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For the perturbed Hamiltonian of a multifrequency resonance harmonic oscillator, a new approach to calculating the coefficients in the procedure of quantum averaging is proposed. The procedure of quantum averaging is transferred to the space of the graded algebra of symbols by using twisted product introduced in the paper. As a result, the averaged Hamiltonian is represented as a function of generators of the quantum symmetry algebra of the harmonic part of the Hamiltonian. The proposed method is applied to the spectral problem for the Hamiltonian of the cylindrical Penning trap. [ABSTRACT FROM AUTHOR]

In this paper, we construct a symbol calculus yielding short exact sequences for the dual Toeplitz algebra generated by all bounded dual Toeplitz operators on the Hardy space associated with the polydisk D n in the unitary space C n , that have been introduced and well studied in our earlier paper (Benaissa and Guediri in Taiwan J Math 19: 31–49, 2015), as well as for the C*-subalgebra generated by dual Toeplitz operators with symbols continuous on the associated hypertorus T n . [ABSTRACT FROM AUTHOR]

In this paper we investigate left ideals as codes in twisted skew group rings. The considered rings, which are in most cases algebras over a finite field, allow us to retrieve many of the well-known codes. The presentation, given here, unifies the concept of group codes, twisted group codes and skew group codes. [ABSTRACT FROM AUTHOR]

In this paper we show how the Functional Separation of Variables (FSoV) method can be applied to the problem of computing overlaps with integrable boundary states in integrable systems. We demonstrate our general method on the example of a particular boundary state, a singlet of the symmetry group, in an su 3 rational spin chain in an alternating fundamental-anti-fundamental representation. The FSoV formalism allows us to compute in determinant form not only the overlaps of the boundary state with the eigenstates of the transfer matrix, but in fact with any factorisable state. This includes off-shell Bethe states, whose overlaps with the boundary state have been out of reach with other methods. Furthermore, we also found determinant representations for insertions of so-called Principal Operators (forming a complete algebra of all observables) between the boundary and the factorisable state as well as certain types of multiple insertions of Principal Operators. Concise formulas for the matrix elements of the boundary state in the SoV basis and su N generalisations are presented. Finally, we managed to construct a complete basis of integrable boundary states by repeated action of conserved charges on the singlet state. As a result, we are also able to compute the overlaps of all of these states with integral of motion eigenstates. [ABSTRACT FROM AUTHOR]

It has been shown that by using a Lagrange multiplier field to ensure that the classical equations of motion are satisfied, radiative effects beyond one-loop order are eliminated. It has also been shown that through the contribution of some additional ghost fields, the effective action becomes form invariant under a redefinition of field variables, and furthermore, the usual one-loop results coincide with the quantum corrections obtained from this effective action. In this paper, we consider the consequences of a gauge invariance being present in the classical action. The resulting gauge transformations for the Lagrange multiplier field as well as for the additional ghost fields are found. These gauge transformations result in a set of Faddeev–Popov ghost fields arising in the effective action. If the gauge algebra is closed, we find the Becci–Rouet–Stora–Tyutin (BRST) transformations that leave the effective action invariant. [ABSTRACT FROM AUTHOR]

In this paper, we study conformal points among the class of E -models. The latter are σ-models formulated in terms of a current Poisson algebra, whose Lie-theoretic definition allows for a purely algebraic description of their dynamics and their 1-loop RG-flow. We use these results to formulate a simple algebraic condition on the defining data of such a model which ensures its 1-loop conformal invariance and the decoupling of its observables into two chiral Poisson algebras, describing the classical left- and right-moving fields of the theory. In the case of so-called non-degenerate E -models, these chiral sectors form two current algebras and the model takes the form of a WZW theory once realised as a σ-model. The case of degenerate E -models, in which a subalgebra of the current algebra is gauged, is more involved: the conformal condition yields a wider class of theories, which includes gauged WZW models but also other examples, seemingly different, which however sometimes turn out to be related to gauged WZW models based on other Lie algebras. For this class, we build non-local chiral fields of parafermionic-type as well as higher-spin local ones, forming classical W -algebras. In particular, we find an explicit and efficient algorithm to build these local chiral fields. These results (and their potential generalisations discussed at the end of the paper) open the way for the quantisation of a large class of conformal E -models using the standard operator formalism of two-dimensional CFT. [ABSTRACT FROM AUTHOR]

In this paper, we consider 3D Young diagrams with at most N layers in z-axis direction, which can be constructed by N 2D Young diagrams on slice z = j, j = 1, 2, · · · , N from the Yang-Baxter equation. Using 2D Bosons {aj,m, m ∈ ℤ} associated to 2D Young diagrams on the slice z = j, we constructed 3D Bosons. Then we show the 3D Boson representation of W1+∞ algebra, and give the method to calculate the Littlewood-Richardson rule for 3-Jack polynomials. [ABSTRACT FROM AUTHOR]

In this paper, we prove that any group occurs as the group of homotopy classes of self-equivalences of a none elliptic Sullivan algebra. [ABSTRACT FROM AUTHOR]

Let Γ ⊆ PSL (2 , R) correspond to the group of units of norm 1 in an Eichler order O of an indefinite quaternion algebra over Q . Closed geodesics on Γ \ H correspond to optimal embeddings of real quadratic orders into O . The weighted intersection numbers of pairs of these closed geodesics conjecturally relates to the work of Darmon-Vonk on a real quadratic analogue to the difference of singular moduli. In this paper, we study the total intersection number over all embeddings of a given pair of discriminants. We precisely describe the arithmetic of each intersection, and produce a formula for the total intersection. This formula is a real quadratic analogue of the work of Gross and Zagier on factorizing the difference of singular moduli. The results are fairly general, allowing for a large class of non-maximal Eichler orders, and non-fundamental/non-coprime discriminants. The paper ends with some explicit examples illustrating the results of the paper. [ABSTRACT FROM AUTHOR]

In this paper we present the Hietarinta Chern–Simons supergravity theory in three space-time dimensions which extends the simplest Poincaré supergravity theory. After approaching the construction of the action using the Chern–Simons formalism, the analysis of the corresponding asymptotic symmetry algebra is considered. For this purpose, we first propose a consistent set of asymptotic boundary conditions for the aforementioned supergravity theory whose underlying symmetry corresponds to the supersymmetric extension of the Hietarinta algebra. We then show that the corresponding charge algebra contains the super- bms 3 algebra as subalgebra, and has three independent central charges. We also show that the obtained asymptotic symmetry algebra can alternatively be recovered as a vanishing cosmological constant limit of three copies of the Virasoro algebra, one of which is augmented by supersymmetry. [ABSTRACT FROM AUTHOR]

In 1993, I. P. Shestakov posed the question of the existence of a central simple finite-dimensional algebra over a field of characteristic zero whose identities are not given by a finite set of identities. In 2012, I. M. Isaev and the author of the present paper constructed an example of a seven-dimensional central simple algebra over any field that does not have a finite basis of identities. In the present paper, we construct an example of a six-dimensional central simple algebra over a field of characteristic zero which has no finite basis of identities. [ABSTRACT FROM AUTHOR]

Similar to gravity, an infinite tower of symmetries generated by higher-spin charges has been identified in Yang-Mills theory by studying collinear limits or celestial operator products of gluons. This work aims to recover this loop symmetry in terms of charge aspects constructed on the gluonic Fock space. We propose an explicit construction for these higher spin charge aspects as operators which are polynomial in the gluonic annihilation and creation operators. The core of the paper consists of a proof that the charges we propose form a closed loop algebra to quadratic order. This closure involves using the commutator of the cubic order expansion of the charges with the linear (soft) charge. Quite remarkably, this shows that this infinite-dimensional symmetry constrains the non-linear structure of Yang-Mills theory. We provide a similar all spin proof in gravity for the so-called global quadratic (hard) charges which form the loop wedge subalgebra of w1+∞. [ABSTRACT FROM AUTHOR]

This is the first in a series of two papers that study monogenicity of number rings from a moduli-theoretic perspective. Given an extension of algebras B/A, when is B generated by a single element θ ∈ B over A? In this paper, we show there is a scheme M B / A parameterizing the choice of a generator θ ∈ B , a "moduli space" of generators. This scheme relates naturally to Hilbert schemes and configuration spaces. We give explicit equations and ample examples. [ABSTRACT FROM AUTHOR]

We perform a complete and systematic analysis of the solution space of six-dimensional Einstein gravity. We show that a particular subclass of solutions — those that are analytic near I + — admit a non-trivial action of the generalised Bondi-Metzner-van der Burg-Sachs (GBMS) group which contains infinite-dimensional supertranslations and superrotations. The latter consists of all smooth volume-preserving Diff×Weyl transformations of the celestial S4. Using the covariant phase space formalism and a new technique which we develop in this paper (phase space renormalization), we are able to renormalize the symplectic potential using counterterms which are local and covariant. The Hamiltonian charges corresponding to GBMS diffeomorphisms are non-integrable. We show that the integrable part of these charges faithfully represent the GBMS algebra and in doing so, settle a long-standing open question regarding the existence of infinite-dimensional asymptotic symmetries in higher even dimensional non-linear gravity. Finally, we show that the semi-classical Ward identities for supertranslations and superrotations are precisely the leading and subleading soft-graviton theorems respectively. [ABSTRACT FROM AUTHOR]

In this paper we consider perturbation theory in generic two-dimensional sigma models in the so-called first-order formalism, using the coordinate regularization approach. Our goal is to analyze the first-order formalism in application to β functions and compare its results with the standard geometric calculations. Already in the second loop, we observe deviations from the geometric results that cannot be explained by the regularization/renormalization scheme choices. Moreover, in certain cases the first-order calculations produce results that are not symmetric under the classical diffeomorphisms of the target space. Although we could not present the full solution to this remarkable phenomenon, we found some indirect arguments indicating that an anomaly similar to that established in supersymmetric Yang-Mills theory might manifest itself starting from the second loop. We discuss why the difference between two answers might be an infrared effect, similar to that in β functions in supersymmetric Yang-Mills theories. In addition to the generic Kähler target spaces we discuss in detail the so-called Lie-algebraic sigma models. In particular, this is the case when the perturbed field G i j ¯ is a product of the holomorphic and antiholomorphic currents satisfying two-dimensional current algebra. [ABSTRACT FROM AUTHOR]

The tensionless limit of string theory has recently been formulated in terms of worldsheet Rindler physics. In this paper, by considering closed strings moving in background Rindler spacetimes, we provide a concrete exemplification of this phenomenon. We first show that strings probing the near-horizon region of a generic non-extremal blackhole become tensionless thereby linking a spacetime Carroll limit to a worldsheet Carroll limit. Then, considering strings in d-dimensional Rindler spacetime we find a Rindler structure induced on the worldsheet. Novelties, including folds, appear on the closed string worldsheet pertaining to the formation of the worldsheet horizon. The closed string becomes segmented at these folding points and different segments go into the formation of closed strings in the different Rindler wedges. The Bondi-Metzner-Sachs (BMS) or the Conformal Carroll algebra emerges from the closed string Virasoro algebra as the horizon is hit. Quantum states on these accelerated worldsheets are discussed and we show the formation of boundary states from gluing conditions of the different segments of the accelerated closed string. [ABSTRACT FROM AUTHOR]

Considering supergravity theory is a natural step in the development of gravity models. This paper follows the "algebraic" path and constructs possible extensions of the Poincaré and Anti-de-Sitter algebras, which inherit their basic commutation structure. Previously achieved results of this type are fragmentary and show only a limited fraction of possible algebraic realizations. Our paper presents the newly obtained symmetry algebras, evaluated within an efficient pattern-based computational method of generating the so-called 'resonating' algebraic structures. These supersymmetric extensions of algebras, going beyond the Poincaré and Anti-de Sitter ones, contain additional bosonic generators Z ab (Lorentz-like), and U a (translational-like) added to the standard Lorentz generator J ab and translation generator P a . Our analysis includes all cases up to two fermionic supercharges, Q α and Y α . The delivered plethora of superalgebras includes few past results and offers a vastness of new examples. The list of the cases is complete and contains all superalgebras up to two of Lorentz-like, translation-like, and supercharge-like generators (J P + Q) + (Z U + Y) = J P Z U + Q Y . In the latter class, among 667 founded superalgebras, the 264 are suitable for direct supergravity construction. For each of them, one can construct a unique supergravity model defined by the Lagrangian. As an example, we consider one of the algebra configurations and provide its Lagrangian realization. [ABSTRACT FROM AUTHOR]

Let be a field of characteristic zero, and let be the first Weyl algebra. In the present paper, we prove the following two results. Assume that there exists a nonzero polynomial such that (i) has a nontrivial solution with ; (ii) the set of solutions of in splits into finitely many -orbits under the natural actuon of the group . Then the Dixmier conjecture holds; i.e., every is an automorphism. Assume that is an endomorphism of monomial type. (In particular, it is not an automorphism; see Theorem 4.1.) Then has no nontrivial fixed points; i.e. there exists no such that . [ABSTRACT FROM AUTHOR]

Let X be an infinite-dimensional complex Banach space, B (X) the algebra of all bounded linear operators on X . Denote the spectral domain by σ γ (T) = { λ ∈ σ a (T) : T that is semi-Fredholm and a s c (T − λ I) < ∞ } . In this paper, we characterize the structure of additive surjective maps φ : B (X) → B (X) with σ γ (φ (T)) = σ γ (T) for all T ∈ B (X) . [ABSTRACT FROM AUTHOR]

In this paper, we reduce the electromagnetic theory to future null infinity and obtain a vector theory at the boundary. We compute the Poincaré flux operators which could be generalized. We quantize the vector theory, and impose normal order on the extended flux operators. It is shown that these flux operators generate the supertranslation and superrotation. When work out the commutators of these operators, we find that a generalized electromagnetic duality operator should be included as the generators to form a closed symmetry algebra. [ABSTRACT FROM AUTHOR]

Let H be a complex Hilbert space and let B (H) be the algebra of all bounded linear operators on H. In this paper, for T ∈ B (H) and a unit vector x ∈ H , we introduce a local version of the reduced minimum modulus of T at x, noted by γ (T , x) . Properties of this quantity are investigated. We study the relations between γ (T , x) and the Moore–Penrose inverse, spectrum of | T | and the local spectrum of | T | at x. At the end of this paper we will be interested in several problems around this quantity (preserving, continuity, local spectral theory). [ABSTRACT FROM AUTHOR]

Both the authors have participated in all three case studies of the GraBaTs 2008 tool contest with the Fujaba tool. This paper reports about our solutions and the improvements we have made to the Fujaba tool suite in order to enhance performance and modeling capabilities and about the lessons we have learned from these case studies. [ABSTRACT FROM AUTHOR]

In the paper, it is proved that the radical of a relatively free associative algebra of countable rank over an infinite field of characteristic p > 0 is a nil ideal of bounded index if the complexity of the corresponding variety is less than p. Moreover, a description of a basis for trace identities for the matrix algebra M n over an infinite field of characteristic p > 0, n < p, is obtained in the paper. [ABSTRACT FROM AUTHOR]

The Kepler's equation of elliptic orbits is one of the most significant fundamental physical equations in Satellite Geodesy. This paper demonstrates symbolic iteration method based on computer algebra analysis (SICAA) to solve the Kepler's equation. The paper presents general symbolic formulas to compute the eccentric anomaly (E) without complex numerical iterative computation at run-time. This approach couples the Taylor series expansion with higher-order trigonometric function reductions during the symbolic iterative progress. Meanwhile, the relationship between our method and the traditional infinite series expansion solution is analyzed in this paper, obtaining a new truncation method of the series expansion solution for the Kepler's equation. We performed substantial tests on a modest laptop computer. Solutions for 1,002,001 pairs of (e, M) has been conducted. Compared with numerical iterative methods, 99.93% of all absolute errors δE of eccentric anomaly (E) obtained by our method is lower than machine precision ϵ over the entire interval. The results show that the accuracy is almost one order of magnitude higher than that of those methods (double precision). Besides, the simple codes make our method well-suited for a wide range of algebraic programming languages and computer hardware (GPU and so on). [ABSTRACT FROM AUTHOR]

In the paper (J Math Soc Jpn 72(1):303–331, 2020), Tse-Chung Yang and the first two current authors computed explicitly the number | SSp 2 (F q) | of isomorphism classes of superspecial abelian surfaces over an arbitrary finite field F q of even degree over the prime field F p . There it was assumed that certain commutative Z p -orders satisfy an étale condition that excludes the primes p = 2 , 3 , 5 . We treat these remaining primes in the present paper, where the computations are more involved because of the ramification. This completes the calculation of | SSp 2 (F q) | in the even degree case. The odd degree case was previous treated by Tse-Chung Yang and the first two current authors in (Doc Math 21:1607–1643, 2016). To complete the proof of our main theorem, we give a classification of lattices over local quaternion Bass orders, which is a new input to our previous works. [ABSTRACT FROM AUTHOR]

Summary This paper investigates the fate of Thomas Harriot's algebra after his death in 1621 and, in particular, the largely unsuccessful efforts of seventeenth-century mathematicians to promote it. The little known surviving manuscripts of Nathaniel Torporley have been used to elucidate the roles of Torporley and Walter Warner in the preparation of the Praxis, and a partial translation of Torporley's important critique of the Praxis is offered here for the first time. The known whereabouts of Harriot's mathematical papers, both originals and copies, during the seventeenth century and later are summarised. John Wallis's controversial 1685 account of Harriot's algebra is examined in detail and it is argued that John Pell's influence on Wallis was far more significant than has previously been realised. The paper ends with a reassessment of Harriot's underrated and important contribution to the development of modern algebra. [ABSTRACT FROM AUTHOR]

In this paper the gradient-free modification of the mirror descent method for convex stochastic online optimization problems is proposed. The crucial assumption in the problem setting is that function realizations are observed with minor noises. The aim of this paper is to derive the convergence rate of the proposed methods and to determine a noise level which does not significantly affect the convergence rate. [ABSTRACT FROM AUTHOR]

The aim of this paper is to determine the λ-linear functionals sending any given polynomial p (x) with coefficients in C p to the p-adic invariant integral of P (x) on Z p and also to that of P (x 1 + ⋯ + x r) on Z p r . We show that the former is given by the generating function of degenerate Bernoulli polynomials and the latter by that of degenerate Bernoulli polynomials of order r. For this purpose, we use the λ-umbral algebra which has been recently introduced by Kim and Kim (J. Math. Anal. Appl. 493(1):124521 2021). [ABSTRACT FROM AUTHOR]

In this paper, we provide a recursive method to construct self-orthogonal and self-dual codes of the type { k 1 , k 2 , ... , k e } and length n over the quasi-Galois ring F 2 r [ u ] / < u e > from a self-orthogonal code of the same length n and dimension k 1 + k 2 + ⋯ + k ⌈ e 2 ⌉ over F 2 r and vice versa, where F 2 r is the finite field of order 2 r , n ≥ 1 , e ≥ 2 are integers, ⌈ e 2 ⌉ is the smallest integer greater than or equal to e 2 , and k 1 , k 2 , ... , k e are non-negative integers satisfying k 1 ≤ n - (k 1 + k 2 + ⋯ + k e) and k i = k e - i + 2 for 2 ≤ i ≤ e. We further apply this recursive method to provide explicit enumeration formulae for self-orthogonal and self-dual codes of an arbitrary length over the ring F 2 r [ u ] / < u e > . With the help of these enumeration formulae and by carrying out computations in the Magma Computational Algebra system, we classify all self-orthogonal and self-dual codes of lengths 2, 3, 4, 5 over the ring F 2 [ u ] / < u 3 > and of lengths 2, 3, 4 over the ring F 4 [ u ] / < u 2 > . [ABSTRACT FROM AUTHOR]

In this paper, approximate Noether and Lie symmetries of 2nd order for Gibbons–Maeda–Garfinkle–Horowitz–Strominger (GMGHS) charged black hole in the Einstein frame are analyzed comprehensively. To explore the approximate Noether symmetries of 2nd order, Noether symmetries of Minkowski spacetime are used which forms a 17 dimensional Lie algebra. It is observed that no new approximate Noether symmetry is obtained at 1st and 2nd order. To examine the 1st and 2nd order approximate Lie symmetries of the GMGHS black hole spacetime, 35 Lie symmetries (exact) of the Minkowski spacetime are used which forms an algebra sl(6, R). It is shown that no new approximate Lie symmetry exists at 1st and 2nd order and only exact 35 symmetries are recouped as trivial approximate Lie symmetries at both orders. Furthermore, no energy rescaling factor is seen in this spacetime. [ABSTRACT FROM AUTHOR]

This paper introduces the counterpart of strong bisimilarity for labelled transition systems extended with timeout transitions. It supports this concept through a modal characterisation, congruence results for a standard process algebra with recursion, and a complete axiomatisation. [ABSTRACT FROM AUTHOR]

In this paper we determine the generalized Weierstrass semigroup H ^ (P ∞ , P 1 , ... , P m) , and consequently the Weierstrass semigroup H (P ∞ , P 1 , ... , P m) , at m + 1 points on the curves X a , b , n , s and Y n , s . These curves has been introduced in Tafazolian et al (J Pure Appl Algebra 220:1122–1132, 2016) as new examples of curves which cannot be covered by the Hermitian curve. [ABSTRACT FROM AUTHOR]

In this paper, we aim at highlighting the significance of the A- and B-properties introduced by Finch (Bull Aust Math Soc 2:57–62, 1970b). These conditions turn out to capture interesting structural features of lattices of closed subspaces of complete inner vector spaces. Moreover, we generalise them to the context of effect algebras, establishing a novel connection between quantum structures (orthomodular posets, orthoalgebras, effect algebras) arising from the logico-algebraic approach to quantum mechanics. [ABSTRACT FROM AUTHOR]

In this paper, we study a family of C*-subalgebras defined by fixed points of generalized gauge actions of a Cuntz--Krieger algebra by introducing a family of étale groupoids whose associated C*-algebras are these C*-subalgebras. We show that topological conjugacy classes of one-sided topological Markov shifts are characterized in terms of the isomorphism classes of these étale groupoids. [ABSTRACT FROM AUTHOR]

In this paper, we consider the problem of constructing semiclassical asymptotics for the tunnel splitting of the spectrum of an operator defined on an irreducible representation of the Lie algebra . It is assumed that the operator is a quadratic function of the generators of the algebra. We present coherent states and a unitary coherent transform that allow us to reduce the problem to the analysis of a second-order differential operator in the space of holomorphic functions. Semiclassical asymptotic spectral series and the corresponding wave functions are constructed as decompositions in coherent states. For some values of the system parameters, the minimal energy corresponds to a pair of nondegenerate equilibria, and the discrete spectrum of the operator has an exponentially small tunnel splitting of the levels. We apply the complex WKB method to prove asymptotic formulas for the tunnel splitting of the energies. We also show that, in contrast to the one-dimensional Schrödinger operator, the tunnel splitting in this problem not only decays exponentially but also contains an oscillating factor, which can be interpreted as tunneling interference between distinct instantons. We also show that, for some parameter values, the tunneling is completely suppressed and some of the spectral levels are doubly degenerate, which is not typical of one-dimensional systems. [ABSTRACT FROM AUTHOR]

Problems are often too complex to solve them in an optimal way. The complexity arises from connections between their elements, such that a change in one element influences the performance of other elements. Kauffman’s NK model offers a way to depict such interdependencies and has therefore often been used in economic investigations of the influence of problem or search decomposition on the attainable results. However, papers on the effect of different decompositions on solution quality come to contradictory conclusions. Some observe an initial advantage of over-modularization where others do not. As they also differ in the employed search procedures, but do not base them on empirical findings, the present paper examines the results of more empirically based search strategies. Using algorithms based on innovation strategies derived from patent data, the paper establishes a clear advantage of correct problem decompositions. [ABSTRACT FROM AUTHOR]

In the paper, we consider the problem of uniform approximation of a continuous function defined on a compact metric space by elements of the sum of two algebras in the space of all continuous functions on . We prove a Chebyshev-type theorem for characterization of best approximation. [ABSTRACT FROM AUTHOR]

Let Q be a subset of a finite distributive lattice D. An algebra A represents the inclusion Q ⊆ D by principal congruences if the congruence lattice of A is isomorphic to D and the ordered set of principal congruences of A corresponds to Q under this isomorphism. If there is such an algebra for every subset Q containing 0, 1, and all join-irreducible elements of D, then D is said to be fully (A1)-representable. We prove that every fully (A1)- representable finite distributive lattice is planar and it has at most one joinreducible coatom. Conversely, we prove that every finite planar distributive lattice with at most one join-reducible coatom is fully chain-representable in the sense of a recent paper of G. Grätzer. Combining the results of this paper with another result of the present author, it follows that every fully (A1)- representable finite distributive lattice is "fully representable" even by principal congruences of finite lattices. Finally, we prove that every chain-representable inclusion Q ⊆ D can be represented by the principal congruences of a finite (and quite small) algebra. [ABSTRACT FROM AUTHOR]

Interpretation methods constitute a foundation of the termination analysis of term rewriting. From time to time, remarkable instances of interpretation methods appeared, such as polynomial interpretations, matrix interpretations, arctic interpretations, and their variants. In this paper, we introduce a general framework, the tuple interpretation method, that subsumes these variants as well as many previously unknown interpretation methods as instances. Employing the notion of derivers, we prove the soundness of the proposed method in an elegant way. We implement the proposed method in the termination prover NaTT and verify its significance through experiments. [ABSTRACT FROM AUTHOR]

Graph neural networks (GNNs) show powerful processing ability on graph structure data for nodes and graph classification. However, existing GNN models may cause information loss with the increasing number of the network layer. To improve the graph-structured data features representation quality, we introduce geometric algebra into graph neural networks. In this paper, we construct a high-dimensional geometric algebra (GA) space in the Non-Euclidean domain to better learn vector embedding for graph nodes. We focus our study on few-shot learning and propose a geometric algebra graph neural network (GA-GNN) as the metric network for cross-domain few-shot classification tasks. In the geometric algebra space, the feature nodes are mapped into hyper-complex vector, which helps reduce the distortion of feature information with the increased hidden layers. The experimental results demonstrate that the approach we proposed achieves the state-of-the-art few-shot cross-domain classification accuracy in five public datasets. [ABSTRACT FROM AUTHOR]

Bondi-Metzner-Sachs (BMS) symmetries, or equivalently Conformal Carroll symmetries, are intrinsically associated to null manifolds and in two dimensions can be obtained as an Inönü-Wigner contraction of the two-dimensional (2d) relativistic conformal algebra. Instead of performing contractions, we demonstrate in this paper how this transmutation of symmetries can be achieved by infinite boosts or degenerate linear transformations on coordinates. Taking explicit cues from the worldsheet theory of null strings, we show boosting the system is equivalent to adding a current-current deformation term to the Hamiltonian. As the strength of this deformation term reaches a critical value, the classical symmetry algebra "flows" from two copies of Virasoro to the BMS algebra. We further explore the situation where the CFT coordinates are asymmetrically transformed, and degenerate limits lead to chiral theories. [ABSTRACT FROM AUTHOR]

In this paper we continue our study of the tree level MHV graviton scattering amplitudes from the point of view of celestial holography. In arXiv:2008.04330 we showed that the celestial OPE of two gravitons in the MHV sector can be written as a linear combination of SL 2 ℂ ¯ current algebra and supertranslation descendants. In this note we show that the OPE is in fact manifestly invariant under the infinite dimensional Virasoro algebra as is expected for a 2-D CFT. This is consistent with the conjecture that the holographic dual in 4-D asymptotically flat space time is a 2-D CFT. Since we get only one copy of the Virasoro algebra we can conclude that the holographic dual theory which computes the MHV amplitudes is a chiral CFT with a host of other infinite dimensional global symmetries including SL 2 ℂ ¯ current algebra, supertranslations and subsubleading soft graviton symmetry. We also discuss some puzzles related to the appearance of the Virasoro symmetry. [ABSTRACT FROM AUTHOR]

Motivated by the Swampland Distance and the Emergent String Conjecture of Quantum Gravity, we analyse the infinite distance degenerations in the complex structure moduli space of elliptic K3 surfaces. All complex degenerations of K3 surfaces are known to be classified according to their associated Kulikov models of Type I (finite distance), Type II or Type III (infinite distance). For elliptic K3 surfaces, we characterise the underlying Weierstrass models in detail. Similarly to the known two classes of Type II Kulikov models for elliptic K3 surfaces we find that the Weierstrass models of the more elusive Type III Kulikov models can be brought into two canonical forms. We furthermore show that all infinite distance limits are related to degenerations of Weierstrass models with non-minimal singularities in codimension one or to models with degenerating generic fibers as in the Sen limit. We explicitly work out the general structure of blowups and base changes required to remove the non-minimal singularities. These results form the basis for a classification of the infinite distance limits of elliptic K3 surfaces as probed by F-theory in the companion paper [1]. The Type III limits, in particular, are (partial) decompactification limits as signalled by an emergent affine enhancement of the symmetry algebra. [ABSTRACT FROM AUTHOR]

In this paper, we employ the well-known Krasnoselskii fixed point theorem to study the existence and n-multiplicity of positive periodic solutions for the periodic impulsive functional differential equations with two parameters. The form including an impulsive term of the equations in this paper is rather general and incorporates as special cases various problems which have been studied extensively in the literature. Easily verifiable sufficient criteria are obtained for the existence and n-multiplicity of positive periodic solutions of the impulsive functional differential equations. [ABSTRACT FROM AUTHOR]

Expressing machine-interpretable statements in the form of subject-predicate-object triples is a well-established practice for capturing semantics of structured data. However, the standard used for representing these triples, RDF, inherently lacks the mechanism to attach provenance data, which would be crucial to make automatically generated and/or processed data authoritative. This paper is a critical review of data models, annotation frameworks, knowledge organization systems, serialization syntaxes, and algebras that enable provenance-aware RDF statements. The various approaches are assessed in terms of standard compliance, formal semantics, tuple type, vocabulary term usage, blank nodes, provenance granularity, and scalability. This can be used to advance existing solutions and help implementers to select the most suitable approach (or a combination of approaches) for their applications. Moreover, the analysis of the mechanisms and their limitations highlighted in this paper can serve as the basis for novel approaches in RDF-powered applications with increasing provenance needs. [ABSTRACT FROM AUTHOR]

Tropical algebra emerges in many fields of mathematics such as algebraic geometry, mathematical physics and combinatorial optimization. In part, its importance is related to the fact that it makes various parameters of mathematical objects computationally accessible. Tropical polynomials play a fundamental role in this, especially for the case of algebraic geometry. On the other hand, many algebraic questions behind tropical polynomials remain open. In this paper, we address four basic questions on tropical polynomials closely related to their computational properties: Given a polynomial with a certain support (set of monomials) and a (finite) set of inputs, when is it possible for the polynomial to vanish on all these inputs? A more precise question, given a polynomial with a certain support and a (finite) set of inputs, how many roots can this polynomial have on this set of inputs? Given an integer k, for which s there is a set of s inputs such that any nonzero polynomial with at most k monomials has a non-root among these inputs? How many integer roots can have a one variable polynomial given by a tropical algebraic circuit? In the classical algebra well-known results in the direction of these questions are Combinatorial Nullstellensatz due to N. Alon, J. Schwartz–R. Zippel Lemma and Universal Testing Set for sparse polynomials, respectively. The classical analog of the last question is known as τ -conjecture due to M. Shub–S. Smale. In this paper, we provide results on these four questions for tropical polynomials. [ABSTRACT FROM AUTHOR]

In this paper we present a constructive method to characterize ideals of the local ring O C n , 0 of germs of holomorphic functions at 0 ∈ C n which arise as the moduli ideal ⟨ f , m j (f) ⟩ , for some f ∈ m ⊂ O C n , 0 . A consequence of our characterization is an effective solution to a problem dating back to the 1980s, called the Reconstruction Problem of the hypersurface singularity from its moduli algebra. Our results work regardless of whether the hypersurface singularity is isolated or not. [ABSTRACT FROM AUTHOR]

Let be an embedded projective variety. The Lie algebra defined by the tangent directions to at smooth points is an interesting algebraic invariant of . In some cases, this algebra is isomorphic to the symbol algebra of a filtered system of distributions on a Fano manifold, which plays an important role in the theory of these manifolds. In addition, algebras defined by tangent directions are interesting on their own right. In this paper, we study the Lie algebra corresponding to a variety that is the projectivization of the orbit of the lowest weight vector of an irreducible representation of a complex semisimple Lie group. We describe these algebras in terms of generators and relations. In many cases, we can describe their structure completely. [ABSTRACT FROM AUTHOR]

In this paper, we deal with the lumpability approach to cope with the state space explosion problem inherent to the computation of the stationary performance indices of large stochastic models. The lumpability method is based on a state aggregation technique and applies to Markov chains exhibiting some structural regularity. Moreover, it allows one to efficiently compute the exact values of the stationary performance indices when the model is actually lumpable. The notion of quasi-lumpability is based on the idea that a Markov chain can be altered by relatively small perturbations of the transition rates in such a way that the new resulting Markov chain is lumpable. In this case, only upper and lower bounds on the performance indices can be derived. Here, we introduce a novel notion of quasi-lumpability, named proportional lumpability, which extends the original definition of lumpability but, differently from the general definition of quasi-lumpability, it allows one to derive exact stationary performance indices for the original process. We then introduce the notion of proportional bisimilarity for the terms of the performance process algebra PEPA. Proportional bisimilarity induces a proportional lumpability on the underlying continuous-time Markov chains. Finally, we prove some compositionality results and show the applicability of our theory through examples. [ABSTRACT FROM AUTHOR]

This paper proposes a novel, probabilistic data model and algebra that improves the modeling and querying of uncertain data in spatial OLAP (SOLAP) to support location-based services. Data warehouses that support location-based services need to combine complex hierarchies, such as road networks or transportation infrastructures, with static and dynamic content, e.g., speed limits and vehicle positions, respectively. Both the hierarchies and the content are often uncertain in real-world applications. Our model supports the use of probability distributions within both facts and dimensions. We give an algebra that correctly aggregates uncertain data over uncertain hierarchies. This paper also describes an implementation of the model and algebra, gives a complexity analysis of the algebra, and reports on an empirical, experimental evaluation of the implementation. The work is motivated with a real-world case study, based on our collaboration with a leading Danish vendor of location-based services. [ABSTRACT FROM AUTHOR]