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Tangent fibrations generate a "multi-oored tower", while raising from one of its oors to the next one, one practically reiterates the previously performed actions. In this way, the "tower" admits a ladder-shaped structure. Raising to the first oors sufaces for iteratively performing the subsequent steps. The paper mainly studies the tangent functor. We describe the structure of multiple vector bundle which naturally appears on the oors, tangent maps, sector-forms, the lift of vector elds to upper oors. Further, we show how tangent groups of Lie groups lead to gauge theory, and explain in this context the meaning of covariant di erentiation. Finally, we will point out within the oors special subbundles - the osculating bundles, which play an essential role in classical theories. [ABSTRACT FROM AUTHOR]

We explore the class of triples (M, ∇, P) where M is a manifold, ∇ is an affine connection in M and P is a G-structure in M. Inside this class there are infinitesimally homogeneous manifolds, characterized by having G-constant curvature, torsion and inner torsion. For each matrix Lie group G ⊆ GL(R n ) there is a class of infinitesimally homogeneous manifolds with structure group G. In this paper we characterize the classes of infinitesimally homogeneous manifolds for some specific values of the structure group G including: identity group, finite groups, diagonal group, special linear group, orthogonal group and unitary group. [ABSTRACT FROM AUTHOR]

We present new examples of accumulation of secants for orbits (of a real analytic three-dimensional vector fields) having the origin as only ω-limit point. These new examples have the structure of a proper algebraic variety of $\mathbb {S}^{2}$ intersected with a cone. In particular, we present explicit examples of accumulation of secants sets which are not in the list of possibilities of the classical Poincaréé-Bendixson Theorem. [ABSTRACT FROM AUTHOR]

While quantum theory cannot be described by a local hidden variable model, it is nevertheless possible to construct such models that exhibit features commonly associated with quantum mechanics. These models are also used to explore the question of $$\psi $$ -ontic versus $$\psi $$ -epistemic theories for quantum mechanics. Spekkens' toy theory is one such model. It arises from classical probabilistic mechanics via a limit on the knowledge an observer may have about the state of a system. The toy theory for the simplest possible underlying system closely resembles stabilizer quantum mechanics, a fragment of quantum theory which is efficiently classically simulable but also non-local. Further analysis of the similarities and differences between those two theories can thus yield new insights into what distinguishes quantum theory from classical theories, and $$\psi $$ -ontic from $$\psi $$ -epistemic theories. In this paper, we develop a graphical language for Spekkens' toy theory. Graphical languages offer intuitive and rigorous formalisms for the analysis of quantum mechanics and similar theories. To compare quantum mechanics and a toy model, it is useful to have similar formalisms for both. We show that our language fully describes Spekkens' toy theory and in particular, that it is complete: meaning any equality that can be derived using other formalisms can also be derived entirely graphically. Our language is inspired by a similar graphical language for quantum mechanics called the ZX-calculus. Thus Spekkens' toy bit theory and stabilizer quantum mechanics can be analysed and compared using analogous graphical formalisms. [ABSTRACT FROM AUTHOR]

We consider the Prandtl-Reuss model of plasticity with kinematic hardening, aiming at a homogenization result. For a sequence of coefficient fields and corresponding solutions [ABSTRACT FROM AUTHOR]

This paper presents the results of a documentary investigation with the intention of substantiate how and why, and the level and depth of the topics used by the teacher in the classroom for the development of the mathematical knowledge on the part of higher level engineering students. The analysis of the mathematical object was made through the construction of conceptual cartography, being the core of the derivative concept. To construct the axes, the socio-formative theory of Sergio Tobón was used, together with the semiotic representation register of Raychmond Duval and Tall's mathematical advanced thought in the engineering context. The topic is a part of the Unit of learning: Differential and Integral Calculus. This corresponds to the first semester. The course lasts for a semester and is intended for students aged between 18 and 20 years. The research shows that by constructing a conceptual cartography involving at least 8 axes of analysis that the socioformation orientates, and taking mathematics in the context of careers offered by the educational institution, the teacher is allowed to place the thematic content in the appropriate level and depth, guiding in a possible treatment of knowledge to be brought into the classroom. [ABSTRACT FROM AUTHOR]

This paper outlines the use of the fractional calculus for dynamic measurements while describing dynamic properties of measuring transducers, which the authors consider to be an original and unique achievement. The aim of this paper is to investigate how models of accelerometers based on the fractional calculus notation convey their dynamic behaviour in comparison to models represented by differential equations of integer orders and to the processing characteristics of their real counterparts. This paper presents state-of-the-art knowledge, simulation and laboratory studies of measuring transducers to measure acceleration (accelerometers), considering them a representative group of the measuring transducers used in transportation facilities. Measurement errors and comparisons of classical and fractional models in terms of dynamic properties were examined. The advantages of fractional calculus in modelling dynamic elements are also indicated. Tests are executed in the MATLAB & Simulink programme. [ABSTRACT FROM AUTHOR]

We study the interplay between infinitesimal deformations of conformal mappings, quasiconformal distortion estimates and integral means spectra. By the work of McMullen, the second derivative of the Hausdorff dimension of the boundary of the image domain is naturally related to asymptotic variance of the Beurling transform. In view of a theorem of Smirnov which states that the dimension of a k-quasicircle is at most $${1 + k^2}$$ , it is natural to expect that the maximum asymptotic variance $${\Sigma^2 = 1}$$ . In this paper, we prove $${0.87913 \le \Sigma^2 \le 1}$$ . For the lower bound, we give examples of polynomial Julia sets which are k-quasicircles with dimensions $${1 + 0.87913 \, k^2}$$ for k small, thereby showing that $${\Sigma^2 \ge 0.87913}$$ . The key ingredient in this construction is a good estimate for the distortion k, which is better than the one given by a straightforward use of the $${\lambda}$$ -lemma in the appropriate parameter space. Finally, we develop a new fractal approximation scheme for evaluating $${\Sigma^2}$$ in terms of nearly circular polynomial Julia sets. [ABSTRACT FROM AUTHOR]

We show that the commutant of the range of the infinitesimal generator of a norm-continuous quantum Markov semigroup on , not consisting of identity maps, with a faithful normal invariant state is trivial whenever the fixed point algebra is atomic. As a consequence, two formulations of the irreversible -KMS condition proposed in Ref. 2 are equivalent for this class of quantum Markov semigroups. [ABSTRACT FROM AUTHOR]

We propose a method to study the magnetic properties of a disordered Ising kagome lattice. The model considers small spin clusters with infinite-range disordered couplings and short-range ferromagnetic (FE) or antiferromagnetic interactions. The correlated cluster mean-field theory is used to obtain an effective single-cluster problem. A finite disorder intensity in FE kagome lattice introduces a cluster spin-glass (CSG) phase. Nevertheless, an infinitesimal disorder stabilizes the CSG behavior in the geometrically frustrated kagome system. Entropy, magnetic susceptibility and spin–spin correlation are used to describe the interplay between disorder and geometric frustration (GF). We find that GF plays an important role in the low-disorder CSG phase. However, the increase of disorder can rule out the effect of GF. [ABSTRACT FROM AUTHOR]

We introduce the notion of Leiko network as a generalization of the geodetic network on the surfaces of nonzero Gaussian curvature in the Euclidian space E and study its characteristics. The conditions of preservation of the Leiko network under infinitesimal deformations of the surfaces are also obtained. [ABSTRACT FROM AUTHOR]

The deformation and extension theory of Lie–Yamaguti algebras is studied. We prove that a 1-parameter infinitesimal deformation of a Lie–Yamaguti algebracorresponds to a Lie–Yamaguti algebra of deformation type and a (2,3)-cocycle ofwith coefficients in the adjoint representation. The notion of Nijenhuis operators for Lie–Yamaguti algebra is introduced to describe trivial deformations. We also prove that equivalence classes of abelian extensions of Lie–Yamaguti algebras are in one-to-one correspondence to elements of the (2,3)-cohomology group. [ABSTRACT FROM PUBLISHER]

Poly-Bernoulli polynomials of the second kind were introduced in Kim et al. (Adv. Differ. Equ. 2014:219, 2014) as a generalization of the Bernoulli polynomial of the second kind. Here we investigate those polynomials and derive further results about them by using umbral calculus. [ABSTRACT FROM AUTHOR]

The phase space of general relativistic test particle is defined as the 1-jet space of motions. A Lorentzian metric and an electromagnetic field define the joined almost-cosymplectic-contact structure on the odd-dimensional phase space. In this paper, we study infinitesimal symmetries (ISs) of this phase structure. We prove that there are no hidden ISs. [ABSTRACT FROM AUTHOR]

Improving the performance of Dynamic Positioning System in such applications as station keeping, position mooring and slow speed references tracking requires improving the position and heading control precision. These goals can be achieved through the improvement of the ship control system. Fractional-order calculus is a very useful tool which extends classical, integer-order calculus and is used in contemporary modeling and control applications. Fractional-order PIλPIλD controller designed to maintain the ship position and heading and the results were compared with classical integer order PID controller. [ABSTRACT FROM AUTHOR]

We consider a combined stochastic control and impulse control problem of forward-backward systems driven by Lévy processes, where both the system coefficients and the objective performance functional are allowed to be random, non-Markovian; the information available to the controller is partial information. Applying a Malliavin calculus approach, we derive a maximum principle for this control problem, where the adjoint processes are explicitly represented by the parameters and the states of the system. Finally, we give two examples of applications. © 2015 Chinese Automatic Control Society and Wiley Publishing Asia Pty Ltd [ABSTRACT FROM AUTHOR]

In this study, cumulative growth of a physical quantity with Fibonacci method and fractional calculus is handled. The development of the growth process is described in terms of Fibonacci numbers, Mittag–Leffler and exponential functions. A compound growth process with the contribution of a constant quantity is also discussed. For the accumulation of residual quantity, equilibrium and lessening cases are discussed. To the best of our knowledge; compound growth process is solved for the first time in the framework of fractional calculus. In this sense, differintegral order of fractional calculus α has been achieved a physical content. It is emphasized that, in the basis of qualification of the fractional calculus for describing genuine complex physical systems with respect to ordinary descriptions is the cumulative growth mechanism with Fibonacci method. It is concluded that compound diminution and growth process mechanisms can be taken as a basis for the comprehension of derivative and integral operations in fractional calculus. [ABSTRACT FROM AUTHOR]

We show that, for negative generators of analytic semigroups, a bounded [ABSTRACT FROM AUTHOR]

The Mechanics of a viscoelastic collision between a solid body and a semi-space at impact at an arbitrary angle of attack have been examined in this paper. The radius of the contact area has been determined, using a geometry of mutual approach between two spherical bodies. It was proposed that the forces of viscosity and the forces of elasticity can be found by integration of the specific forces acting inside an elementary volume of the contact zone, and on this basis the principally new method of defining the viscoelastic forces has been developed. In this method, the volume of deformation is considered as a system comprising an infinitely large number of infinitesimal discrete elements connected to each other in a definite way. It is shown here that this method allows finding the viscoelastic forces for any theoretical or experimental dependencies between the distance of mutual approach of two solid bodies and the diameter of the contact area. Also the differential equations of the displacement (the movement) of the centre of mass of the body have been obtained. Equations for the calculation of work and energy in the compression and in the restitution phases, and also in the rolling shear phase have been derived. Approximate solutions for the differential equations of movement (displacement) by using the method of equivalent work have been derived. Equations for the normal contact stresses have been obtained. Also, equations for kinematic and dynamic parameters of the viscoelastic collision have been derived in this article. Examples of the comparison of theoretical results and conclusions have been given in the paper. [ABSTRACT FROM AUTHOR]

To study infinitesimal deformation problems with cohomology constraints, we introduce and study cohomology jump functors for differential graded Lie algebra (DGLA) pairs. We apply this to local systems, vector bundles, Higgs bundles, and representations of fundamental groups. The results obtained describe the analytic germs of the cohomology jump loci inside the corresponding moduli space, extending previous results of Goldman–Millson, Green–Lazarsfeld, Nadel, Simpson, Dimca–Papadima, and of the second author. [ABSTRACT FROM AUTHOR]

Infinitesimal supersymmetries over classical Lie groups that are not necessarily induced by a Lie supergroup are described. They yield a notion of supersymmetry that is less rigid than the assumption of a Lie supergroup action but still implies an underlying action of a Lie group. In contrast to Lie supergroups, the arising representation-theoretical Lie supergroups (RTLSG) occur as families associated to Harish–Chandra superpairs. However morphisms of RTLSGs directly correspond to morphisms of Harish–Chandra superpairs. Particular RTLSGs can be derived from the explicit constructions of Lie supergroups given by Kostant and Koszul. The Lie superalgebras g l ( 1 , 1 ) or s l ( 1 , 1 ) appearing also in higher dimensional classical Lie superalgebras, provide interesting first examples of RTLSGs. A classification of RTLSGs associated to real and complex g l ( 1 , 1 ) - and s l ( 1 , 1 ) -Harish–Chandra superpairs is given by parameter spaces and complete sets of invariants. The underlying Lie group is assumed to be connected but possibly not simply connected. [ABSTRACT FROM AUTHOR]

We consider a family of isotropic volumetric-isochoric decoupled strain energies based on the Hencky-logarithmic (true, natural) strain tensor log U, where μ > 0 is the infinitesimal shear modulus, $${\kappa=\frac{2\mu+3\lambda}{3} > 0}$$ is the infinitesimal bulk modulus with $${\lambda}$$ the first Lamé constant, $${k,\hat{k}}$$ are dimensionless parameters, $${F=\nabla \varphi}$$ is the gradient of deformation, $${U=\sqrt{F^T F}}$$ is the right stretch tensor and $${{\rm dev}_n{\rm log} {U} ={\rm log} {U}-\frac{1}{n}{\rm tr}({\rm log} {U})\cdot{1\!\!1}}$$ is the deviatoric part (the projection onto the traceless tensors) of the strain tensor log U. For small elastic strains, the energies reduce to first order to the classical quadratic Hencky energy which is known to be not rank-one convex. The main result in this paper is that in plane elastostatics the energies of the family $${W_{_{\rm eH}}}$$ are polyconvex for $${k\geq \frac{1}{3},\,\widehat{k}\geq \frac{1}{8}}$$ , extending a previous finding on its rank-one convexity. Our method uses a judicious application of Steigmann's polyconvexity criteria based on the representation of the energy in terms of the principal invariants of the stretch tensor U. These energies also satisfy suitable growth and coercivity conditions. We formulate the equilibrium equations, and we prove the existence of minimizers by the direct methods of the calculus of variations. [ABSTRACT FROM AUTHOR]

Perception systems can measure the orientation of a solid 3D object; however, their measurements will contain some uncertainties. In many robotic applications, it is important to propagate the orientation uncertainties of a rigid object onto the uncertainties of specific points on its surface. The orientation uncertainty can be reported as a 3 × 3 covariance matrix. We show that the off-diagonal elements of this matrix provide important clues about the angular uncertainties of points on the object’s surface. Specifically, large off-diagonal elements correspond to a highly concentrated distribution of axes of random infinitesimal rotations which causes large variability in the angular uncertainties of surface points. In particular, experimental data indicate that the ratio of maximum to minimum angular uncertainties can exceed three. In contrast, small off-diagonal elements correspond to a uniform distribution of axes which causes the angular uncertainty of all points on the object’s surface to be almost constant. [ABSTRACT FROM AUTHOR]

Given a multisymplectic manifold ( M , ω ) and a Lie algebra g acting on it by infinitesimal symmetries, Fregier–Rogers–Zambon define a homotopy (co-)moment as an L ∞ -algebra-homomorphism from g to the observable algebra L ( M , ω ) associated to ( M , ω ) , in analogy with and generalizing the notion of a co-moment map in symplectic geometry. We give a cohomological characterization of existence and unicity for homotopy co-moment maps and show its utility in multisymplectic geometry by applying it to special cases as exact multisymplectic manifolds and simple Lie groups and by deriving from it existence results concerning partial co-moment maps, as e.g. covariant multimomentum maps and multi-moment maps. [ABSTRACT FROM AUTHOR]

We develop a finite element method for a large deformation membrane elasticity problem on meshed curved surfaces using a tangential differential calculus approach that avoids the use of classical differential geometric methods. The method is also applied to form finding problems. [ABSTRACT FROM AUTHOR]

We categorify the notion of an infinitesimal braiding in a linear strict symmetric monoidal category, leading to the notion of a (strict) infinitesimal 2-braiding in a linear symmetric strict monoidal 2-category. We describe the associated categorification of the 4-term relations, leading to six categorified relations. We prove that any infinitesimal 2-braiding gives rise to a flat and fake flat 2-connection in the configuration space of n particles in the complex plane, hence to a categorification of the Knizhnik–Zamolodchikov connection. We discuss infinitesimal 2-braidings in a certain monoidal 2-category naturally assigned to every differential crossed module, leading to the notion of a symmetric quasi-invariant tensor in a differential crossed module. Finally, we prove that symmetric quasi-invariant tensors exist in the differential crossed module associated to Wagemann's version of the String Lie-2-algebra. As a corollary, we obtain a more conceptual proof of the flatness of a previously constructed categorified Knizhnik–Zamolodchikov connection with values in the String Lie-2-algebra. [ABSTRACT FROM AUTHOR]

This paper studies the correctness of automated synthesis for concurrent monitors. We adapt a subset of the Hennessy-Milner logic with recursion (a reformulation of the modal $$\mu $$ -calculus) to specify safety properties for Erlang programs. We also define an automated translation from formulas in this sub-logic to concurrent Erlang monitors that detect formula violations at runtime. Subsequently, we formalise a novel definition for monitor correctness that incorporates monitor behaviour when instrumented with the program being monitored. Finally, we devise a sound technique that allows us to prove monitor correctness in stages; this technique is used to prove the correctness of our automated monitor synthesis. [ABSTRACT FROM AUTHOR]

Let G be an infinitesimal group scheme of finite height r and V ( G ) the scheme which represents 1-parameter subgroups of G . We consider sheaves over the projective support variety P ( G ) constructed from a G -module M . We show that if P ( G ) is regular then the sheaf H [ 1 ] ( M ) is zero if and only if M is projective. In general, H [ 1 ] defines a functor from the stable module category and we prove that its kernel is a thick triangulated subcategory. Finally, we give examples of G such that P ( G ) is regular and indicate, in characteristic 2, the connection to the BGG correspondence. Along the way we will provide new proofs of some known results and correct some errors in the literature. [ABSTRACT FROM AUTHOR]

The paper addresses a state space realization problem of a set of higher order delta-differential input–output equations, defined on a homogeneous time scale. The algebraic framework of differential one-forms is applied to formulate necessary and sufficient solvability conditions. This approach applies the total differential operator to analytic system equations to obtain the infinitesimal system description in terms of one-forms. This representation can be converted into polynomial system description by interpreting the polynomial indeterminate as the delta derivative acting on one-forms. The system description in terms of two matrices over skew polynomial ring is then used to derive explicit formulas for the differentials of state coordinates that significantly simplify the calculations. The formulas are found from the left quotients computed by the left Euclidean division algorithm. [ABSTRACT FROM AUTHOR]

In this paper, impulsive Lasota-Wazewska model with infinite delay is studied. By using fixed point theorem of decreasing operator, we obtain sufficient conditions for the existence of unique almost periodic positive solution. Particularly, we give iterative sequence, which converges to the almost periodic positive solution. Moreover, we investigate exponential stability of the almost periodic positive solution by Liapunov functional. Copyright © 2014 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

In this paper, a unified strategy is developed to simultaneously insert inclusions or holes of regular shape as well as redistribute the material to effect optimal topologies of solids. We demonstrate the unified optimal design strategy through three possible choices of design variables: (1) purely geometrical, (2) purely material, and (3) geometrical-material. We couple the geometrical approach with the topological derivative of the objective function and a condition derived for optimally inserting an infinitesimal ellipsoidal heterogeneity (hole or inclusion) into the structure. The approximations of the geometry, material and behavioral fields are isoparametric (or 'isogeometric') and are composed consistent with the Hierarchical Partition of Unity Field Compositions (HPFC) theory (Rayasam et al., Int J Numer Methods Eng 72(12):1452-1489, ). Specifically, analogous to the constructive solid geometry procedure of CAD, the complex material as well as the behavioral field is modeled hierarchically through a series of pair-wise compositions of primitive fields defined on the primitive geometrical domains. The geometrical, material and behavioral approximations are made using Non-Uniform Rational B-Splines (NURBS) basis functions. Thus, the proposed approach seamlessly unifies the explicit representation of boundary shapes with the implicit representations of boundaries arising out of material redistribution, and is termed ISOCOMP, or isoparametric compositions for topology optimization. The methodology is demonstrated first on a set of example problems that increase in complexity of design variable choice culminating in simultaneous optimization of hole location, hole shape and material distribution within the domain. This is followed by a detailed case study involving topology optimization of a bicycle 'dropout.' [ABSTRACT FROM AUTHOR]

We study the relative equilibria of the limit case of the planar Newtonian 4-body problem when three masses tend to zero, the so-called (1+3)-body problem. Depending on the values of the infinitesimal masses the number of relative equilibria varies from ten to fourteen. Six of these relative equilibria are always convex, and the others are concave. Each convex relative equilibrium of the (1+3)-body problem can be continued to a unique family of relative equilibria of the general 4-body problem when three of the masses are sufficiently small and every convex relative equilibrium for these masses belongs to one of these six families. [ABSTRACT FROM AUTHOR]

This article applies nonstandard analysis - a modern theory of infinitesimals - to study generalized-function perturbations of contact discontinuities in compressible, inviscid fluids. Nonstandard analysis is an area of modern mathematics that studies extensions of the real number system to nonstandard number systems that contain infinitely large and infinitely small numbers. Perturbations of a contact discontinuity are considered that represent one-dimensional analogs of the two-dimensional perturbations observed in the initial evolution of a Richtmyer-Meshkov instability on a density interface. Nonstandard predistributions of the Dirac delta function, δ, and its derivatives, δ (n), are applied as the perturbations of a contact discontinuity. The one-dimensional Euler equations are used to model the flow field of a fluid containing a perturbed density interface and generalized solutions are constructed for the perturbed flow field. [ABSTRACT FROM AUTHOR]

The problem of phase regulation for a population of oscillating systems is considered. The proposed control strategy is based on a phase response curve (PRC) model of an oscillator (the first-order reduced model obtained for linearized system and inputs with infinitesimal amplitude). It is proven that the control provides phase resetting for the original nonlinear system. Next, the problem of phase resetting for a network of oscillators is considered when applying a common control input. Performance of the obtained solutions is demonstrated via computer simulation for three different models of circadian/neural oscillators. [ABSTRACT FROM AUTHOR]