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Using Chern character, we construct a natural transformation from the local Hilbert functor to a functor of Artin rings defined from Hochschild homology. This enables us to realize (after slight modification) the infinitesimal Abel-Jacobi map as a morphism between tangent spaces of two functors of Artin rings and also enables us to reconstruct the semi-regularity map together with giving a different proof of a theorem of Bloch stating that the semi-regularity map annihilates certain obstructions to embedded deformations of a closed subvariety which is a locally complete intersection. [ABSTRACT FROM AUTHOR]

Classifying the nonflat hypersurfaces in Euclidean space f\colon M^n\to \mathbb {R}^{n+1} that locally admit smooth infinitesimal deformations that preserve the Gauss map infinitesimally was a problem only considered by Schouten in 1928 [Proceedings Amsterdam 31 (1928), pp. 208–218]. He found two conditions that are necessary and sufficient, with the first one being the minimality of the submanifold. The second is a technical condition that does not clarify much about the geometric nature of the hypersurface. In that respect, the parametric solution of the problem given in this note yields that the submanifold has to be Kaehler. [ABSTRACT FROM AUTHOR]

We consider mappings distorting the modulus of families of paths by means of a Poletsky type inequality. At boundary points of a domain, we prove the Hölder continuity of mappings such that integral averages of their characteristics over infinitesimal balls are finite. We separately study the classes of homeomorphisms and the classes of mappings with branching, as well as domains with good boundaries and domains with prime ends. [ABSTRACT FROM AUTHOR]

We study the equivalence between the infinitesimal Torelli theorem for smooth hypersurfaces in rational homogeneous varieties with Picard number 1 and the theory of generalized Massey products. This equivalence shows that the differential of the period map vanishes on an infinitesimal deformation if and only if certain twisted differential forms are elements of the Jacobian ideal of the hypersurface. We also prove an infinitesimal Torelli theorem result for smooth hypersurfaces in log parallelizable varieties. [ABSTRACT FROM AUTHOR]

We prove that the Hodge–Tate spectral sequence of a proper smooth rigid analytic variety can be reconstructed from its infinitesimal $\mathbb{B}_{\text{dR}}^+$ -cohomology through the Bialynicki–Birula map. We also give a new proof of the torsion-freeness of the infinitesimal $\mathbb{B}_{\text{dR}}^+$ -cohomology independent of Conrad–Gabber spreading theorem, and a conceptual explanation that the degeneration of Hodge–Tate spectral sequences is equivalent to that of Hodge–de Rham spectral sequences. [ABSTRACT FROM AUTHOR]

In this article,we prove that nonlinear resolvents of infinitesimal generators on bounded and convex subdomains of C n are decreasing Loewner chains. Furthermore, we consider the problem of the existence of nonlinear resolvents on unbounded convex domains in C . In the case of the upper half-plane, we obtain a complete solution by using that nonlinear resolvents of certain generators correspond to semigroups of probability measures with respect to free convolution. [ABSTRACT FROM AUTHOR]

We prove that the bi-invariant Einstein metric on |$SU_{2n+1}$| is isolated in the moduli space of Einstein metrics, even though it admits infinitesimal deformations. This gives a non-Kähler, non-product example of this phenomenon adding to the famous example of |$\mathbb{C}\mathbb{P}^{2n}\times \mathbb{C}\mathbb{P}^{1}$| found by Koiso. We apply our methods to derive similar solitonic rigidity results for the Kähler–Einstein metrics on "odd" Grassmannians. We also make explicit a connection between non-integrable deformations and the dynamical instability of metrics under Ricci flow. [ABSTRACT FROM AUTHOR]

In the present paper, we prove vanishing theorems for the null space of the Lichnerowicz Laplacian acting on symmetric two tensors on complete and closed Riemannian manifolds and further estimate its lowest eigenvalue on closed Riemannian manifolds. In addition, we give an application of the obtained results to the theory of infinitesimal Einstein deformations. [ABSTRACT FROM AUTHOR]

Korn's first inequality states that there exists a constant such that the L 2 -norm of the infinitesimal displacement gradient is bounded above by this constant times the L 2 -norm of the infinitesimal strain, i.e., the symmetric part of the gradient, for all infinitesimal displacements that are equal to zero on the boundary of a body ℬ. This inequality is known to hold when the L 2 -norm is replaced by the L p -norm for any p ∈ (1 , ∞) . However, if p = 1 or p = ∞ the resulting inequality is false. It was previously shown that if one replaces the L ∞ -norm by the BMO -seminorm (Bounded Mean Oscillation) then one maintains Korn's inequality. (Recall that L ∞ (B) ⊂ BMO (B) ⊂ L p (B) ⊂ L 1 (B) , 1 < p < ∞ .) In this manuscript it is shown that Korn's inequality is also maintained if one replaces the L 1 -norm by the norm in the Hardy space H 1 , the predual of BMO . One caveat: the results herein are only applicable to the pure-displacement problem with the displacement equal to zero on the entire boundary of ℬ. [ABSTRACT FROM AUTHOR]

Upper and lower bounds for the height of proofs in sequent calculus for intuitionistic logic are proved for the case when cut formulas may only contain essentially positive occurrences of the existential quantifier. The considered cases include both proofs with and proofs without function symbols. [ABSTRACT FROM AUTHOR]

In this paper, we generalize known results of nonabelian embedding tensor to the Hom setting. We introduce the concept of Hom-Leibniz-Lie algebra, which is the basic algebraic structure of nonabelian embedded tensors on Hom-Lie algebras and can also be regarded as a nonabelian generalization of Hom-Leibniz algebra. Moreover, we define a cohomology of nonabelian embedding tensors on Hom-Lie algebras with coeffcients in a suitable representation. The first cohomology group is used to describe infinitesimal deformations as an application. In addition, Nijenhuis elements are used to describe trivial infinitesimal deformations. [ABSTRACT FROM AUTHOR]

This article reports an experimental work that unveils some interesting yet unknown phenomena underneath all smooth nonlinear maps. The findings are based on the fact that, generalizing the conventional gradient dynamics, the right singular vectors of the Jacobian matrix of any differentiable map point in directions that are most pertinent to the infinitesimal deformation of the underlying function and that the singular values measure the rate of deformation in the corresponding directions. A continuous adaption of these singular vectors, therefore, constitutes a natural moving frame that carries indwelling information of the variation. This structure exists in any dimensional space, but the development of the fundamental theory and algorithm for surface exploration is an important first step for immediate application and further generalization. In this case, trajectories of these singular vectors, referred to as singular curves, unveil some intriguing patterns per the given function. At points where singular values coalesce, curious and complex behaviors occur, manifesting specific landmarks for the function. Upon analyzing the dynamics, it is discovered that there is a remarkably simple and universal structure underneath all smooth two-parameter maps. This work delineates graphs with this interesting dynamical system and the possible new discovery that, analogous to the double helix with two base parings in DNA, two strands of critical curves and eight base pairings could encode properties of a generic and arbitrary surface. This innate structure suggests that this approach could provide a unifying paradigm in functional genetics, where all smooth surfaces could be genome-sequenced and classified accordingly. Such a concept has sparked curiosity and warrants further investigation. [ABSTRACT FROM AUTHOR]

In this paper, first, we provide a graded Lie algebra whose Maurer–Cartan elements characterize Lie triple system structures. Then, we use it to study cohomology and deformations of O -operators on Lie triple systems by constructing a Lie 3-algebra whose Maurer–Cartan elements are O -operators. Furthermore, we define a cohomology of an O -operator T as the Lie–Yamaguti cohomology of a certain Lie triple system induced by T with coefficients in a suitable representation. Therefore, we consider infinitesimal and formal deformations of O -operators from a cohomological viewpoint. Moreover, we provide relationships between O -operators on Lie algebras and associated Lie triple systems. [ABSTRACT FROM AUTHOR]

In this article, we investigate the long-time behavior of solutions to a class of infinitely many master equations defined from transition rates that are suitable for the description of a quantum system approaching thermodynamical equilibrium with a heat bath at fixed temperature and a reservoir consisting of one species of particles characterized by a fixed chemical potential. We do so by proving a result which pertains to the spectral resolution of the semigroup generated by the equations, whose infinitesimal generator is realized as a trace-class self-adjoint operator defined in a suitably weighted sequence space. This allows us to prove the existence of global solutions which all stabilize toward the grand canonical equilibrium probability distribution as the time variable becomes large, some of them doing so exponentially rapidly but not all. When we set the chemical potential equal to zero, the stability statements continue to hold in the sense that all solutions converge toward the Gibbs probability distribution of the canonical ensemble which characterizes the equilibrium of the given system with a heat bath at fixed temperature. [ABSTRACT FROM AUTHOR]

We first present an analysis of infinitesimal null deformations for the entanglement entropy, which leads to a major simplification of the proof of the C, F and A-theorems in quantum field theory. Next, we study the quantum null energy condition (QNEC) on the light-cone for a CFT. Finally, we combine these tools in order to establish the irreversibility of renormalization group flows on planar d-dimensional defects, embedded in D-dimensional conformal field theories. This proof completes and unifies all known defect irreversibility theorems for defect dimensions d ≤ 4. The F-theorem on defects (d = 3) is a new result using information-theoretic methods. For d ≥ 4 we also establish the monotonicity of the relative entropy coefficient proportional to Rd−4. The geometric construction connects the proof of irreversibility with and without defects through the QNEC inequality in the bulk, and makes contact with the proof of strong subadditivity of holographic entropy taking into account quantum corrections. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to find some conditions under which the tangent bundle T M has a dualistic structure. Then, we introduce infinitesimal affine transformations on statistical manifolds and investigate these structures on a special statistical distribution and the tangent bundle of a statistical manifold too. Moreover, we also study the mutual curvatures of a statistical manifold M and its tangent bundle T M and we investigate their relations. More precisely, we obtain the mutual curvatures of well-known connections on the tangent bundle T M (the complete, horizontal, and Sasaki connections) and we study the vanishing of them. [ABSTRACT FROM AUTHOR]

In this paper we study part of the dynamics of a circular restricted $ (N+2) $-body problem on the sphere $ \mathbb{S}^2 $ and considering the logarithmic potential, where $ N $ primaries remain in a ring type configuration (identical masses placed at the vertices of a regular polygon in a fixed parallel and rotating uniformly with respect to the $ Z $-axis) and a $ (N+1) $-th primary of mass $ M\in \mathbb{R} $ fixed at the south pole of $ \mathbb{S}^2 $. Such a particular configuration will be called ring-pole configuration (RP). An infinitesimal mass particle has an equilibrium position at the north pole for any value of $ M $, any parallel where the ring has been fixed (we use as parameter $ z = \cos\theta $, where $ \theta $ is the polar angle of the ring) and any number $ N\geq 2 $ of masses forming the ring. We study the non-linear stability of the north pole in terms of the parameters $ (z, M, N) $ and some bifurcations near the north pole. [ABSTRACT FROM AUTHOR]

The Quantified argument calculus (Quarc) has received a lot of attention recently as an interesting system of quantified logic which eschews the use of variables and unrestricted quantification, but nonetheless achieves results similar to the Predicate calculus (PC) by employing quantifiers applied directly to predicates instead. Despite this noted similarity, the issue of the relationship between Quarc and PC has so far not been definitively resolved. We address this question in the present paper, and then expand upon that result. Utilizing recent developments in structural proof theory, we develop a G3-style sequent calculus for Quarc and briefly demonstrate its structural properties. We put these properties to use immediately to construct direct proofs of the meta-theoretical properties of the system. We then incorporate an abstract (and, as we shall see, logical) predicate into the system in a way that preserves all the structural properties. This allows us to identify a system of Quarc which is deductively equivalent to PC, and also yields a constructive method of demonstrating the Craig interpolation theorem (which speaks in favor of the aforementioned predicate being logical). We further generalize this extension to develop a bivalent system of Quarc with defining clauses that still maintains all the desirable properties of a good proof system. [ABSTRACT FROM AUTHOR]

As educational systems move from onsite to online due to the COVID-19 pandemic, teachers face the difficulty of designing online examination formats which minimize opportunities for dishonesty. In this paper, we expose our design of such a format: a protected Microsoft Excel spreadsheet containing short-answer questions, which was implemented in a calculus module taught by us. This format allows examiners to randomize questions with the aim that each student receives each question with different numerical details, making plagiarism impossible, while keeping the marking effort very low. [ABSTRACT FROM AUTHOR]

Taylor series play a ubiquitous role in calculus courses, and their applications as approximants to functions are widely taught and used everywhere. However, it is not common to present the students with other types of approximations besides Taylor polynomials. These notes show that polynomials construed to satisfy certain boundary conditions at an interval of definition can be represented by rapidly converging Fourier series. These polynomials are shifted and re-scaled versions of the Bernoulli polynomials. From this construction, an argument to 'invert' the Fourier series to obtain an approximation to sines and cosines in terms of the Bernoulli polynomials is presented. We then show that approximating sines and cosines by the Bernoulli polynomials might be much better than using the truncated Taylor series, especially in problems where global proprieties are desired. These contents can be taught in advanced Calculus classes approaching series of functions and their applications. [ABSTRACT FROM AUTHOR]

The restricted concave kite five-body problem is a problem in which four positive masses, called the primaries, rotate in the concave kite configuration with a mass at the center of the triangle formed by three of the primaries. The fifth body has negligible mass and does not influence the motion of the four primaries. It is assumed that the fifth mass is in the same plane of the primaries and that the masses of the primaries are m 1 , m 2 , m 3 , and m 4 , respectively. Three different types of concave kite configurations are considered based on the masses of the primaries. In case I, one pair of primaries has equal masses; in case II, two pairs of primaries have equal masses; in case III, three of the primaries have equal masses. For all three cases, the regions of central configuration are obtained using both analytical and numerical techniques. The existence and uniqueness of equilibrium positions of the infinitesimal mass are investigated in the gravitational field of the four primaries. It is numerically confirmed that none of the equilibrium points are linearly stable. The Jacobian constant C is used to investigate the regions of possible motion of the infinitesimal mass. [ABSTRACT FROM AUTHOR]

With the advancement of computing, representations of Land in contemporary markets are becoming increasingly indeterminate. Citing two examples from Australia, the author identifies the manipulation of system-specific measurements of space and time as the methodological process behind these changing representations while examining their close relationship to the settler colonial project. They argue that a new conceptual formulation is required to de-standardize how markets perceive Land to situate potential sites of value extraction and their calculi and advance movements to reclaim space and time amidst changing digital landscapes. [ABSTRACT FROM AUTHOR]

The notion of a context in formal concept analysis and that of an approximation space in rough set theory are unified in this study to define a Kripke context. For any context (G,M,I), a relation on the set G of objects and a relation on the set M of properties are included, giving a structure of the form ((G,R), (M,S), I). A Kripke context gives rise to complex algebras based on the collections of protoconcepts and semiconcepts of the underlying context. On abstraction, double Boolean algebras (dBas) with operators and topological dBas are defined. Representation results for these algebras are established in terms of the complex algebras of an appropriate Kripke context. As a natural next step, logics corresponding to classes of these algebras are formulated. A sequent calculus is proposed for contextual dBas, modal extensions of which give logics for contextual dBas with operators and topological contextual dBas. The representation theorems for the algebras result in a protoconcept-based semantics for these logics. [ABSTRACT FROM AUTHOR]

Flextegrity lattices are spatial grids composed of stiff segments kept in contact by compliant pre-tensioned tendons. The kinematic skeleton is sensible to the orientation of the segments, since their relative rotation produces the straining of the tendons to an amount that depends upon the angle of rotation and the shape of the pitch surfaces of the contact joints: these dictate the constitutive properties of the lattice in response to external actions. Two- and three-dimensional lattices are investigated, in which the contact pitch surfaces, obtained with axial-symmetric toothed conjugate profiles, mimic the kinematics of spheres, centred at the nodes of a simple cubic lattice, in pure rolling motion. The allowed mechanisms are discussed under infinitesimal deformation, to recognize possible eigenstress states in the lattice. The response under finite deformations is worked out for two-dimensional lattices under symmetric and asymmetric loading. The theoretical predictions are compared with experimental results on 3D-printed physical models. Possible extensions are discussed for lattices with segments of varying size, different arrangements and multi-stable contact joints. The flextegrity microstructure can represent a mesoscopic model for homogeneous crystals composed of non-pointwise molecules, but it could actually be manufactured in metamaterials with peculiar properties. [ABSTRACT FROM AUTHOR]

Representations and cohomologies of Hom- δ -Jordan Lie supertriple systems are established. As an application, Nijenhuis operators and abelian extensions of Hom- δ -Jordan Lie supertriple systems are discussed. We obtain the infinitesimal deformation generated by virtue of a Nijenhuis operator. It is obtained that the sufficient and necessary condition for the equivalence of abelian extensions of Hom- δ -Jordan Lie supertriple systems. [ABSTRACT FROM AUTHOR]

In this paper, an analytical form of the Unit Step Function (or Heaviside Step Function) is presented. In particular, this piecewise - defined function, which constitutes a fundamental concept of Operational Calculus and is also involved in many other areas of applied and engineering mathematics, is explicitly performed in a very simple manner by the aid of purely algebraic representations. The novelty of this work, when compared with other analytical approximations to this discontinuous function, is that the proposed exact formula is not performed in terms of non - elementary special functions, e.g. Gamma function or Error function. In addition, this function does not contain any infinitesimal quantities and also is not the limit of a sequence of functions with a pointwise or uniform convergence. Hence, it may be much more appropriate and useful to the computational procedures which are inserted into Operational Calculus techniques and other engineering practices. [ABSTRACT FROM AUTHOR]

Our goal in this article is to prove a form of p-adic Birch and Swinnerton-Dyer formula for the second derivative of the p-adic L-function associated to a newform f which is non-crystalline semistable at p at its central critical point, by expressing this quantity in terms of a p-adic (cyclotomic) regulator defined on an extended trianguline Selmer group. We also prove a two-variable version of this result for height pairings we construct by considering infinitesimal deformations afforded by a Coleman family passing through f. This, among other things, leads us to a proof of an appropriate version of Perrin-Riou's conjecture in this set up. [ABSTRACT FROM AUTHOR]

We study the energy decay of a damped Euler-Bernoulli beam which is subject to a point-wise feedback force representing a Shape Memory Alloy (SMA) cable. The problem we consider is that of the paper by [Liu et al. Int. J. Struct. Stab. Dyn.17 (2017) 1750076] but, for simplicity, our mod-elization does not take into account the additional stiffness term they considered. An explicit expression is given for the resolvent of the underlying operator as well as its eigenvalues and eigenfunctions. We show the exponential decay of the energy. The fastest decay rate is given by the supremum of the real part of the spectrum of the infinitesimal generator of the underlying semigroup since we prove the existence of a Riesz basis. To the question "Do Shape Memory Alloy cables restrain the vibrations of girder bridges?", the experiments in [Liu et al. Int. J. Struct. Stab. Dyn.17 (2017) 1750076] answer positively. Our study does not allow to give a definite answer yet. The only presence of these cables may not be enough. Some physical parameters have to be chosen carefully. [ABSTRACT FROM AUTHOR]

We study the deformation theory of G2${\mathrm{G}}_2$‐instantons on the round 7‐sphere, specifically those obtained from instantons on the 4‐sphere via the quaternionic Hopf fibration. We find that the pullback of the standard ASD instanton lies in a smooth, complete, 15‐dimensional family of G2${\mathrm{G}}_2$‐instantons. In general, the space of infinitesimal G2${\mathrm{G}}_2$‐instanton deformations on S7$S^7$ is identified with three copies of the space of ASD deformations on S4$S^4$. [ABSTRACT FROM AUTHOR]

One of the most fundamental properties of a proof system is analyticity , expressing the fact that a proof of a given formula F only uses subformulas of F. In sequent calculus, this property is usually proved by showing that the $\mathsf{cut}$ rule is admissible, i.e., the introduction of the auxiliary lemma H in the reasoning "if H follows from G and F follows from H , then F follows from G " can be eliminated. The proof of cut admissibility is usually a tedious, error-prone process through several proof transformations, thus requiring the assistance of (semi-)automatic procedures. In a previous work by Miller and Pimentel, linear logic ( $\mathsf{LL}$) was used as a logical framework for establishing sufficient conditions for cut admissibility of object logical systems (OL). The OL's inference rules are specified as an $\mathsf{LL}$ theory and an easy-to-verify criterion sufficed to establish the cut-admissibility theorem for the OL at hand. However, there are many logical systems that cannot be adequately encoded in $\mathsf{LL}$ , the most symptomatic cases being sequent systems for modal logics. In this paper, we use a linear-nested sequent ( $\mathsf{LNS}$) presentation of $\mathsf{MMLL}$ (a variant of LL with subexponentials), and show that it is possible to establish a cut-admissibility criterion for $\mathsf{LNS}$ systems for (classical or substructural) multimodal logics. We show that the same approach is suitable for handling the $\mathsf{LNS}$ system for intuitionistic logic. [ABSTRACT FROM AUTHOR]

By exploring the uniform integrability of a sequence of some order statistics (OSs), we obtain the moment convergence conclusion of the sequence under some weak conditions even when the corresponding population of interest has no moment of any positive order. As an application, we embody the range of applications of a theorem presented in a reference dealing with the approximation of the difference between the moment of a sequence of normalized OSs and the corresponding moment of a standard normal distribution. By the aid of the embodied theorem, we explore the infinitesimal type of the moments of errors when we estimate some population quantiles by relative OSs. Finally, by the obtained conclusion, we can easily get a combination formula which seems hard to be proved in other methods. [ABSTRACT FROM AUTHOR]

I propose a theory of space with infinitesimal regions called smooth infinitesimal geometry (SIG) based on certain algebraic objects (i.e., rings), which regiments a mode of reasoning heuristically used by geometricists and physicists (e.g., circle is composed of infinitely many straight lines). I argue that SIG has the following utilities. (1) It provides a simple metaphysics of vector fields and tangent space that are otherwise perplexing. A tangent space can be considered an infinitesimal region of space. (2) It generalizes a standard implementation of spacetime algebraicism (according to which physical fields exist fundamentally without an underlying manifold) called Einstein algebras. (3) It solves the long-standing problem of interpreting smooth infinitesimal analysis (SIA) realistically, an alternative foundation of spacetime theories to real analysis (Lawvere Cahiers de Topologie et Géométrie Différentielle Catégoriques, 21(4), 277–392, 1980). SIA is formulated in intuitionistic logic and is thought to have no classical reformulations (Hellman Journal of Philosophical Logic, 35, 621–651, 2006). Against this, I argue that SIG is (part of) such a reformulation. But SIG has an unorthodox mereology, in which the principle of supplementation fails. [ABSTRACT FROM AUTHOR]

(It is trivially remarked that the conformal invariance is for the tensor HT ht and not for its associated tensor HT ht .) Thus HT ht is a linear map on HT ht and may be classified by its Jordan form/Segre type structure. For example, suppose I C i satisfies HT ht at I p i for HT ht . (I have taken the liberty of changing a sign in the HT ht form HT ht from that in [[1]]). [Extracted from the article]

On the Euclidean geographical map, and using rectangular coordinates, the coordinates of the images of these two points are ( I x i , I y i ) and HT ht , respectively. Transforming it into HT ht Graph Chebyshev observed that the sum of the first two terms satisfies Laplace's equation HT ht . [Extracted from the article]

We investigate four-dimensional Heterotic solitons, defined as a particular class of solutions of the equations of motion of Heterotic supergravity on a four-manifold M or, equivalently, as self-similar points of the renormalization group flow of the NS-NS sector of the Heterotic world-sheet. Heterotic solitons depend on a parameter x and consist of a Riemannian metric g, a metric connection with skew torsion Hon TM and a closed 1-form φ on M satisfying a differential system that generalizes the celebrated Hull-Strominger system. In the limit x → 0, Heterotic solitons reduce to a class of generalized Ricci solitons and can be considered as a higher-order curvature modification of the latter. If the torsion His equal to the Hodge dual of φ, Heterotic solitons consist of either flat tori or Ricci-flat Weyl structures on manifolds of type S¹ x S³ as introduced by P. Gauduchon. We prove that the moduli space of such Ricci-flat Weyl structures is isomorphic to the product of R with a certain finite quotient of the Cartan torus of the isometry group of the typical fiber of a natural fibration M → S¹. We also consider the associated space of essential infinitesimal deformations, which we prove to be obstructed. More generally, we characterize several families of Heterotic solitons as suspensions of certain three-manifolds with prescribed constant principal Ricci curvatures, amongst which we find hyperbolic manifolds, manifolds covered by Sl(2,R) and E(1,1)or certain Sasakian three-manifolds. These solutions exhibit a topological dependence in the string slope parameter xand yield, to the best of our knowledge, the first examples of Heterotic compactification backgrounds not locally isomorphic to supersymmetric compactification backgrounds. [ABSTRACT FROM AUTHOR]

In this paper, we propose necessary and sufficient conditions for a scalar function to be nonincreasing along solutions to general differential inclusions with state constraints. The problem of determining if a function is nonincreasing appears in the study of stability and safety, typically using Lyapunov and barrier functions, respectively. The results in this paper present infinitesimal conditions that do not require any knowledge about the solutions to the system. Results under different regularity properties of the considered scalar function are provided. This includes when the scalar function is lower semicontinuous, locally Lipschitz and regular, or continuously differentiable. [ABSTRACT FROM AUTHOR]

This paper is concerned with the status of mathematical fictions in Leibniz's work and especially with infinitary quantities as fictions. Thus, it is maintained that mathematical fictions constitute a kind of symbolic notion that implies various degrees of impossibility. With this framework, different kinds of notions of possibility and impossibility are proposed, reviewing the usual interpretation of both modal concepts, which appeals to the consistency property. Thus, three concepts of the possibility/impossibility pair are distinguished; they give rise, in turn, to three concepts of mathematical fictions. Moreover, such a distinction is the base for the claim that infinitesimal quantities, as mathematical fictions, do not imply an absolute impossibility, resulting from self-contradiction, but a relative impossibility, founded on irrepresentability and on the fact that it does not conform to architectural principles. In conclusion, this "soft" impossibility of infinitesimals yields them, in Leibniz view, a presumptive or "conjectural" status. [ABSTRACT FROM AUTHOR]

Mathematical logic was first introduced to China in early 1920s. Although, the process of introduction was facilitated by the lectures of Bertrand Russel at Peking University in 1921 and continued by China's most passionate adherents of Russell's philosophy, the establishment of mathematical logic as an academic discipline occurred only in late 1920s, in the framework of a recently reorganised Qinghua University in Peking. The main aim of this paper is to shed some light on the process of establishment of mathematical logic at the department of philosophy at Qinghua University, between the years 1926 and 1945. In its main line of discussion, the article highlights the curricular developments at the department, connecting them with concrete theoretical endeavours by the leading members of the department. Furthermore, in its later parts, the article summarises the main advances made in the context of studies of modern logic at Qinghua University, from the expositions on the quintessential work Principia Mathematica, to the subsequent integration of criticisms coming from circles of logicians at Harvard University and ground-breaking contributions of Kurt Gödel in the mid-1930s. [ABSTRACT FROM AUTHOR]

In the years following Bertrand Russell's visit in China, fragments from his work on mathematical logic and the foundations of mathematics started to enter the Chinese intellectual world. While up until 1925 Chinese intellectuals like Zhang Shenfu, Zhang Dongsun and others mainly contributed to the dissemination of general notions from Russell's logicism, epistemology and mathematical philosophy, two aspiring Chinese mathematicians, Fu Zhongsun and Zhang Bangming, introduced to Chinese readers the first Chinese translation of Russell's Introduction to Mathematical Philosophy. One year after the second edition of the book was published in 1924, Chinese intellectuals started to respond to Russell's 'mathematical logic' and philosophy of mathematics, as contained in the work. Consequently, in 1925 the first public debate on this topic broke out in Chinese periodicals. This article aims at providing a summary of these 1925 discussions, from criticisms of Russell's work to a general debate on 'infinitesimals'. Through close examination of the arguments and discursive strategies used in the debate, the article tries to highlight the state of Chinese discourse on mathematical logic during the time in question, while also shedding some light on the mechanisms and strategies in the undergoing process of the adoption of Western scientific ideas. [ABSTRACT FROM AUTHOR]

One of the consequential problems in the theory of infinitesimal bendings of surfaces is the problem of the rigidity of surfaces in various classes of their deformations. The magnitude of this is dictated not only by the internal development of the very theory of infinitesimal bendings of surfaces and its connection with the analytical rigidity of the surface, but also by its applied side since the strength conditions of structures containing circular shells as elements require, as a rule, the geometric mean rigidity shell surface. This paper investigates infinitesimal bendings of convex surfaces, which along with a certain curve on the surface, are fixed simultaneously with respect to a point and a plane. It is proved that such surfaces in the indicated class of deformations enable rigidity not higher than the second order, and, therefore, are analytically non-bendable. [ABSTRACT FROM AUTHOR]

In this paper, we study a class of infinitesimal deformations of the centerless higher rank Heisenberg–Virasoro algebras. Explicitly, the universal central extensions, derivations and isomorphism classes of these algebras are determined. [ABSTRACT FROM AUTHOR]

We determine the L ∞ -algebra that controls deformations of a relative Rota–Baxter Lie algebra and show that it is an extension of the dg Lie algebra controlling deformations of the underlying Lie Rep pair by the dg Lie algebra controlling deformations of the relative Rota–Baxter operator. Consequently, we define the cohomology of relative Rota–Baxter Lie algebras and relate it to their infinitesimal deformations. A large class of relative Rota–Baxter Lie algebras is obtained from triangular Lie bialgebras and we construct a map between the corresponding deformation complexes. Next, the notion of a homotopy relative Rota–Baxter Lie algebra is introduced. We show that a class of homotopy relative Rota–Baxter Lie algebras is intimately related to pre-Lie ∞ -algebras. [ABSTRACT FROM AUTHOR]

We describe the geometry of generic heterotic backgrounds preserving minimal supersymmetry in four dimensions using the language of generalised geometry. They are characterised by an SU(3) × Spin(6 + n) structure within O(6, 6 + n) × ℝ+ generalised geometry. Supersymmetry of the background is encoded in the existence of an involutive subbundle of the generalised tangent bundle and the vanishing of a moment map for the action of diffeomorphisms and gauge symmetries. We give both the superpotential and the Kähler potential for a generic background, showing that the latter defines a natural Hitchin functional for heterotic geometries. Intriguingly, this formulation suggests new connections to geometric invariant theory and an extended notion of stability. Finally we show that the analysis of infinitesimal deformations of these geometric structures naturally reproduces the known cohomologies that count the massless moduli of supersymmetric heterotic backgrounds. [ABSTRACT FROM AUTHOR]

Based on JB*-triple theory, we refine earlier results on the structure of strongly continuous one-parameter semigroups (C0-SGR) of holomorphic Carathéodory isometries of the unit ball in infinite dimensional spin factors, resulting in finite algebraic formulas in terms of joint boundary fixed points, Mobius charts and inverse Laplace transform. [ABSTRACT FROM AUTHOR]

On the one hand, we prove that the Clifford torus in ℂ 2 {\mathbb{C}^{2}} is unstable for Lagrangian mean curvature flow under arbitrarily small Hamiltonian perturbations, even though it is Hamiltonian F-stable and locally area minimising under Hamiltonian variations. On the other hand, we show that the Clifford torus is rigid: it is locally unique as a self-shrinker for mean curvature flow, despite having infinitesimal deformations which do not arise from rigid motions. The proofs rely on analysing higher order phenomena: specifically, showing that the Clifford torus is not a local entropy minimiser even under Hamiltonian variations, and demonstrating that infinitesimal deformations which do not generate rigid motions are genuinely obstructed. [ABSTRACT FROM AUTHOR]

We study the deformation theory of pre-symplectic structures, that is, closed 2-forms of fixed rank. The main result is a parametrization of nearby deformations of a given pre-symplectic structure in terms of an |$L_{\infty }$| -algebra, which we call the Koszul |$L_{\infty }$| -algebra. This |$L_{\infty }$| -algebra is a cousin of the Koszul dg Lie algebra associated to a Poisson manifold. In addition, we show that a quotient of the Koszul |$L_{\infty }$| -algebra is isomorphic to the |$L_{\infty }$| -algebra that controls the deformations of the underlying characteristic foliation. Finally, we show that the infinitesimal deformations of pre-symplectic structures and of foliations are both obstructed. [ABSTRACT FROM AUTHOR]

The cut-free Gentzen-type sequent calculus LLK for the logic of likelihood (LL) is introduced in the paper. It is proved that the calculus is sound and complete for LL. Using the introduced calculus LLK , a decision procedure for LL is presented. [ABSTRACT FROM AUTHOR]

The solution of the Eisenhart equation for pseudo-Riemannian manifolds (Mn,g) of arbitrary signature and any dimension is obtained. Thereby, pseudo-Riemannian h-spaces (i.e., spaces admitting nontrivial solutions h ≠ cg of the Eisenhart equation) of all possible types determined by the Segrè characteristic χ of the bilinear form h are found. Necessary and sufficient conditions for the existence of an infinitesimal projective transformation in (Mn,g) are given. The curvature 2-form of a (rigid) h-space of type χ = {r1, ..., rk} is calculated and necessary and sufficient conditions for this space to have constant curvature are obtained. [ABSTRACT FROM AUTHOR]

We have obtained the approximate analytical solutions of the nonrelativistic Hydrogen-like atoms such as ( He + , Li 2 + and Be +) and neutral atoms such as ( 2 2 Na , 12 C and 158 Au) atoms with a newly proposed generalized perturbed Yukawa potential with centrifugal barrier (GPYPCB) model using the generalized Bopp's shift method and standard perturbation theory in the symmetries of noncommutative three-dimensional real space phase (NC: 3D-RSP). By approximating the centrifugal term through the Greene–Aldrich approximation scheme, we have obtained the energy eigenvalues and generalized Hamiltonian operator for all orbital quantum numbers ℓ in the symmetries of NC: 3D-RSP. The potential is a superposition of the perturbed Yukawa potential and new terms proportional with (exp (− 2 a r) r 4 , exp (− 2 a r) r 3 , exp (− a r) r 3  and  exp (− a r) r 2 ) appear as a result of the effects of noncommutativity properties of space and phase on the perturbed Yukawa potential model. The obtained energy eigenvalues appear as functions of the generalized Gamma function, the discreet atomic quantum numbers (j , n , l , s  and  m) , two infinitesimal parameters (Θ , 𝜃 ¯) , which are induced by (position–position and phase–phase). In addition, the dimensional parameters (a , V 0 , V 1 , α) of perturbed Yukawa potential with centrifugal barrier model in NC: 3D-RSP. Furthermore, we have shown that the corresponding Hamiltonian operator in (NC: 3D-RSP) symmetries is the sum of the Hamiltonian operator of perturbed Yukawa potential model and the two operators are modified spin–orbit interaction and the modified Zeeman operator for the previous Hydrogenic and neutral atoms. [ABSTRACT FROM AUTHOR]

Ceramic balls represent a new type of damaging element, and studies on their damaging power of composite armor are required for a comprehensive evaluation of the effectiveness of various types of weapons. The goal of this study was to determine the impact of ϕ7 mm toughened Al2O3 ceramic balls on a composite ceramic/metal armor. The influences of the ceramic panel and the thickness of the metal backing material on the destroying power of the ceramic balls were first determined. Based on the agreement between numerical simulation, experimental results, and calculation models of the target plate resistance, the response mechanism of the ceramic balls was further analyzed. The results indicate that for a back plate of Q235 steel, with an increasing thickness of the ceramic panel, the piercing speed limit of the ceramic balls gradually increases and the diameter of the out-going hole on the metal back decreases. Different conditions were tested to assess the effects on the piercing speed, the diameter of the out-going hole, the micro-element stress, and the integrity of the recovered ceramic bowl. [ABSTRACT FROM AUTHOR]