deformable partial surface matching problem . The problem is labeled as partial matching because we seek the best match for a given smaller query site (e.g. a small molecule fragment binding site) with respect to each larger binding site (in the search dataset). The question we address is "how can a given part of a binding site be realistically deformed to obtain the best partial match between that part and a full binding site"? This goal is addressed by optimizing the similarity between the site representations subject to modeling the proteins as kinematic chains. The preliminary results of our implementation (ArtSurf) show that the proposed approach is feasible, and that the implementation gives physically reasonable results which improve the detection of partial binding site similarity.

Law Number 20 of 2003 emphasizes character education in the Indonesian National Education System, which functions to develop competence and build the character of students who are devout and have faith in God Almighty. The material should be combined with a religious approach to fulfill these expectations. This research explores EFL students' perceptions of utilizing Islamic-based texts for teaching reading. The respondent was a Muhammadiyah University student, Prof. Dr. HAMKA. 36 students participated in responding to questionnaires and interviews. The results show that most EFL students benefit from using Islamic-based texts on their knowledge, attitudes, and skills. It has implications for using Islamic-based texts that can shape student character and behavior. With this method, in the end, students can read critically and understand the text, which also greatly influences their character and behavior.

We propose a real-time approach for indoor scene reconstruction. It is capable of producing a ready-to-use 3D geometric model even while the user is still scanning the environment with a consumer depth camera. Our approach features explicit representations of planar regions and nonplanar objects extracted from the noisy feed of the depth camera, via an online structure analysis on the dynamic, incomplete data. The structural information is incorporated into the volumetric representation of the scene, resulting in a seamless integration with KinectFusion's global data structure and an efficient implementation of the whole reconstruction process. Moreover, heuristics based on rectilinear shapes in typical indoor scenes effectively eliminate camera tracking drift and further improve reconstruction accuracy. The instantaneous feedback enabled by our on-the-fly structure analysis, including repeated object recognition, allows the user to selectively scan the scene and produce high-fi...

We propose a compact data structure for volumetric meshes of arbitrary topology and bounded valence, which offers cell-face, faceedge, and edge-vertex incidence queries in constant time. Our structure is simple to implement, easy to use, and allows for arbitrary, user-defined volume cells, while remaining very efficient in memory usage compared to previous work.

We propose a compact data structure for volumetric meshes of arbitrary topology and bounded valence that offers cell-face, face-edge, and edge-vertex incidence queries in constant time. Our structure is simple to implement, easy to use, and allows for arbitrary, user-defined 3-cells such as prisms and hexahedra, while remaining very efficient in memory usage compared to previous work. Its time complexity for commonly-used incidence and adjacency queries such as vertex and dart one-rings is analyzed.

We propose a texture mapping technique that allows user to directly manipulate texture coordinates of subdivision surfaces through adding feature correspondences. After features, or constraints, are specified by user on the subdivision surface, the constraints are projected back to the control mesh and a polygon matching/embedding algorithm is performed to generate polygon regions that embed texture coordinates of control mesh into different regions. After this step, some Steiner points are added to the control mesh. The generated texture coordinates exactly satisfy the input constraints but with high distortions. Then a constrained smoothing algorithm is performed to minimize distortions of the subdivision surface via updating texture coordinates of the control mesh. Finally, an Iterative Closest Point (ICP)-based deformation algorithm is performed to remove subdivision errors caused by the added Steiner points.

The geometric nature of Euler fluids has been clearly identified and extensively studied in mathematics. However computational approaches to fluid mechanics, mostly derived from a numerical-analytic point of view, are rarely designed with structure preservation in mind, and often suffer from spurious numerical artifacts. In contrast, we geometrically derive discrete equations of motion for fluid dynamics from first principles. Our approach uses a finite-dimensional Lie group to discretize the group of volume-preserving diffeomorphisms, and the discrete Euler equations are derived from a variational principle with non-holonomic constraints. The resulting discrete equations of motion induce a structure-preserving time integrator with good longterm energy behavior, for which an exact discrete Kelvin circulation theorem holds. Possible extensions of our method to magnetohydrodynamics, viscous flows, optimal transport and a link to Brenier's generalized flows are also discussed. vi

Geometric flows are ubiquitous in mesh processing. Curve and surface evolutions based on functional minimization have been used in the context of surface diffusion, denoising, shape optimization, minimal surfaces, and geodesic paths to mention a few. Such gradient flows are nearly always, yet often implicitly, based on the canonical L2 inner product of vector fields. In this paper, we point out that changing this inner product provides a simple, powerful, and untapped approach to extend current flows. We demonstrate the value of such a norm alteration for regularization and volume-preservation purposes and in the context of shape matching, where deformation priors (ranging from rigid motion to articulated motion) can be incorporated into a gradient flow to drastically improve results. Implementation details, including a differentiable approximation of the Hausdorff distance between irregular meshes, are presented.

We present a general-purpose numerical scheme for time integration of Lagrangian dynamical systems—an im- portant computational tool at the core of most physics-based animation techniques. Several features make this particular time integrator highly desirable for computer animation: it numerically preserves important invariants, such as linear and angular momenta; the symplectic nature of the integrator also guarantees a correct energy behavior,

In this paper, we present a numerical technique for performing Lie advection of arbitrary differential forms. Leveraging advances in high-resolution finite volume methods for scalar hyperbolic conservation laws, we first discretize the interior product (also called contraction) through integrals over Eulerian approximations of extrusions. This, along with Cartan's homotopy formula and a discrete exterior derivative, can then be used to derive a discrete Lie derivative. The usefulness of this operator is demonstrated through the numerical advection of scalar fields and 1-forms on regular grids.

Widely used for morphing between objects with arbitrary topology, distance field interpolation (DFI) handles topological transition naturally without the need for correspondence or remeshing, unlike surface-based interpolation approaches. However, lack of correspondence in DFI also leads to ineffective control over the morphing process. In particular, unless the user specifies a dense set of landmarks, it is not even possible to measure the distortion of intermediate shapes during interpolation, let alone control it. To remedy such issues, we introduce an approach for establishing correspondence between the interior of two arbitrary objects, formulated as an optimal mass transport problem with a sparse set of landmarks. This correspondence enables us to compute non-rigid warping functions that better align the source and target objects as well as to incorporate local rigidity constraints to perform as-rigid-as-possible DFI. We demonstrate how our approach helps achieve flexible morphing results with a small number of landmarks.

Numerical viscosity has long been a problem in fluid animation. Existing methods suffer from intrinsic artificial dissipation and often apply complicated computational mechanisms to combat such effects. Consequently, dissipative behavior cannot be controlled or modeled explicitly in a manner independent of time step size, complicating the use of coarse previews and adaptive-time stepping methods. This paper proposes simple, unconditionally stable, fully Eulerian integration schemes with no numerical viscosity that are capable of maintaining the liveliness of fluid motion without recourse to corrective devices. Pressure and fluxes are solved efficiently and simultaneously in a time-reversible manner on simplicial grids, and the energy is preserved exactly over long time scales in the case of inviscid fluids. These integrators can be viewed as an extension of the classical energy-preserving Harlow-Welch / Crank-Nicolson scheme to simplicial grids.

We present a purely Eulerian framework for geometry processing of surfaces and foliations. Contrary to current Eulerian methods used in graphics, we use conservative methods and a variational interpretation, offering a unified framework for routine surface operations such as smoothing, offsetting, and animation. Computations are performed on a fixed volumetric grid without recourse to Lagrangian techniques such as triangle meshes, particles, or path tracing. At the core of our approach is the use of the Coarea Formula to express area integrals over isosurfaces as volume integrals. This enables the simultaneous processing of multiple isosurfaces, while a single interface can be treated as the special case of a dense foliation. We show that our method is a powerful alternative to conventional geometric representations in delicate cases such as the handling of high-genus surfaces, weighted offsetting, foliation smoothing of medical datasets, and incompressible fluid animation.