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Algebra environments provide requisites for studying objects of interest in operator algebra theory, group representation theory, spin geometry, Clifford analysis, and several variable operator theory. The concept is analyzed by developing an algebraic geometry approach. Specific algebraic sets, called structure manifolds of algebra environments, and their Zariski tangent spaces are introduced and described by using as critical tools derivations on algebras. Structure manifolds of tensor environments in particular yield spaces of algebra homomorphisms. Consequently, such spaces could be investigated as algebraic manifolds. Related issues include characterizations of their Zariski tangent spaces and of derivations that preserve algebra homomorphisms. [ABSTRACT FROM AUTHOR]

A linear group is called unisingular if every element of it has eigenvalue 1. In this paper we develop some general machinery for the study of unisingular irreducible linear groups. A motivation for the study of such groups comes from several sources, including algebraic geometry, Galois theory, finite group theory and representation theory. In particular, a certain aspect of the theory of abelian varieties requires the knowledge of unisingular irreducible subgroups of the symplectic groups over the field of two elements, and in this paper we concentrate on this special case of the general problem. A more special but important question is that of the existence of such subgroups in the symplectic groups of particular degrees. We answer this question for almost all degrees 2 n < 250 , specifically, the question remains open only 7 values of n. [ABSTRACT FROM AUTHOR]

In a general L 2 extension theorem of Demailly for log canonical pairs, the L 2 criterion with respect to a measure called the Ohsawa measure determines when a given holomorphic function can be extended. Despite the analytic nature of the Ohsawa measure, we establish a geometric characterization of this analytic criterion using the theory of log canonical centers from algebraic geometry. Along the way, we characterize when the Ohsawa measure fails to have generically smooth positive density, which answers an essential question arising from Demailly's work. [ABSTRACT FROM AUTHOR]

Motivated by questions in algebraic geometry, Yifeng Huang recently derived generating functions for counting mutually annihilating matrices and mutually annihilating nilpotent matrices over a finite field. We give a different derivation of his results using statistical properties of random partitions chosen from the Cohen-Lenstra measure. [ABSTRACT FROM AUTHOR]

Linear complementary dual (LCD) codes and linear complementary pairs (LCP) of codes have been proposed for new applications as countermeasures against side-channel attacks (SCA) and fault injection attacks (FIA) in the context of direct sum masking (DSM). The countermeasure against FIA may lead to a vulnerability for SCA when the whole algorithm needs to be masked (in environments like smart cards). This led to a variant of the LCD and LCP problems, where several results were obtained intensively for LCD codes, but only partial results were derived for LCP codes. Given the gap between the thin results and their particular importance, this paper aims to reduce this by further studying the LCP of codes in special code families and, precisely, the characterization and construction mechanism of LCP codes of algebraic geometry codes over finite fields. Notably, we propose constructing explicit LCP of codes from elliptic curves. Besides, we also study the security parameters of the derived LCP of codes (C , D) (notably for cyclic codes), which are given by the minimum distances d (C) and d (D ⊥). Further, we show that for LCP algebraic geometry codes (C , D) , the dual code C ⊥ is equivalent to D under some specific conditions we exhibit. Finally, we investigate whether MDS LCP of algebraic geometry codes exist (MDS codes are among the most important in coding theory due to their theoretical significance and practical interests). Construction schemes for obtaining LCD codes from any algebraic curve were given in 2018 by Mesnager, Tang and Qi in [11]. To our knowledge, it is the first time LCP of algebraic geometry codes has been studied. [ABSTRACT FROM AUTHOR]

Let F q denote the finite field of size q and F q n denote the set of n -tuples of elements from F q. A family of k -dimensional subspaces of F q n , which forms a partial spread, is called L -almost affinely disjoint (or briefly [ n , k , L ] q -AAD) if each affine coset of a member of this family intersects with only at most L subspaces from the family. Polyanskii and Vorobyev introduced AAD subspace families in [23] (2019) in order to construct some types of primitive batch codes. For this purpose, they made use of Reed-Solomon codes and hence they presented [ n = 2 k + 1 , k , L = k ] -AAD subspace families of size ⌊ q / k ⌋. Later on, AAD spaces received a notable attention and have been studied for various parameters, see Liu et al. [18] (2021) and Otal and Arıkan [21] (2022) for example. In Arıkan et al. (2023) [1] , we presented a construction of [ 2 k + 1 , k , k ] -AAD subspace families of size q , which improves the lower bound ⌊ q / k ⌋ given by Polyanskii and Vorobyev in [23] (2019) k times. We also proved that our construction yields AAD subspace families for k = 2 and k = 3 , and left the proof of the case k ≥ 4 as a conjecture. In this paper, we now prove this conjecture by a newer and more abstract approach. For the proof, we carefully utilize several sophisticated arguments coming from coding theory, algebraic geometry, and linear algebra. In particular, we interpret each subspace as a suitable generator matrix (the lifting idea in random network coding), then algebraically manipulate their AAD-ness relation and produce suitable Van der Monde matrices (the fundamental tools for the BCH bound in coding theory), later apply Cramer's rule (a well-known key tool in linear algebra), and lastly obtain low-degree polynomials as contradictions (the trick coming from algebraic geometry utilized for Goppa codes). We present our constructive proof step-by-step to make it more easy to understand. [ABSTRACT FROM AUTHOR]

Avoiding singularities is a crucial task in robotics and path planning. This paper proposes a novel algorithm for detecting the closest singularity to a given pose for nine interpretations of the 3-RPR manipulator. The algorithm utilizes intrinsic metrics based on the framework's total elastic strain energy density, employing the physical concept of Green-Lagrange strain. The constrained optimization problem for detecting the closest singular configuration with respect to these metrics is solved globally using tools from numerical algebraic geometry implemented in the software package Bertini. The effectiveness of the proposed algorithm is demonstrated on a 3-RPR manipulator executing a one-parametric motion. Additionally, the obtained intrinsic singularity distances are compared with extrinsic metrics. Finally, the paper illustrates the advantage of employing a well-defined metric for identifying the closest singularities in comparison with the existing methods in the literature, highlighting its application in design optimization. [ABSTRACT FROM AUTHOR]

It is well-known that parallel manipulators are prone to singularities. However, there is still a lack of distance evaluation functions, referred to as metrics, for computing the distance between two 3-RPR configurations. The proposed extrinsic metrics take the combinatorial structure of the manipulator into account as well as different design options. Utilizing these extrinsic metrics, we formulate constrained optimization problems. These problems are aimed at identifying the closest singular configurations for a given non-singular configuration. The solution to the associated system of polynomial equations relies on algorithms from numerical algebraic geometry implemented in the software package Bertini. Furthermore, we have developed a computational pipeline for determining the distance to singularity during a one-parametric motion of the manipulator. To facilitate these computations for the user, an open-source interface is developed between software packages Maple, Bertini, and Paramotopy. The effectiveness of the presented approach is demonstrated based on numerical examples and compared with existing indices evaluating the singularity closeness. • Extrinsic metrics are presented considering the combinatorial structure of the 3-RPR. • They are used for computing the closest singularities along a 1-parametric motion. • The constrained minimizations are solved with tools from numerical algebraic geometry. • Geometric characterization of singular points of the related constrained varieties. • The algorithm has the capability to be executed in real-time using parallel computing. [ABSTRACT FROM AUTHOR]

A real univariate polynomial of degree n is called hyperbolic if all of its n roots are on the real line. Such polynomials appear quite naturally in different applications, for example, in combinatorics and optimization. The focus of this article is on families of hyperbolic polynomials which are determined through k linear conditions on the coefficients. The coefficients corresponding to such a family of hyperbolic polynomials form a semi-algebraic set which we call a hyperbolic slice. We initiate here the study of the geometry of these objects in more detail. The set of hyperbolic polynomials is naturally stratified with respect to the multiplicities of the real zeros and this stratification induces also a stratification on the hyperbolic slices. Our main focus here is on the local extreme points of hyperbolic slices, i.e., the local extreme points of linear functionals, and we show that these correspond precisely to those hyperbolic polynomials in the hyperbolic slice which have at most k distinct roots and we can show that generically the convex hull of such a family is a polyhedron. Building on these results, we give consequences of our results to the study of symmetric real varieties and symmetric semi-algebraic sets. Here, we show that sets defined by symmetric polynomials which can be expressed sparsely in terms of elementary symmetric polynomials can be sampled on points with few distinct coordinates. This in turn allows for algorithmic simplifications, for example, to verify that such polynomials are non-negative or that a semi-algebraic set defined by such polynomials is empty. [ABSTRACT FROM AUTHOR]

In the last two decades, remarkable progress has been done in singular learning machine theories on the basis of algebraic geometry. These theories reveal that we need to find resolution maps of singularities for analyzing asymptotic behavior of state probability functions when the number of data increases. In particular, it is essential to construct normal crossing divisors of average log loss functions. However, there are few examples for obtaining these for singular models. In this paper, we determine the resolution map and normal crossing divisors for multiple-layered neural networks with linear units. Moreover, we have the exact values for the learning efficiency, which is so called learning coefficients. Multiple-layered neural networks with linear units are simple, however, very important models because these models give the essential information from data of input–output pairs. Moreover, these models are very close to multiple-layered neural networks with rectified linear units (ReLU). We show the learning coefficients of multiple-layered neural networks with linear units are bounded even though the number of layers goes to infinity, which means that the main term of asymptotic expansion of the free energy and generalization error of singular models are much smaller than the dimension of its parameter space. [ABSTRACT FROM AUTHOR]

The paper explores the relation between Perling's toric Proj of a multigraded ring A associated with a toric variety (tProj A) and Brenner and Schröer's homogeneous spectrum Proj MH A of the same ring. We show that there is always a canonical open embedding and study a criterion for them to be isomorphic. [ABSTRACT FROM AUTHOR]

We study unbendable rational curves, i.e., nonsingular rational curves in a complex manifold of dimension n with normal bundles isomorphic to O P 1 (1) ⊕ p ⊕ O P 1 ⊕ (n − 1 − p) for some nonnegative integer p. Well-known examples arise from algebraic geometry as general minimal rational curves of uniruled projective manifolds. After describing the relations between the differential geometric properties of the natural distributions on the deformation spaces of unbendable rational curves and the projective geometric properties of their varieties of minimal rational tangents, we concentrate on the case of p = 1 and n ≤ 5 , which is the simplest nontrivial situation. In this case, the families of unbendable rational curves fall essentially into two classes: Goursat type or Cartan type. Those of Goursat type arise from ordinary differential equations and those of Cartan type have special features related to contact geometry. We show that the family of lines on any nonsingular cubic 4-fold is of Goursat type, whereas the family of lines on a general quartic 5-fold is of Cartan type, in the proof of which the projective geometry of varieties of minimal rational tangents plays a key role. [ABSTRACT FROM AUTHOR]

Latent Dirichlet allocation (LDA) obtains essential information from data by using Bayesian inference. It is applied to knowledge discovery via dimension reducing and clustering in many fields. However, its generalization error had not been yet clarified since it is a singular statistical model where there is no one-to-one mapping from parameters to probability distributions. In this paper, we give the exact asymptotic form of its generalization error and marginal likelihood, by theoretical analysis of its learning coefficient using algebraic geometry. The theoretical result shows that the Bayesian generalization error in LDA is expressed in terms of that in matrix factorization and a penalty from the simplex restriction of LDA's parameter region. A numerical experiment is consistent with the theoretical result. • Latent Dirichlet Allocation (LDA) is a widely used statistical model in many fields. • The Bayesian generalization error (GE) in LDA has been unknown since it is singular. • We determine the learning coefficient of GE in LDA with resolution of singularity. • The main result shows the behavior of GE in LDA when topics or sample size increases. • The experimental result is consistent with Main Theorem when sample size is finite. [ABSTRACT FROM AUTHOR]

CM-type projective varieties X of complex dimension n are characterized by their CM-type rational Hodge structures on the cohomology groups. One may impose such a condition in a weakest form when the canonical bundle of X is trivial; the rational Hodge structure on the level- n subspace of H n (X ; Q) is required to be of CM-type. This brief note addresses the question whether this weak condition implies that the Hodge structure on the entire H ⁎ (X ; Q) is of CM-type. We study in particular abelian varieties when the dimension of the level- n subspace is two or four, and K3 × T 2. It turns out that the answer is affirmative. Moreover, such an abelian variety is always isogenous to a product of CM-type elliptic curves or abelian surfaces. This extends a result of Shioda and Mitani in 1974. [ABSTRACT FROM AUTHOR]

To incorporate quantum nonlocality into general relativity, we propose that the preparation and measurement of a quantum system are simultaneous events. To make progress in realizing this proposal, we introduce a spacetime geometry that is endowed with particles which have no distinct points in their worldlines; we call these particles 'pointons'. This new geometry recently arose in nonnoetherian algebraic geometry. We show that on such a spacetime, metrics are degenerate and tangent spaces have variable dimension. This variability then implies that pointons are spin- 1 2 fermions that satisfy the Born rule, where a projective measurement of spin corresponds to an actual projection of tangent spaces of different dimensions. Furthermore, the 4-velocities of pointons are necessarily replaced by their Hodge duals, and this transfer from vector to pseudo-tensor introduces a free choice of orientation that we identify with electric charge. Finally, a simple composite model of electrons and photons results from the metric degeneracy, and from this we obtain a new ontological model of photon polarization. [ABSTRACT FROM AUTHOR]

The positive semidefinite Gram matrices of a form f with real coefficients parametrize the sum-of-squares representations of f. The convex body formed by the entirety of these matrices is the so-called Gram spectrahedron of f. We analyze the facial structures of symmetric and Hermitian Gram spectrahedra in the case of binary forms. We give upper bounds for the dimensions of polyhedral faces in Hermitian Gram spectrahedra and show that, if the form f is sufficiently generic, they can be realized by faces that are simplices and whose extreme points are rank-one tensors. We use our construction to prove a similar statement for the real symmetric case. [ABSTRACT FROM AUTHOR]

The classical Morse–Sard theorem claims that for a mapping v : R n → R m + 1 of class C k the measure of critical values v (Z v , m) is zero under condition k ≥ n − m. Here the critical set, or m -critical set is defined as Z v , m = { x ∈ R n : rank ∇ v (x) ≤ m }. Further Dubovitskiĭ in 1957 and independently Federer and Dubovitskiĭ in 1967 found some elegant extensions of this theorem to the case of other (e.g., lower) smoothness assumptions. They also established the sharpness of their results within the C k category. Here we formulate and prove a bridge theorem that includes all the above results as particular cases: namely, if a function v : R n → R d belongs to the Hölder class C k , α , 0 ≤ α ≤ 1 , then for every q > m the identity H μ (Z v , m ∩ v − 1 (y)) = 0 holds for H q -almost all y ∈ R d , where μ = n − m − (k + α) (q − m). Intuitively, the sense of this bridge theorem is very close to Heisenberg's uncertainty principle in theoretical physics: the more precise is the information we receive on measure of the image of the critical set, the less precisely the preimages are described, and vice versa. The result is new even for the classical C k -case (when α = 0); similar result is established for the Sobolev classes of mappings W p k (R n , R d) with minimal integrability assumptions p = max ⁡ (1 , n / k) , i.e., it guarantees in general only the continuity (not everywhere differentiability) of a mapping. However, using some N -properties for Sobolev mappings, established in our previous paper, we obtained that the sets of nondifferentiability points of Sobolev mappings are fortunately negligible in the above bridge theorem. We cover also the case of fractional Sobolev spaces. The proofs of the most results are based on our previous joint papers with J. Bourgain and J. Kristensen (2013, 2015). We also crucially use very deep Y. Yomdin's entropy estimates of near critical values for polynomials (based on algebraic geometry tools). [ABSTRACT FROM AUTHOR]

Non-negative matrix factorization (NMF) is a knowledge discovery method that is used in many fields. Variational inference and Gibbs sampling methods for it are also well-known. However, the variational approximation error has not been clarified yet, because NMF is not statistically regular and the prior distribution used in variational Bayesian NMF (VBNMF) has zero or divergence points. In this paper, using algebraic geometrical methods, we theoretically analyze the difference in negative log evidence (a.k.a. free energy) between VBNMF and Bayesian NMF, i.e., the Kullback–Leibler divergence between the variational posterior and the true posterior. We derive an upper bound for the learning coefficient (a.k.a. the real log canonical threshold) in Bayesian NMF. By using the upper bound, we find a lower bound for the approximation error, asymptotically. The result quantitatively shows how well the VBNMF algorithm can approximate Bayesian NMF; the lower bound depends on the hyperparameters and the true non-negative rank. A numerical experiment demonstrates the theoretical result. • Variational inference approximates the posterior using mean-field approximation. • We give a lower bound for the approximation error by using algebraic geometry. • The lower bound depends on the true non-negative rank and the hyperparameters. [ABSTRACT FROM AUTHOR]

Bounding the number of morphisms between compact Riemann surfaces is a long standing problem coming from complex and algebraic geometry. We show that linear algebra techniques allow to improve the known results when we assume a kind of condition number bound for the period matrix. [ABSTRACT FROM AUTHOR]

Tropical varieties capture combinatorial information about how coordinates of points in a classical variety approach zero or infinity. We present algorithms for computing the rays of a complex and real tropical curve defined by polynomials with constant coefficients. These algorithms rely on homotopy continuation, monodromy loops, and Cauchy integrals. Several examples are presented which are computed using an implementation that builds on the numerical algebraic geometry software Bertini. [ABSTRACT FROM AUTHOR]

We generalize the result of Voronov (1988) to give an expression for the super Mumford form μ on the moduli spaces of super Riemann surfaces with Ramond and Neveu–Schwarz punctures. In the Ramond case we take the number of punctures to be large compared to the genus. We consider for the case of Neveu–Schwarz punctures the super Mumford form over the component of the moduli space corresponding to an odd spin structure. The super Mumford form μ can be used to create a measure whose integral computes scattering amplitudes of superstring theory. We express μ in terms of local bases of H 0 (X , ω j) for ω the Berezinian line bundle of a family of super Riemann surfaces. [ABSTRACT FROM AUTHOR]

We generalize Artin's three main algebraicity theorems to the setting of supergeometry: approximation [2] , algebraization of formal deformations [3] , and algebraization of stacks [4]. [ABSTRACT FROM AUTHOR]

The prevalence of neural networks in applications is expanding at an increasing rate. It is becoming clear that providing robust guarantees on systems that use neural networks is very important, especially in safety-critical applications. A trained neural network's sensitivity to adversarial attacks is one of its greatest shortcomings. To provide robust guarantees, one popular method that has seen success is to describe and bound the activation functions in the neural network using equality and inequality constraints. However, there are numerous ways to form these bounds, providing a trade-off between conservativeness and complexity. We approach the problem from a different perspective, using sparse polynomial optimisation theory and Positivstellensatz, a key result in real algebraic geometry. The former exploits the natural cascading structure of the neural network using ideas from chordal sparsity while the latter tests the emptiness of a semi-algebraic set using algebra, to provide tight bounds. We show that bounds can be tightened significantly, whilst the computational time remains reasonable. We compare the solve times of different solvers and show how accuracy can be improved at the expense of increased computation time. In particular, we show that with the sparse polynomial framework the solve time and accuracy can be improved over other methods for neural network verification, for networks with ReLU, sigmoid and tanh activation functions. [ABSTRACT FROM AUTHOR]

Given a real projective curve with homogeneous coordinate ring R and a nonnegative homogeneous element f ∈ R , we bound the degree of a nonzero homogeneous sum of squares g ∈ R such that the product fg is again a sum of squares. Better yet, our degree bounds only depend on geometric invariants of the curve and we show that there exist smooth curves and nonnegative elements for which our bounds are sharp. We deduce the existence of a multiplier g from a new Bertini Theorem in convex algebraic geometry and prove sharpness by deforming rational Harnack curves on toric surfaces. Our techniques also yield similar bounds for multipliers on surfaces of minimal degree, generalizing Hilbert's work on ternary forms. [ABSTRACT FROM AUTHOR]

• We show that algebraic methods of Cárdenas-Barrón [2] are questionable. • We provide our improvements for algebraic methods of Cárdenas-Barrón [2]. • We indicate that Sphicas [1] already solved these models by algebraic methods. This study examines the paper of Cárdenas-Barrón, entitled "The derivation of EOQ/EPQ inventory models with two backorders costs using analytic geometry and algebra." that was published in Applied Mathematical Modelling at 2011 to point out it contains questionable results. The algebraic geometry and algebra applied by Cárdenas-Barrón [2] violated the rule of the arithmetic-geometric mean (AGM) mentioned by Cárdenas-Barrón [74]. Moreover, we point out Sphicas [1] had already solved this kind of EOQ and EPQ models using algebraic methods. [ABSTRACT FROM AUTHOR]

Abstract Let H be the Hermitian curve defined over a finite field F q 2 . In this paper we complete the geometrical characterization of the supports of the minimum-weight codewords of the algebraic-geometry codes over H , started in [26] : if d is the distance of the code, the supports are all the sets of d distinct F q 2 -points on H complete intersection of two curves defined by polynomials with prescribed initial monomials w.r.t. DegRevLex. For most Hermitian codes, and especially for all those with distance d ≥ q 2 − q studied in [26] , one of the two curves is always the Hermitian curve H itself, while if d < q the supports are complete intersection of two curves none of which can be H. Finally, for some special codes among those with intermediate distance between q and q 2 − q , both possibilities occur. We provide simple and explicit numerical criteria that allow to decide for each code what kind of supports its minimum-weight codewords have and to obtain a parametric description of the family (or the two families) of the supports. [ABSTRACT FROM AUTHOR]

A novel surface interrogation technique is proposed to compute the intersection of curves with spline surfaces in isogeometric analysis. The intersection points are determined in one-shot without resorting to a Newton–Raphson iteration or successive refinement. Surface-curve intersection is required in a wide range of applications, including contact, immersed boundary methods and lattice-skin structures, and requires usually the solution of a system of nonlinear equations. It is assumed that the surface is given in form of a spline, such as a NURBS, T-spline or Catmull–Clark subdivision surface, and is convertible into a collection of Bézier patches. First, a hierarchical bounding volume tree is used to efficiently identify the Bézier patches with a convex-hull intersecting the convex-hull of a given curve segment. For ease of implementation convex-hulls are approximated with k-dops (discrete orientation polytopes). Subsequently, the intersections of the identified Bézier patches with the curve segment are determined with a matrix-based implicit representation leading to the computation of a sequence of small singular value decompositions (SVDs). As an application of the developed interrogation technique the isogeometric design and analysis of lattice-skin structures is investigated. Although such structures have been common in large-scale civil engineering, current additive manufacturing, or 3d printing, technologies make it possible to produce up to metre size lattice-skin structures with designed geometric features reaching down to submillimetre scale. The skin is a spline surface that is usually created in a computer-aided design (CAD) system and the periodic lattice to be fitted consists of unit cells, each containing a small number of struts. The lattice-skin structure is generated by projecting selected lattice nodes onto the surface after determining the intersection of unit cell edges with the surface. For mechanical analysis, the skin is modelled as a Kirchhoff–Love thin-shell and the lattice as a pin-jointed truss. The two types of structures are coupled with a standard Lagrange multiplier approach. • A surface interrogation technique for isogeometric design and analysis is proposed. • Intersections between curves and surfaces are determined using algebraic geometry. • Possible intersections are first identified with a (k-dop) bounding volume tree. • Intersection coordinates are computed by solving a sequence of small SVDs. • Applied to large-scale lattice-skin structures with several thousands of intersections. [ABSTRACT FROM AUTHOR]

Abstract We introduce and study a new way to categorize supersingular abelian varieties defined over a finite field by classifying them as fully maximal , mixed or fully minimal. The type of A depends on the normalized Weil numbers of A and its twists. We analyze these types for supersingular abelian varieties and curves under conditions on the automorphism group. In particular, we present a complete analysis of these properties for supersingular elliptic curves and supersingular abelian surfaces in arbitrary characteristic, and for a one-dimensional family of supersingular curves of genus 3 in characteristic 2. [ABSTRACT FROM AUTHOR]

Abstract The subject of this paper is the higher structure of the strictification adjunction, which relates the two fundamental bases of three-dimensional category theory: the Gray -category of 2-categories and the tricategory of bicategories. We show that – far from requiring the full weakness provided by the definitions of tricategory theory – this adjunction can be strictly enriched over the symmetric closed multicategory of bicategories defined by Verity. Moreover, we show that this adjunction underlies an adjunction of bicategory-enriched symmetric multicategories. An appendix introduces the symmetric closed multicategory of pseudo double categories, into which Verity's symmetric multicategory of bicategories embeds fully. [ABSTRACT FROM AUTHOR]

Abstract We compute the group of K 1 -zero-cycles on the second generalized involution variety for an algebra of degree 4 with symplectic involution. This description is given in terms of the group of multipliers of similitudes associated to the algebra with involution. Our method utilizes the framework of Chernousov and Merkurjev for computing K 1 -zero-cycles in terms of R -equivalence classes of prescribed algebraic groups. This gives a computation of K 1 -zero-cycles for some homogeneous varieties of type C 2. [ABSTRACT FROM AUTHOR]

Highlights • Algebraic description of the manipulator's workspace. • Detailed analysis of the input and output singularities. • Description of singularities in Study space and joint space. • Possible transition between different operation modes. • Investigation of a self motion. Abstract The aim of this paper is to give a detailed examination of the input and output singularities of a 3- R UU parallel manipulator in the translational operation mode. This task is achieved by using algebraic constraint equations. For this type of manipulator a complete workspace representation in Study coordinates is presented after elimination of the input parameters. Both, input and output singularities are mapped into a Study subspace as well as into the joint space. Therewith a detailed singularity investigation of the translational operation mode of a 3- R UU parallel manipulator is provided. This paper is an extended version of a previous publication. The addendum comprises the discovery of a possible transition between two operation modes as well as a self motion and an examination of another component of the output singularity surface, most of them for arbitrary design parameters. [ABSTRACT FROM AUTHOR]

Abstract We study a numerical semigroup ring as an algebra over another numerical semigroup ring. The complete intersection property of numerical semigroup algebras is investigated using factorizations of monomials into minimal ones. The goal is to study whether a flat rectangular algebra is a complete intersection. Along this direction, special types of algebras generated by few monomials are worked out in detail. [ABSTRACT FROM AUTHOR]

Abstract Classical results concerning slenderness for commutative integral domains are generalized to commutative rings with zero divisors. This is done by extending the methods from the domain case and bringing them in connection with results on the linear topologies associated to non-discrete Hausdorff filtrations. In many cases a weakened notion "almost slenderness" of slenderness is appropriate for rings with zero divisors. Special results for countable rings are extended to rings said to be of "bounded type" (including countable rings, 'small' rings, and, for instance, rings that are countably generated as algebras over an Artinian ring). More precisely, for a ring R of bounded type it is proved that R is slender if R is reduced and has no simple ideals, or if R is Noetherian and has no simple ideals; moreover, R is almost slender if R is not perfect (in the sense of H. Bass). We use our methods to study various special classes of rings, for instance von Neumann regular rings and valuation rings. Among other results we show that the following two rings are slender: the ring of Puiseux series over a field and the von Neumann regular ring k N / k (N) over a von Neumann regular ring k. For a Noetherian ring R we prove that R is a finite product of local complete rings iff R satisfies one of several (equivalent) conditions of algebraic compactness. A 1-dimensional Noetherian ring is outside this 'compact' class precisely when it is almost slender. For the rings of classical algebraic geometry we prove that a localization of an algebra finitely generated over a field is either Artinian or almost slender. Finally, we show that a Noetherian ring R is a finite product of local complete rings with finite residue fields exactly when there exists a map of R -algebras R N → R vanishing on R (N). [ABSTRACT FROM AUTHOR]

Abstract In this paper, an approach is proposed to estimate the open-circuit voltage of a battery – and consequently the battery State-of-Charge during normal operation of electric vehicles – by only measuring the charge/discharge current and the voltage on the load. The result is achieved, without any knowledge of the characteristic values of the battery model, by combining a system embedding with a high-gain observer design strategy. Interestingly, the method is based on genuinely nonlinear estimation strategies, hence it does not suffer the typical drawbacks of linear approaches, including the requirement for persistence of excitation of the charge/discharge current. [ABSTRACT FROM AUTHOR]

Abstract We prove that the set of non-degenerate second order maximally superintegrable systems in the complex Euclidean plane carries a natural structure of a projective variety, equipped with a linear isometry group action. This is done by deriving an explicit system of homogeneous algebraic equations. We then solve these equations and give a detailed analysis of the algebraic geometric structure of the corresponding projective variety. This naturally associates a unique planar line triple arrangement to every superintegrable system, providing a geometric realisation of this variety and an intrinsic labelling scheme. In particular, our results confirm the known classification by independent, purely algebraic means. [ABSTRACT FROM AUTHOR]

Abstract Let X = P (E 1) × C P (E 2) , where C is any smooth irreducible curve and E 1 , E 2 are vector bundles over C. In this paper, we calculate the nef cone and pseudoeffective cone of X whenever E 1 and E 2 are semistable, and show that they coincide. We also calculate the nef cone and pseudoeffective cone of X in case both E 1 and E 2 have rank 2, and neither E 1 nor E 2 is semistable or at least one of them is semistable. As a result, in the case when both E 1 and E 2 are of rank 2 bundles on C , we get that nef cone and pseudoeffective cone of X coincide iff both E 1 and E 2 are semistable. [ABSTRACT FROM AUTHOR]

Abstract The GM-switching , invented by Godsil and McKay, is a powerful method to construct cospectral graphs. In its simplest version, the method can be roughly described as follows: let A (G) be the adjacency matrix of a graph G on n vertices, let Q = diag (R , I n − 2 m) be a block diagonal matrix, where R = 1 m J − I 2 m is an orthogonal matrix of order 2 m , J is the all-one matrix and I k denotes the identity matrix of order k. If G satisfies certain conditions, then Q T A (G) Q is an adjacency matrix of a graph G ′ which is cospectral with G (with cospectral complements). In this paper, we consider the problem of replacing the above R with another rational orthogonal matrix U of order 2 p (p an odd prime). We characterize all graphs G admitting a GM-switching corresponding to Q = diag (U , I n − 2 p) , i.e., such that Q T A (G) Q is an adjacency matrix of a graph G ′ , and hence giving a counterpart of the GM-switching method for generating cospectral graphs (with cospectral complements), which can be used to construct pairs of non-isomorphic cospectral graphs that are not obtainable by the original GM-switching method. Next, we consider the problem of how to determine whether a given graph has a non-trivial GM-switching associated with some U. A simple algebraic condition is presented which partially answers this question. [ABSTRACT FROM AUTHOR]

Abstract Consider zero-dimensional Donaldson–Thomas invariants of a toric threefold or toric Calabi–Yau fourfold. In the second case, invariants can be defined using a tautological insertion. In both cases, the generating series can be expressed in terms of the MacMahon function. In the first case, this follows from a theorem of Maulik–Nekrasov–Okounkov–Pandharipande. In the second case, this follows from a conjecture of the authors and a (more general K -theoretic) conjecture of Nekrasov. In this paper, we consider formal analogues of these invariants in any dimension d ⁄ ≡ 2 mod 4. The direct analogues of the above-mentioned conjectures fail in general when d > 4 , showing that dimensions 3 and 4 are special. Surprisingly, after appropriate specialization of the equivariant parameters, the conjectures seem to hold in all dimensions. [ABSTRACT FROM AUTHOR]

Abstract Compressed sensing is a relatively new mathematical paradigm that shows a small number of linear measurements are enough to efficiently reconstruct a large dimensional signal under the assumption the signal is sparse. Applications for this technology are ubiquitous, ranging from wireless communications to medical imaging, and there is now a solid foundation of mathematical theory and algorithms to robustly and efficiently reconstruct such signals. However, in many of these applications, the signals of interest do not only have a sparse representation, but have other structure such as lattice-valued coefficients. While there has been a small amount of work in this setting, it is still not very well understood how such extra information can be utilized during sampling and reconstruction. Here, we explore the problem of integer sparse reconstruction, lending insight into when this knowledge can be useful, and what types of sampling designs lead to robust reconstruction guarantees. We use a combination of combinatorial, probabilistic and number-theoretic methods to discuss existence and some constructions of such sensing matrices with concrete examples. We also prove sparse versions of Minkowski’s Convex Body and Linear Forms theorems that exhibit some limitations of this framework. [ABSTRACT FROM AUTHOR]

Highlights • It is necessary to adapt SQL query processing to evaluate percentage cubes efficiently. • A percentage cube is significantly more difficult to compute than a standard cube due to a higher exponential complexity. • Percentage cubes should be computed at low cube dimensionality with dimensions having low cardinality. • Selecting top-k percentages across all cuboids is the most difficult analysis, harder than selecting minimum percentages. • Incremental materialized view algorithms are feasible for one percentage query, but not for the percentage cube. Abstract OLAP cubes provide exploratory query capabilities combining joins and aggregations at multiple granularity levels. However, cubes cannot intuitively or directly show the relationship between measures aggregated at different grouping levels. One prominent example is the percentage, which is widely used in most analytical applications. Considering this limitation, we introduce percentage cube as a generalized data cube that takes percentages as its basic measure. More precisely, a percentage cube shows the fractional relationship in every cuboid between each aggregated measure on several dimensions and its rolled-up measure aggregated by fewer dimensions. We propose the syntax and introduce query optimizations to materialize the percentage cube. We justify that percentage cubes are significantly harder to evaluate than standard data cubes because in addition to the exponential number of cuboids, there is an additional exponential number of grouping column pairs (grouping columns at the individual level and the total level) on which percentages are computed. We propose alternative methods to prune the cube to identify interesting percentages including a row count threshold, a percentage threshold, and selecting the top k percentages. We study percentage aggregations within the classification of distributive, algebraic, and holistic functions. Finally, we also consider the problem of incremental computation of percentage cube. Experiments compare our query optimizations with existing SQL functions, evaluate the impact and speed of lattice pruning methods and study the effectiveness of the incremental computation. [ABSTRACT FROM AUTHOR]

Recently, there have been found relations between the algebraic co-rank and the zero forcing number along with the minimum rank. We continue on this direction by giving a characterization of the graphs with real algebraic co-rank at most 2. This implies that for any graph with minimum rank at most 3, its minimum rank is bounded from above by its real algebraic co-rank. This sheds some light on the conjecture that the real minimum rank is bounded from above by the real algebraic co-rank. [ABSTRACT FROM AUTHOR]

Nilpotence has been studied in stable homotopy theory and algebraic geometry. We study the corresponding notion in modular representation theory of finite groups, and apply the discussion to the study of ghosts, and generation of the stable module category. In particular, we show that for a finitely generated kG -module M , the tensor M -generation number and the tensor M -ghost number are both equal to the degree of tensor nilpotence of a certain map associated with M . [ABSTRACT FROM AUTHOR]

For a graph G , let λ 2 ( G ) denote its second smallest Laplacian eigenvalue. It was conjectured that λ 2 ( G ) + λ 2 ( G ‾ ) ≥ 1 , where G ‾ is the complement of G . In this paper, it is shown that max ⁡ { λ 2 ( G ) , λ 2 ( G ‾ ) } ≥ 2 5 . [ABSTRACT FROM AUTHOR]

Abstract The latest techniques from Neural Networks and Support Vector Machines (SVM) are used to investigate geometric properties of Complete Intersection Calabi–Yau (CICY) threefolds, a class of manifolds that facilitate string model building. An advanced neural network classifier and SVM are employed to (1) learn Hodge numbers and report a remarkable improvement over previous efforts, (2) query for favourability, and (3) predict discrete symmetries, a highly imbalanced problem to which both Synthetic Minority Oversampling Technique (SMOTE) and permutations of the CICY matrix are used to decrease the class imbalance and improve performance. In each case study, we employ a genetic algorithm to optimise the hyperparameters of the neural network. We demonstrate that our approach provides quick diagnostic tools capable of shortlisting quasi-realistic string models based on compactification over smooth CICYs and further supports the paradigm that classes of problems in algebraic geometry can be machine learned. [ABSTRACT FROM AUTHOR]

C ∞ -Rings are R -algebras equipped with operations ϕ f for every f ∈ C ∞ (R n) and every n ∈ N. Therefore, a C ∞ -version of algebraic geometry can be developed using C ∞ -rings instead of ordinary rings and many classical constructions can be performed in this context. In particular, C ∞ -schemes are the C ∞ counterpart of classical schemes. Examples of schemes are often obtained by gluing schemes or using fiber products. Another useful way to give examples of schemes is looking for representable functors F : Schemes → Sets. In this work, we show that constructions such as gluing schemes and fiber products can be done in the context of C ∞ -algebraic geometry and they can be used to exhibit some examples of C ∞ -schemes such as projective spaces and Grassmannians as well as necessary and sufficient conditions for a functor F : C ∞ − Schemes → Sets to be representable. [ABSTRACT FROM AUTHOR]