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We construct a compact convex generating set C n of the moduli set of closed connected projective special real manifolds of fixed dimension n. We show that a closed connected projective special real manifold corresponds to an inner point of C n if and only if it has regular boundary behaviour. Our results can be used to describe deformations of 5d supergravity theories with complete scalar geometries. [ABSTRACT FROM AUTHOR]

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Equations governing mechanical systems with nonholonomic constraints can be developed in two ways: (1) using the physical principles of Newtonian mechanics; (2) using a constrained variational principle. Generally, the two sets of resulting equations are not equivalent. While mechanics arises from the first of these methods, sub-Riemannian geometry is a special case of the second. Thus both sets of equations are of independent interest.The equations in both cases are carefully derived using a novel Sobolev analysis where infinite-dimensional Hilbert manifolds are replaced with infinite-dimensional Hilbert spaces for the purposes of analysis. A useful representation of these equations is given using the so-called constrained connection derived from the system's Riemannian metric, and the constraint distribution and its orthogonal complement. In the special case of sub-Riemannian geometry, some observations are made about the affine connection formulation of the equations for extremals.Using the affine connection formulation of the equations, the physical and variational equations are compared and conditions are given that characterise when all physical solutions arise as extremals in the variational formulation. The characterisation is complete in the real analytic case, while in the smooth case a locally constant rank assumption must be made. The main construction is that of the largest affine subbundle variety of a subbundle that is invariant under the flow of an affine vector field on the total space of a vector bundle. [ABSTRACT FROM AUTHOR]

In this work we study the affine principal lines on surfaces in 3-space. We consider the binary differential equation of the affine curvature lines and obtain the topological models of these curves near the affine umbilic points (elliptic and hyperbolic). We also describe the local generic behavior of affine curvature lines at points with double eigenvalues of the affine shape operator and at parabolic points. [ABSTRACT FROM AUTHOR]

Microcanonical molecular dynamics simulations have been performed in the smectic A phase of an elementary liquid-crystal model. Smectic order parameters and diffusion coefficients along directions parallel and perpendicular to the director have been calculated during the same trajectory for a number of state points. This has permitted the satisfactory testing of a procedure, adopted in the analysis of experimental self-diffusion coefficients, leading to an estimate of the temperature dependence of the smectic order parameters. This methodology has been then confidently applied to two smectogenic thermotropic liquid crystals belonging to the 4,4′-di-n-alkyl-azoxybenzene series. The derived smectic order parameters are larger for the homologue compound with the longest alkyl chains. This is consistent with the well-established increased tendency, for members of a homologue series, to form a smectic phase as their alkyl chains become longer. [ABSTRACT FROM AUTHOR]

We introduce a system of monadic affine sized types, which substantially generalizes usual sized types and allows in this way to capture probabilistic higher-order programs that terminate almost surely. Going beyond plain, strong normalization without losing soundness turns out to be a hard task, which cannot be accomplished without a richer, quantitative notion of types, but also without imposing some affinity constraints. The proposed type system is powerful enough to type classic examples of probabilistically terminating programs such as random walks. The way typable programs are proved to be almost surely terminating is based on reducibility but requires a substantial adaptation of the technique. [ABSTRACT FROM AUTHOR]

We study the properties of the image of a rational surface of revolution under a nonsingular affine mapping. We prove that this image has a notable property, namely that all the affine normal lines, a concept that appears in the context of affine differential geometry, created by Blaschke in the first decades of the 20th century, intersect a fixed line. Given a rational surface with this property, which can be algorithmically checked, we provide an algorithmic method to find a surface of revolution, if it exists, whose image under an affine mapping is the given surface; the algorithm also finds the affine transformation mapping one surface onto the other. Finally, we also prove that the only rational affine surfaces of rotation, a generalization of surfaces of revolution that arises in the context of affine differential geometry, and which includes surfaces of revolution as a subtype, affinely transforming into a surface of revolution are the surfaces of revolution, and that in that case the affine mapping must be a similarity.

Abstract An object-oriented framework is developed to implement discrete models in a monolithic solution of coupled systems of partial differential equations by combining different kinds of finite element approximation spaces chosen for each field. In this sense, we aim at contributing to applications of what is currently classified as "multiphysics simulation", by composing numerical techniques where each physical phenomenon or scale component is approximated by its most appropriate numerical scheme. The integration of the methods is discussed, in a systematic and generic manner, based on an existing object oriented finite element computational framework, allowing any of the usual kinds of affine and/or curved element geometry (point, segment, triangle, quadrilateral, tetrahedral, hexahedral, prismatic or pyramidal). They can be used in several finite element formulations that require continuous, discontinuous, H (div)-conforming functions, or interactions with lower dimensional approximations, which typically occurs in hybrid methods or reduced models. Furthermore, to improve accuracy and/or efficiency, when necessary and/or allowed by the adopted mathematical formulation, different levels of mesh refinements can be adopted for each field, with different refinement configurations (h, p, hp, and directional refinements), as long as the meshes are nested. The generality and flexibility of the proposed framework is verified against a particular set of two dimensional test problems modeled by different formulations - Darcy's problem coupled with tracer transport, fluid flow coupled with geomechanical interaction, multiscale hybrid formulation for linear elasticity, and a hybrid mixed formulation for discrete fracture networks. The main implementation ideas can also be applied to three dimensional problems, as well as to other kinds of approximation space combinations. Highlights • A multiphysics framework for the finite element method using object-oriented programming. • Combination of different finite element approximation space configurations: H 1, H (div) and L 2 approximation spaces. • Code verification tests for different coupled problems. • Tracer injection problem in porous media; Linear poroelastic problem. • Multiscale hybrid formulation for elasticity problems; Discrete fracture network. [ABSTRACT FROM AUTHOR]

We consider Chern–Simons theory on a 3-manifold M that is the total space of a circle bundle over a 2d base Σ . We show that this theory is equivalent to a new 2d TQFT on the base, which we call Caloron BF theory, that can be obtained by an appropriate type of push-forward. This is a gauge theory on a bundle with structure group given by the full affine level k central extension of the loop group L G . The space of fields of this 2d theory is naturally symplectic, and this provides a new formulation of a result of Beasley–Witten about the equivariant localization of the Chern–Simons path integral. The main tool that we employ is the Caloron correspondence, originally due to Murray–Garland, that relates the space of gauge fields on M with a certain enlarged space of connections on an equivariant version of the loop space of the G -bundle. We show that the symplectic structure that Beasley–Witten found is related to a looped version of the Atiyah–Bott construction in 2-dimensional Yang–Mills theory. We also show that Wilson loops that wrap a single circle fiber are also described very naturally in this framework. [ABSTRACT FROM AUTHOR]

We provide a description of the space of continuous and translation invariant Minkowski valuations Φ : K n → K n for which there is an upper and a lower bound for the volume of Φ ( K ) in terms of the volume of the convex body K itself. Although no invariance with respect to a group acting on the space of convex bodies is imposed, we prove that only two types of operators appear: a family of operators having only cylinders over ( n − 1 ) -dimensional convex bodies as images, and a second family consisting essentially of 1-homogeneous operators. Using this description, we give improvements of some known characterization results for the difference body. [ABSTRACT FROM AUTHOR]

Affine matrix rank minimization problem is a fundamental problem in many important applications. It is well known that this problem is combinatorial and NP-hard in general. In this paper, a continuous promoting low rank non-convex fraction function is studied to replace the rank function in this NP-hard problem. An iterative singular value thresholding algorithm is proposed to solve the regularization transformed affine matrix rank minimization problem. With the change of the parameter in non-convex fraction function, we could get some much better results, which is one of the advantages for the iterative singular value thresholding algorithm compared with some state-of-art methods. Some convergence results are established. Moreover, we proved that the value of the regularization parameter λ > 0 cannot be chosen too large. Indeed, there exists λ ̄ > 0 such that the optimal solution of the regularization transformed affine matrix rank minimization problem is equal to zero for any λ > λ ̄ . Numerical experiments on matrix completion problems and image inpainting problems show that our method performs effective in finding a low-rank matrix compared with some state-of-art methods. [ABSTRACT FROM AUTHOR]

We consider the problem of deciding whether or not a zero locus, X , of multivariate real analytic functions crosses a given r -norm ball in the real n -dimensional affine space. We perform a local study of the problem, and we provide both necessary and sufficient conditions to answer the question. Our conditions derive from the analysis of differential geometric properties of X at the center of the ball. An algorithm to evaluate r -norms distances is proposed. [ABSTRACT FROM AUTHOR]

In this paper we give sufficient conditions for a Pontryagin extremal trajectory, consisting of two bang arcs followed by a partially or totally singular one, to be a strong local minimizer for a Mayer problem. The problem is defined on Rn and the end-points constraints are of fixed-free type. We use a Hamiltonian approach and its connection with the second order conditions in the form of a linear quadratic accessory problem. An example is proposed. All the results are coordinate free so they also hold on a manifold. [ABSTRACT FROM AUTHOR]

Conventional grid-stiffened composite structures are designed with straight and uniformly distributed stiffeners. In this paper, a new concept of non-uniform curved grid-stiffened composite structures (NCGCs) with curved non-uniformly distributed stiffeners is proposed, which can significantly boost the design space and flexibility for high efficient composite structures. Under the framework of homogenization-based global/local analysis, optimization design of NCGCs can be solved as material/sizing and shape optimization of local representative cell configurations (RCCs). In the presented contribution, efficient analytical sensitivities with respect to both the skin/stiffener material/sizing parameters and the RCC shape parameters are derived systemically. Sensitivities with respect to the RCC shape parameters are highlighted and derived based on linear membrane and bending transformations between physical and master domains by an affine mapping for the assumed parallelogram RCCs. Optimal material properties, sizes or curved stiffener layout can be efficiently obtained by using the proposed sensitivity calculation method. The accuracy is validated using numerical examples with analytical and finite difference solutions. Finally, a design study of a NCGC panel with linearly varying stiffener angles is presented to demonstrate the feasibility and efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

In this paper, we obtain generators of differential invariants for a curve family in GL(n,R). Then we define GL(n,R) -equivalence of the curve families and develop a point of view for equivalence problem. Using these generators, we give a solution to the problem. [ABSTRACT FROM AUTHOR]

In this study, we determine generating differential invariants for n curves, which is shown to be fonctionally independent. In addition, using these diffenrial invariants, the equivalence problem of two families of n curves in R2 is investigated. [ABSTRACT FROM AUTHOR]

We introduce a geometric property called complementary-finite asymptotic dimension (coasdim). Similar to the case of asymptotic dimension, we prove the corresponding coarse invariant theorem, union theorem and Hurewicz-type theorem. Moreover, we show that coasdim ( X ) ≤ ω + k implies trasdim ( X ) ≤ ω + k and transfinite asymptotic dimension of the shift union sh ⋃ ⨁ i = 1 ∞ 2 i Z is no more than ω + 1 , i.e. trasdim ( sh ⋃ ⨁ i = 1 ∞ 2 i Z ) ≤ ω + 1 . [ABSTRACT FROM AUTHOR]

In this paper, we explore the existence and nonlinearity of affine -cycle permutations on for different values of and . It is proved that a transposition on has nonlinearity for , a -cycle permutation on has nonlinearity for , a prime. The number of affine -cycle permutations on and a bound for their nonlinearity have been obtained. Some results on the cyclic decomposition of a permutation on , a prime, have also been proved. [ABSTRACT FROM AUTHOR]

In [6] , it was asked whether all flat holomorphic Cartan geometries ( G , H ) on a complex torus are translation invariant. We answer this affirmatively under the assumption that the complex Lie group G is affine. More precisely, we show that every holomorphic Cartan geometry of type ( G , H ) , with G a complex affine Lie group, on any complex torus is translation invariant. [ABSTRACT FROM AUTHOR]

Summary: This paper proposes a new methodology for modeling and controlling quality degradation of perishable foods when zero‐order kinetics are considered. This methodology approximates the nonlinear model of the zero‐order quality kinetics using the piecewise affine (PWA) modeling representation. For obtaining a proper PWA model, two state‐of‐the‐art methods are discussed, and eventually, a hybrid identification‐based PWA model is considered after the comparison. This PWA model is then transformed into a computational mixed logical dynamical model, based on which an optimal control strategy is proposed that balances food quality and associated energy consumption. Furthermore, a model predictive control is proposed for improving energy efficiency when a dynamical weather environment is considered. Simulation experiments illustrate the potentials of the proposed optimal controller and the model predictive controller in a case study involving the bighead carp. [ABSTRACT FROM AUTHOR]

The object of the present paper is to study 3-dimensional β- Kenmotsu manifolds whose metric is Ricci soliton with respect to Schouten- van Kampen connection. We found the condition for the Ricci soliton structure to be invariant under Schouten-van Kampen connection. We have also showed that the Ricci soliton structure with respect to usual Levi-Civita connection transforms to a η-Ricci soliton structure under D-homothetic deformation. Finally we have shown that if a 3-dimensional β-Kenmotsu manifold admits a Ricci soliton structure with respect to Schouten-van Kampen connection and potential vector field as the Reeb vector field, then the manifold becomes K -contact Einstein. [ABSTRACT FROM AUTHOR]

In affine differential geometry of hypersurface, C. Scharlach et al. found an inequality involving intrinsic and extrinsic curvatures, and classified elliptic and hyperbolic affine hyperspheres realizing the equality if an affine invariant 2-dimensional distribution D 2 is integrable. In this paper, we continue to study affine hyperspheres realizing the equality, including parabolic affine hyperspheres. As main results, firstly we classify parabolic affine hyperspheres realizing the equality if its scalar curvature is constant, or D 2 is integrable. Next, by introducing a well-defined 3-dimensional distribution D 3 when D 2 is not integrable, we complete the classification of locally strongly convex affine hyperspheres realizing the equality if D 3 is integrable. Finally, we pose a conjecture and a problem in order to determine all affine hyperspheres attaining the equality. [ABSTRACT FROM AUTHOR]

Reduced affine arithmetic (RAA) eliminates the main deficiency of the standard affine arithmetic (AA), i.e. a gradual increase of the number of noise symbols, which makes AA inefficient in a long computation chain. To further reduce overestimation in RAA computation, a new algorithm for the Chebyshev minimum-error multiplication of reduced affine forms is proposed. The algorithm yields the minimum Chebyshev-type bounds and works in linear time, which is asymptotically optimal. We also propose a simplified $\mathcal {O}(n\log n)$ version of the algorithm, which performs better for low dimensional problems. Illustrative examples show that the presented approach significantly improves solutions of many numerical problems, such as the problem of solving parametric interval linear systems or parametric linear programming, and also improves the efficiency of interval global optimisation. [ABSTRACT FROM AUTHOR]

Given two points of a smooth hypersurface, their mid-hyperplane is the hyperplane passing through their mid-point and the intersection of their tangent spaces. In this paper we study the envelope of these mid-hyperplanes (EMH) at pairs whose tangent spaces are transversal. We prove that this envelope consists of centers of conics having contact of order at least three with the hypersurface at both points. Moreover, we describe general conditions for the EMH to be a smooth hypersurface. These results are extensions of the corresponding well-known results for curves. In the case of curves, if the EMH is contained in a straight line, the curve is necessarily affinely symmetric with respect to the line. We show through a counter-example that this property does not hold for hypersurfaces. [ABSTRACT FROM AUTHOR]

The main topic of this paper is to compute the first relative cohomology group of the Lie algebra of smooth vector fields , with coefficients in the space of trilinear differential operators that act on tensor densities, , vanishing on the Lie algebra . [ABSTRACT FROM AUTHOR]

In this paper we study those bent functions which are linear on elements of spreads, their connections with ovals and line ovals, and we give descriptions of their dual bent functions. In particular, we give a geometric characterization of Niho bent functions and of their duals, we give explicit formula for the dual bent function and present direct connections with ovals and line ovals. We also show that bent functions which are linear on elements of inequivalent spreads can be EA-equivalent. [ABSTRACT FROM AUTHOR]

We present methods for computing the explicit decomposition of the minimal simple affine W-algebra $${W_k(\mathfrak{g}, \theta)}$$ as a module for its maximal affine subalgebra $${\mathscr{V}_k(\mathfrak{g}^{\natural})}$$ at a conformal level k, that is, whenever the Virasoro vectors of $${W_k(\mathfrak{g}, \theta)}$$ and $${\mathscr{V}_k(\mathfrak{g}^\natural)}$$ coincide. A particular emphasis is given on the application of affine fusion rules to the determination of branching rules. In almost all cases when $${\mathfrak{g}^{\natural}}$$ is a semisimple Lie algebra, we show that, for a suitable conformal level k, $${W_k(\mathfrak{g}, \theta)}$$ is isomorphic to an extension of $${\mathscr{V}_k(\mathfrak{g}^{\natural})}$$ by its simple module. We are able to prove that in certain cases $${W_k(\mathfrak{g}, \theta)}$$ is a simple current extension of $${\mathscr{V}_k(\mathfrak{g}^{\natural})}$$ . In order to analyze more complicated non simple current extensions at conformal levels, we present an explicit realization of the simple W-algebra $${W_{k}(\mathit{sl}(4), \theta)}$$ at k = −8/3. We prove, as conjectured in [3], that $${W_{k}(\mathit{sl}(4), \theta)}$$ is isomorphic to the vertex algebra $${\mathscr{R}^{(3)}}$$ , and construct infinitely many singular vectors using screening operators. We also construct a new family of simple current modules for the vertex algebra $${V_k (\mathit{sl}(n))}$$ at certain admissible levels and for $${V_k (\mathit{sl}(m \vert n)), m\ne n, m,n\geq 1}$$ at arbitrary levels. [ABSTRACT FROM AUTHOR]

We consider the cyclic parallel Ricci tensor condition, which is a necessary condition for an affine manifold to be Szabó. We show that, in three dimension, there are affine manifolds which satisfy the cyclic parallel Ricci tensor but are not Szabó. Conversely, it is known that in two dimension, the cyclic parallel Ricci tensor forces the affine manifold to be Szabó. Examples of 3-dimensional affine Szabó manifolds are also given. We prove that an affine surface with skew-symmetric Ricci tensor is affine Szabó. Finally, we give some properties of Riemann extensions defined on the cotangent bundle over an affine Szabó manifold. [ABSTRACT FROM AUTHOR]

This paper is concerned with the problem of finding a low-rank solution of an arbitrary sparse linear matrix inequality (LMI). To this end, we map the sparsity of the LMI problem into a graph. We develop a mathematical framework to relate the rank of the minimum-rank solution of the LMI problem to the sparsity of its underlying graph. Furthermore, we propose three graph-theoretic convex programs to obtain a low-rank solution. Two of these convex optimization problems are based on a tree decomposition of the sparsity graph. The third one does not rely on any computationally expensive graph analysis and is always polynomial-time solvable, at the cost of offering a milder theoretical guarantee on the rank of the obtained solution compared to the other two methods. The results of this work can be readily applied to three separate problems of minimum- rank matrix completion, conic relaxation for polynomial optimization, and affine rank minimization. The results are finally illustrated on two applications of optimal distributed control and nonlinear optimization for electrical networks. [ABSTRACT FROM AUTHOR]

Consider the abstract algebraic-delay differential system, xi (t) = Ax(t) + F(x(t), a(t)), a(t) = H(xt, at). Here A is a linear operator on D(A) ⊆ X satisfying the Hille-Yosida conditions, x(t) ∊ D(A) ⊆ X, a(t) ∊ Rn, and X is a real Banach space. Let C0 ⊆ D(A) be closed and convex, and K ⊆ Rn be a compact set contained in the ball of radius h > 0 centered at 0. Under suitable Lipschitz conditions on the nonlinearities F and H and a subtangential condition, the system generates a continuous semiflow on a subset of the space of continuous functions C([-h, 0], C0 ×Rn), which is induced by the algebraic constraint. The object of this paper is to find conditions under which this semiflow is also differentiable with respect to initial data. In the motivating example coming from modelling the dynamics of an age structured population, the nonlinearities F and H are not Fré-chet differentiable on the sets C0 × K and C([-h, 0], C0 × K), respectively. The main challenge of obtaining the differentiability of the semiflow is to determine the right type of differentiability and the right phase space. We develop a novel approach to address this problem which also shows how the spaces on which the derivatives of solution operators act reflect the model structure [ABSTRACT FROM AUTHOR]

Motivated by the concept of Sierpinski object for topological systems of S. Vickers, presented recently by R. Noor and A.K. Srivastava, this paper introduces the Sierpinski object for many-valued topological systems and shows that it has three important properties of the crisp Sierpinski space of general topology. [ABSTRACT FROM AUTHOR]

We study four-dimensional locally strongly convex, locally homogeneous, hypersurfaces whose affine shape operator has two distinct principal curvatures. In case that one of the eigenvalues has dimension 1 these hypersurfaces have been previously studied in Dillen and Vrancken (Math Z 212:61-72, 1993, J Math Soc Jpn 46:477-502, 1994) and Hu et al. (Differ Geom Appl 33:46-74, 2014) in which a classification of such submanifolds was obtained in dimension 4 and 5 under the additional assumption that the multiplicity of one of the eigenvalues is 1. In this paper we complete the classification in dimension 4 by considering the case that the multiplicity of both eigenvalues is 2. [ABSTRACT FROM AUTHOR]

The article discusses the mathematical problems of tail correlation and its realization in the stochastic process. The authors discuss various ways in which the problem could be solved, including the max-stable process and the system of affine inequalities. Further mentioned are the geometric derivations, convex polytope matrices and simplification techniques used to solve the crisis of tail correlation realization.

Neuroscientific studies of drawing-like movements usually analyze neural representation of either geometric (e.g., direction, shape) or temporal (e.g., speed) parameters of trajectories rather than trajectory's representation as a whole. This work is about identifying geometric building blocks of movements by unifying different empirically supported mathematical descriptions that characterize relationship between geometric and temporal aspects of biological motion. Movement primitives supposedly facilitate the efficiency of movements' representation in the brain and comply with such criteria for biological movements as kinematic smoothness and geometric constraint. The minimum-jerk model formalizes criterion for trajectories' maximal smoothness of order 3. I derive a class of differential equations obeyed by movement paths whose nth-order maximally smooth trajectories accumulate path measurement with constant rate. Constant rate of accumulating equi-affine arc complies with the 2/3 power-law model. Candidate primitive shapes identified as equations' solutions for arcs in different geometries in plane and in space are presented. Connection between geometric invariance, motion smoothness, compositionality and performance of the compromised motor control system is proposed within single invariance-smoothness framework. The derived class of differential equations is a novel tool for discovering candidates for geometric movement primitives. [ABSTRACT FROM AUTHOR]

We describe a construction of explicit affine extractors over large finite fields with exponentially small error and linear output length. Our construction relies on a deep theorem of Deligne giving tight estimates for exponential sums over smooth varieties in high dimensions. [ABSTRACT FROM AUTHOR]