Search

This paper consists of four parts. It begins by using the authors' generalized Schauder formula, [41] , and the algebraic multiplicity, χ , of Esquinas and López-Gómez [15,14,31] to package and sharpening all existing results in local and global bifurcation theory for Fredholm operators through the recent author's axiomatization of the Fitzpatrick–Pejsachowicz–Rabier degree, [42]. This facilitates reformulating and refining all existing results in a compact and unifying way. Then, the local structure of the solution set of analytic nonlinearities F (λ , u) = 0 at a simple degenerate eigenvalue is ascertained by means of some concepts and devices of Algebraic Geometry and Galois Theory, which establishes a bisociation between Bifurcation Theory and Algebraic Geometry. Finally, the unilateral theorems of [31,33] , as well as the refinement of Shi and Wang [53] , are substantially generalized. This paper also analyzes two important examples to illustrate and discuss the relevance of the abstract theory. The second one studies the regular positive solutions of a multidimensional quasilinear boundary value problem of mixed type related to the mean curvature operator. [ABSTRACT FROM AUTHOR]

This paper introduces state polynomials, i.e., polynomials in noncommuting variables and formal states of their products. A state analog of Artin's solution to Hilbert's 17th problem is proved showing that state polynomials, positive over all matrices and matricial states, are sums of squares with denominators. Somewhat surprisingly, it is also established that a Krivine–Stengle Positivstellensatz fails to hold in the state polynomial setting. Further, archimedean Positivstellensätze in the spirit of Putinar and Helton–McCullough are presented leading to a hierarchy of semidefinite relaxations converging monotonically to the optimum of a state polynomial subject to state constraints. This hierarchy can be seen as a state analog of the Lasserre hierarchy for optimization of polynomials, and the Navascués–Pironio–Acín scheme for optimization of noncommutative polynomials. The motivation behind this theory arises from the study of correlations in quantum networks. Determining the maximal quantum violation of a polynomial Bell inequality for an arbitrary network is reformulated as a state polynomial optimization problem. Several examples of quadratic Bell inequalities in the bipartite and the bilocal tripartite scenario are analyzed. To reduce the size of the constructed SDPs, sparsity, sign symmetry and conditional expectation of the observables' group structure are exploited. To obtain the above-mentioned results, techniques from noncommutative algebra, real algebraic geometry, operator theory, and convex optimization are employed. [ABSTRACT FROM AUTHOR]

The goal of this paper is to describe the group of strongly stable automorphisms for categories of free finitely generated non-associative algebras with unit for classical varieties of non-associative algebras, in particular for the varieties of commutative, Jordan, alternative and power associative algebras. The motivation of this research is tightly connected with some problems in Universal Algebraic Geometry. [ABSTRACT FROM AUTHOR]

The text is a summary of the Proceedings of the 17th International Workshop on Real and Complex Singularities, which took place in 2022. The workshop is a renowned event in singularity theory and related fields, bringing together experts and young researchers to share their achievements and discuss research trends. The proceedings contain 29 research papers that have been selected and refereed according to the standards of the journal Research in the Mathematical Sciences. The collection aims to provide readers, including graduate students and researchers, with an opportunity to explore the field of singularities. The text also highlights the contributions and achievements of Professor Osamu Saeki, a leading researcher in singularity theory and topology, who was honored during the workshop. [Extracted from the article]

Recently, Yu et al. derived a factorization procedure for detecting and computing the potential self-intersection of 3D integral Bézier cubics, claiming that their proposal distinctly outperforms existing methodologies. First, we recall that in the 2D case, explicit formulas already exist for the parameter values at the self-intersection (the singularity called crunode in algebraic geometry). Such values are the solutions of a quadratic equation, and affine invariants depend only on the curve hodograph. Also, the factorization procedure for cubics is well known. Second, we note that only planar Bézier cubics can display a self-intersection, so there is no need to address the problem in the more involved 3D setting. Finally, we elucidate the connections with the previous literature and provide a geometric interpretation, in terms of the affine classification of cubics, of the algebraic conditions necessary for the existence of a self-intersection. Cubics with a self-intersection are affine versions of the celebrated Tschirnhausen cubic. [ABSTRACT FROM AUTHOR]

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]

This work investigates the noncommutative affine geometry of matrix polynomial algebra extensions of a coefficient algebra by (elementary) matrix variables. A precise description of the spectrum (of maximal one-sided or bilateral ideals) of general matrix algebras is required. It results that the Zariski space of the irreducible representations of a matrix algebra is obtained by a natural gluing of the Zariski spaces of the irreducible representations of its diagonal components. An important step for the geometry of matrix polynomial algebras in commuting variables is achieved by a generalization of the Amitsur–Small Nullstellensatz, from which follows a precise description of their primitive quotients. We also characterize which of them are geometric algebras (in the sense of noncommutative deformation theory), reconstructible as algebras of observables from the scheme of irreducible representations. We then prove that each diagonal component of a matrix polynomial algebra in commuting variables is a Jacobson ring, whose non-Noetherian commutative geometry is efficiently described by the geometry of an affine essential subextension. And in the spirit of nonlocal algebraic geometry and addressing an open question by Charlie Beil, we obtain a class of non-Noetherian commutative monoid rings admitting closed points with positive geometric dimension. [ABSTRACT FROM AUTHOR]

In this paper, we develop an extended space-time geometry that includes an internal space built with division octonions with realization via Pauli matrices and Zorn (vectorial) matrices. Here, we extend a former field theory that used split octonion algebra to a division octonion algebra on a non-Riemannian manifold. The octonionic algebra is the last possible algebra allowed by the Hurwitz theorem. The interpretation of the “octonionic field” in this division octonionic algebra is not straightforward, however it behaves in a similar way to the quaternionic Yang–Mills fields. Former work suggests that this Yang–Mills-like octonionic field is associated with the field governing quarks within nucleons. In this work, we discover that the inclusion of octonionic fields in the geometry of an internal space necessarily excludes the quaternionic (Yang–Mills) fields in an extended non-Riemannian geometry. This is not what is expected from the standpoint of a “unified” field theory, which leads us to propose a different approach to Einstein’s unified field theory. [ABSTRACT FROM AUTHOR]

This article presents an educational overview of the latest mathematical developments in coupled cluster (CC) theory, beginning with Schneider's seminal work from 2009 that introduced the first local analysis of CC theory. We provide a tutorial review of second quantization and the CC ansatz, laying the groundwork for understanding the mathematical basis of the theory. This is followed by a detailed exploration of the most recent mathematical advancements in CC theory. Our review starts with an in‐depth look at the local analysis pioneered by Schneider which has since been applied to various CC methods. We subsequently discuss the graph‐based framework for CC methods developed by Csirik and Laestadius. This framework provides a comprehensive platform for comparing different CC methods, including multireference approaches. Next, we delve into the latest numerical analysis results analyzing the single reference CC method developed by Hassan, Maday, and Wang. This very general approach is based on the invertibility of the CC function's Fréchet derivative. We conclude the article with a discussion on the recent incorporation of algebraic geometry into CC theory, highlighting how this novel and fundamentally different mathematical perspective has furthered our understanding and provides exciting pathways to new computational approaches. [ABSTRACT FROM AUTHOR]

In recent years, algebraic tools have been proven useful in phylogenetic reconstruction and model selection through the study of phylogenetic invariants. However, up to now, the models studied from an algebraic viewpoint are either too general or too restrictive (as group-based models with a uniform stationary distribution) to be used in practice. In this paper we provide a new framework to study time-reversible models, which are the most widely used by biologists. In our approach we consider algebraic time-reversible models on phylogenetic trees (as defined by Allman and Rhodes) and introduce a new inner product to make all transition matrices of the process diagonalizable through the same orthogonal eigenbasis. This framework generalizes the Fourier transform widely used to work with group-based models and recovers some of the well-known results. As illustration, we combine our technique with algebraic geometry tools to provide relevant phylogenetic invariants for trees evolving under the Tamura-Nei model of nucleotide substitution. [ABSTRACT FROM AUTHOR]

The moduli space Y=Y(E6)$Y = Y(E_6)$ of marked cubic surfaces is one of the most classical moduli spaces in algebraic geometry, dating back to the nineteenth‐century work of Cayley and Salmon. Modern interest in Y$Y$ was restored in the 1980s by Naruki's explicit construction of a W(E6)$W(E_6)$‐equivariant smooth projective compactification Y¯${\overline{Y}}$ of Y$Y$, and in the 2000s by Hacking, Keel, and Tevelev's construction of the Kollár–Shepherd‐Barron–Alexeev (KSBA) stable pair compactification Y∼${\widetilde{Y}}$ of Y$Y$ as a natural sequence of blowups of Y¯${\overline{Y}}$. We describe generators for the cones of W(E6)$W(E_6)$‐invariant effective divisors and curves of both Y¯${\overline{Y}}$ and Y∼${\widetilde{Y}}$. For Naruki's compactification Y¯${\overline{Y}}$, we further obtain a complete stable base locus decomposition of the W(E6)$W(E_6)$‐invariant effective cone, and as a consequence find several new W(E6)$W(E_6)$‐equivariant birational models of Y¯${\overline{Y}}$. Furthermore, we fully describe the log minimal model program for the KSBA compactification Y∼${\widetilde{Y}}$, with respect to the divisor KY∼+cB+dE$K_{{\widetilde{Y}}} + cB + dE$, where B$B$ is the boundary and E$E$ is the sum of the divisors parameterizing marked cubic surfaces with Eckardt points. [ABSTRACT FROM AUTHOR]

Young diagrams are fundamental combinatorial objects in representation theory and algebraic geometry. Many constructions that rely on these objects depend on variations of a straightening process that expresses a filling of a Young diagram as a sum of semistandard tableaux subject to certain relations. This paper solves the long standing open problem of giving a non-iterative formula for straightening a filling. We apply our formula to give a complete generalization of a theorem of Gonciulea and Lakshmibai. [ABSTRACT FROM AUTHOR]

In previous papers, we were able to prove that, much like in classical algebraic geometry, it is possible to recover our monoid scheme X from the topos Qcoh(X). This was achieved using topos points and localisations of Qcoh(X). With this philosophy in mind, the aim of this paper is to study the topos of M-sets for non-commutative monoids, especially their points and localisations. We will classify the points and localisations of M-sets for finite monoids in terms of the idempotent elements of M and idempotent ideals of M, respectively. Some of the results obtained in this paper can already be found in previous works, in direct or indirect forms. See the last part of the introduction. [ABSTRACT FROM AUTHOR]

We apply computations of twisted Hodge diamonds to construct an infinite number of non-Fourier–Mukai functors with well-behaved target and source spaces. To accomplish this, we first study the characteristic morphism introduced in Buchweitz and Flenner [Adv. Math. 217 (2008), 205–242] in order to control it for tilting bundles. Then, we continue by applying twisted Hodge diamonds of hypersurfaces embedded in projective space to compute the Hochschild dimension of these spaces. This allows us to compute the kernel of the embedding into the projective space in Hochschild cohomology. Finally, we use the above computations to apply the construction in Rizzardo, Van den Bergh, and Neeman [Invent. Math. 216 (2019), 927–1004] of non-Fourier–Mukai functors and verify that the constructed functors indeed cannot be Fourier–Mukai for odd-dimensional quadrics. Using this approach, we prove that there are a large number of Hochschild cohomology classes that can be used for the construction of Rizzardo, Van den Bergh, and Neeman [Invent. Math. 216 (2019), 927–1004]. Furthermore, our results allow the application of computer-based calculations to construct candidate functors for arbitrary degree hypersurfaces in arbitrary high dimensions. Verifying that these are not Fourier–Mukai still requires the existence of a tilting bundle. In particular, we prove that there is at least one non-Fourier–Mukai functor for every odddimensional smooth quadric. [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]

This paper is concerned with the construction of a small, but non-trivial, example of a polynomial identity algebra, which we call the Jackson algebra, that will be used in sequels to this paper to study non-commutative arithmetic geometry. In this paper this algebra is studied from a ring-theoretic and geometric viewpoint. Among other things it turns out that this algebra is a "non-commutative family" of central simple algebras and thus parameterizes Brauer classes over extensions of the base. [ABSTRACT FROM AUTHOR]

Affine schemes can be understood as objects of the opposite of the category of commutative and unital algebras. Similarly, P -affine schemes can be defined as objects of the opposite of the category of algebras over an operad P . An example is the opposite of the category of associative algebras. The category of operadic schemes of an operad carries a canonical tangent structure. This paper aims to initiate the study of the geometry of operadic affine schemes via this tangent category. For example, we expect the tangent structure over the opposite of the category of associative algebras to describe algebraic non-commutative geometry. In order to initiate such a program, the first step is to classify differential bundles, which are the analogs of vector bundles for differential geometry. In this paper, we prove that the tangent category of affine schemes of the enveloping operad P (A) over a P -affine scheme A is precisely the slice tangent category over A of P -affine schemes. We are going to employ this result to show that differential bundles over a P -affine scheme A are precisely A-modules in the operadic sense. [ABSTRACT FROM AUTHOR]

We study the following problem in computer vision from the perspective of algebraic geometry: Using m pinhole cameras we take m pictures of a line in P 3 . This produces m lines in P 2 and the question is which m-tuples of lines can arise that way. We are interested in polynomial equations and therefore study the complex Zariski closure of all such tuples of lines. The resulting algebraic variety is a subvariety of (P 2) m and is called line multiview variety. In this article, we study its ideal. We show that for generic cameras the ideal is generated by 3 × 3 -minors of a specific matrix. We also compute Gröbner bases and discuss to what extent our results carry over to the non-generic case. [ABSTRACT FROM AUTHOR]

Poset-blowdown of subgroup posets of groups is an analog of blowdown in algebraic geometry. It is a poset map obtained by contracting normal subgroups. For finite groups, this is considered as a map between the Hasse diagrams of the subgroup posets. Poset-blowdowns are classified into three types: tame, wild, and hybrid depending on the sizes of their fibers. In this paper we describe the poset-blowdowns for generalized quaternion groups Q2n (n ≥ 3). They have distinguished nature in that all types (tame, wild, and hybrid) appear in the successive poset-blowdowns associated with the three chief series of Q2n. [ABSTRACT FROM AUTHOR]

We introduce a new class of commutative noetherian DG-rings which generalizes the class of regular local rings. These are defined to be local DG-rings (A , m ¯) such that the maximal ideal m ¯ ⊆ H 0 (A) can be generated by an A -regular sequence. We call these DG-rings sequence-regular DG-rings, and make a detailed study of them. Using methods of Cohen-Macaulay differential graded algebra, we prove that the Auslander-Buchsbaum-Serre theorem about localization generalizes to this setting. This allows us to define global sequence-regular DG-rings, and to introduce this regularity condition to derived algebraic geometry. It is shown that these DG-rings share many properties of classical regular local rings, and in particular we are able to construct canonical residue DG-fields in this context. Finally, we show that sequence-regular DG-rings are ubiquitous, and in particular, any eventually coconnective derived algebraic variety over a perfect field is generically sequence-regular. [ABSTRACT FROM AUTHOR]

We equip integral graded-polarized mixed period spaces with a natural Ralg-definable analytic structure, and prove that any period map associated to an admissible variation of integral graded-polarized mixed Hodge structures is definable in Ran,exp with respect to this structure. As a consequence we re-prove that the zero loci of admissible normal functions are algebraic. [ABSTRACT FROM AUTHOR]

We introduce a conjecture on homological mirror symmetry relating the symplectic topology of the complement of a smooth ample divisor in a K3 surface to the algebraic geometry of type III degenerations. We prove it when the degree of the divisor is either 2 or 4. [ABSTRACT FROM AUTHOR]

Let GRSk(α, υ) be a k-dimensional generalized Reed-Solomon (GRS) code over Fq associated with α = (α1,. . ., αn) and υ = (υ1, . . ., υn). In this paper, we determined the dimension of the Euclidean hull GRSk(α, υ) ∩ GRSk(α, υ)⊥, which addresses an open problem posed in [Chen et al., IEEE-TIT, 2023]. We also presentd a new approach to generating all self-dual RS codes. [ABSTRACT FROM AUTHOR]

The theory of extensive categories determines in particular the notion of separable MV-algebra (equivalently, of separable unital lattice-ordered Abelian group). We establish the following structure theorem: An MV-algebra is separable if, and only if, it is a finite product of algebras of rational numbers—i.e., of subalgebras of the MV-algebra [ 0 , 1 ] ∩ Q. Beyond its intrinsic algebraic interest, this research is motivated by the long-term programme of developing the algebraic geometry of the opposite of the category of MV-algebras, in analogy with the classical case of commutative K -algebras over a field K. [ABSTRACT FROM AUTHOR]

Let f:(Cn,S)→(Cp,0)$f: (\mathbb {C}^n,S)\rightarrow (\mathbb {C}^p,0)$ be a finite map‐germ with n

The aim of this paper is to prove a theorem for holomorphic twisted quiver bundles over a special non-compact Gauduchon manifold, connecting the existence of (σ , τ) -Hermite–Yang–Mills metric in differential geometry and the analytic (σ , τ) -stability in algebraic geometry. The proof of the theorem relies on the flow method and the Uhlenbeck–Yau's continuity method. [ABSTRACT FROM AUTHOR]

K3 surfaces play a prominent role in string theory and algebraic geometry. The properties of their enumerative invariants have important consequences in black hole physics and in number theory. To a K3 surface, string theory associates an Elliptic genus, a certain partition function directly related to the theory of Jacobi modular forms. A multiplicative lift of the Elliptic genus produces another modular object, an Igusa cusp form, which is the generating function of BPS invariants of K3 × E . In this note, we will discuss a refinement of this chain of ideas. The Elliptic genus can be generalized to the so-called Hodge-Elliptic genus which is then related to the counting of refined BPS states of K3 × E . We show how such BPS invariants can be computed explicitly in terms of different versions of the Hodge-Elliptic genus, sometimes in closed form, and discuss some generalizations. [ABSTRACT FROM AUTHOR]

The interactions between topology and algebraic geometry expose various interesting properties. This paper proposes the deformations of topological n-manifolds over the automorphic polynomial ring maps and associated isomorphic imbedding of locally flat submanifolds within the n-manifolds. The manifold deformations include topologically homeomorphic bending of submanifolds at multiple directions under algebraic operations. This paper introduces the concept of a topological equivalence class of manifolds and the associated equivalent class of polynomials in a real ring. The concepts of algebraic compositions in a real polynomial ring and the resulting topological properties (homeomorphism, isomorphism and deformation) of manifolds under algebraic compositions are introduced. It is shown that a set of ideals in a polynomial ring generates manifolds retaining topological isomorphism under algebraic compositions. The numerical simulations are presented in this paper to illustrate the interplay of topological properties and the respective real algebraic sets generated by polynomials in a ring within affine 3-spaces. It is shown that the coefficients of polynomials generated by a periodic smooth function can induce mirror symmetry in manifolds. The proposed formulations do not consider the simplectic class of manifolds and associated quantizable deformations. However, the proposed formulations preserve the properties of Nash representations of real algebraic manifolds including Nash isomorphism. [ABSTRACT FROM AUTHOR]

In this note, we study certain sufficient conditions for a set of minimal klt pairs (X, Δ) with κ(X, Δ) = dim(X) - 1 to be bounded. [ABSTRACT FROM AUTHOR]

We classify RDP del Pezzo surfaces with global vector fields over arbitrary algebraically closed fields of characteristic p ≉ 2. In characteristic 0, every RDP del Pezzo surface X is equivariant, that is, AutX = AuteX, where eX is the minimal resolution of X; hence the classification of RDP del Pezzo surfaces with global vector fields is equivalent to the classification of weak del Pezzo surfaces with global vector fields. In this article, we show that if p ≉ 2, 3, 5, 7, then it is still true that every RDP del Pezzo surface is equivariant. We classify the non-equivariant RDP del Pezzo surfaces in characteristic p = 3, 5, 7, giving explicit equations for every such RDP del Pezzo surface in all possible degrees. As an application, we construct regular non-smooth RDP del Pezzo surfaces over imperfect fields of characteristic 7, thereby showing that the known bound p ⩽ 7 for the characteristics, where such a surface can exist, is sharp. [ABSTRACT FROM AUTHOR]

For an arbitrary function field, from the knowledge of the minimal generating set of the Weierstrass semigroup at two rational places, the set of pure gaps is characterized. Furthermore, we determine the Weierstrass semigroup at one and two totally ramified places in a Kummer extension defined by the affine equation y m = ∏ i = 1 r (x - α i) λ i over K, the algebraic closure of F q , where α 1 , ⋯ , α r ∈ K are pairwise distinct elements, 1 ≤ λ i < m , and gcd (m , ∑ i = 1 r λ i) = 1 . We apply these results to construct algebraic geometry codes over certain function fields with many rational places. For one-point codes we obtain families of codes with exact parameters. [ABSTRACT FROM AUTHOR]

The decreasing norm-trace codes are evaluation codes defined by a set of monomials closed under divisibility and the rational points of the extended norm-trace curve. In particular, the decreasing norm-trace codes contain the one-point algebraic geometry (AG) codes over the extended norm-trace curve. We use Gröbner basis theory and find the indicator functions on the rational points of the curve to determine the basic parameters of the decreasing norm-trace codes: length, dimension, and minimum distance. We also obtain their dual codes. We give conditions for a decreasing norm-trace code to be a self-orthogonal or a self-dual code. We provide a linear exact repair scheme to correct single erasures for decreasing norm-trace codes, which applies to higher rate codes than the scheme developed by Jin et al. (IEEE Trans Inf Theory 64(2):900–908, 2018) when applied to the one-point AG codes over the extended norm-trace curve. [ABSTRACT FROM AUTHOR]

We report new examples of Sidon sets in abelian groups arising from generalized Jacobians of curves, and discuss some of their properties with respect to size and structure. [ABSTRACT FROM AUTHOR]

In this paper, we study the action of diamond operators on Hilbert modular forms with coefficients in a general commutative ring. In particular, we generalize a result of Chai on the surjectivity of the constant term map for Hilbert modular forms with nebentype to the setting of group ring valued modular forms. As an application, we construct certain Hilbert modular forms required for Dasgupta–Kakde's proof of the Brumer–Stark conjecture at odd primes. Since the forms required for the Brumer–Stark conjecture live on the non-PEL Shimura variety associated to the reductive group G = Res F / Q (GL 2) , as opposed to the PEL Shimura variety associated to the subgroup G ∗ ⊂ G studied by Chai, we give a detailed explanation of theory of algebraic diamond operators for G, as well as how the theory of toroidal and minimal compactifications for G may be deduced from the analogous theory for G ∗ . [ABSTRACT FROM AUTHOR]

We discuss various applications of a uniform vanishing result for the graded components of the finite length Koszul module associated to a subspace K ⊆ ⋀ 2 V , where V is a vector space. Previously Koszul modules of finite length have been used to give a proof of Green's Conjecture on syzygies of generic canonical curves. We now give applications to effective stabilization of cohomology of thickenings of algebraic varieties, divisors on moduli spaces of curves, enumerative geometry of curves on K3 surfaces and to skew-symmetric degeneracy loci. We also show that the instability of sufficiently positive rank 2 vector bundles on curves is governed by resonance and give a splitting criterion. [ABSTRACT FROM AUTHOR]

Quantum Entanglement is one of the key manifestations of quantum mechanics that separate the quantum realm from the classical one. Characterization of entanglement as a physical resource for quantum technology became of uppermost importance. While the entanglement of bipartite systems is already well understood, the ultimate goal to cope with the properties of entanglement of multipartite systems is still far from being realized. This paper covers characterization of multipartite entanglement using algebraic-geometric tools. First, we establish an algorithm to classify multipartite entanglement by k -secant varieties of the Segre variety and ℓ -multilinear ranks that are invariant under Stochastic Local Operations with Classical Communication (SLOCC). We present a fine-structure classification of multiqubit and tripartite entanglement based on this algorithm. Another fundamental problem in quantum information theory is entanglement transformation that is quite challenging regarding to multipartite systems. It is captivating that the proposed entanglement classification by algebraic geometry can be considered as a reference to study SLOCC and asymptotic SLOCC interconversions among different resources based on tensor rank and border rank, respectively. In this regard, we also introduce a new class of tensors that we call persistent tensors and construct a lower bound for their tensor rank. We further cover SLOCC convertibility of multipartite systems considering several families of persistent tensors. [ABSTRACT FROM AUTHOR]

The geometry of 4-string contact interaction of closed string field theory is characterized using machine learning. We obtain Strebel quadratic differentials on 4-punctured spheres as a neural network by performing unsupervised learning with a custom-built loss function. This allows us to solve for local coordinates and compute their associated mapping radii numerically. We also train a neural network distinguishing vertex from Feynman region. As a check, 4-tachyon contact term in the tachyon potential is computed and a good agreement with the results in the literature is observed. We argue that our algorithm is manifestly independent of number of punctures and scaling it to characterize the geometry of n-string contact interaction is feasible. [ABSTRACT FROM AUTHOR]

In this study, we aim to prove the generalized Hyers-Ulam-Rassias (GHUR) stability for the following Euler-Lagrange (EL) type cubic functional equation f(bx+y)+f(x+by)=(b+1)(b-1)²[f(x)+f(y)]+b(b+1)f(x+y) in non-Archimedean (n.A) quasi-Banach spaces and n.A (n,β) Banach spaces. In recent decades, the stability of functional equations has emerged as one of the most intriguing and engaging topics, as it leads to the applications of functional equations in various domains such as algebraic geometry, Group theory, Mechanics etc., This study is to investigate the GHUR stability for the above equation using Hyers direct method. Furthermore, we obtain the stability results for the aforementioned equation with an illustrative example for the n.A case. With the study of the example one may easily understand how the stability result of functional equations in n.A case differs from the setting of classical Banach spaces. [ABSTRACT FROM AUTHOR]

We give the explicit equations for a P 3 × P 3 embedding of the Jacobian of a curve of genus 2, which gives a natural analog for abelian surfaces of the Edwards curve model of elliptic curves. This gives a much more succinct description of the Jacobian variety than the standard version in P 15 . We also give a condition under which, as for the Edwards curve, the abelian surfaces have a universal group law. [ABSTRACT FROM AUTHOR]

Sheaves are mathematical objects that describe the globally compatible data associated with open sets of a topological space. Original examples of sheaves were continuous functions; later they also became powerful tools in algebraic geometry, as well as logic and set theory. More recently, sheaves have been applied to the theory of contextuality in quantum mechanics. Whenever the local data are not necessarily compatible, sheaves are replaced by the simpler setting of presheaves. In previous work, we used presheaves to model lexically ambiguous phrases in natural language and identified the order of their disambiguation. In the work presented here, we model syntactic ambiguities and study a phenomenon in human parsing called garden-pathing. It has been shown that the information-theoretic quantity known as 'surprisal' correlates with human reading times in natural language but fails to do so in garden-path sentences. We compute the degree of signalling in our presheaves using probabilities from the large language model BERT and evaluate predictions on two psycholinguistic datasets. Our degree of signalling outperforms surprisal in two ways: (i) it distinguishes between hard and easy garden-path sentences (with a p -value <10−5), whereas existing work could not, (ii) its garden-path effect is larger in one of the datasets (32 ms versus 8.75 ms per word), leading to better prediction accuracies. This article is part of the theme issue 'Quantum contextuality, causality and freedom of choice'. [ABSTRACT FROM AUTHOR]

We introduce the Cox homotopy algorithm for solving a sparse system of polynomial equations on a compact toric variety X Σ . The algorithm lends its name from a construction, described by Cox, of X Σ as a GIT quotient X Σ = (ℂ k ∖ Z) / / G of a quasi-affine variety by the action of a reductive group. Our algorithm tracks paths in the total coordinate space ℂ k of X Σ and can be seen as a homogeneous version of the standard polyhedral homotopy, which works on the dense torus of X Σ . It furthermore generalizes the commonly used path tracking algorithms in (multi)projective spaces in that it tracks a set of homogeneous coordinates contained in the G -orbit corresponding to each solution. The Cox homotopy combines the advantages of polyhedral homotopies and (multi)homogeneous homotopies, tracking only mixed volume many solutions and providing an elegant way to deal with solutions on or near the special divisors of X Σ . In addition, the strategy may help to understand the deficiency of the root count for certain families of systems with respect to the BKK bound. [ABSTRACT FROM AUTHOR]

We study the cone of completely positive (cp) matrices for the first interesting case n = 5 . This is a semialgebraic set for which the polynomial equalities and inequlities that define its boundary can be derived. We characterize the different loci of this boundary and we examine the two open sets with cp-rank 5 or 6. A numerical algorithm is presented that is fast and able to compute the cp-factorization even for matrices in the boundary. With our results, many new example cases can be produced and several insightful numerical experiments are performed that illustrate the difficulty of the cp-factorization problem. [ABSTRACT FROM AUTHOR]

Using tools from algebraic topology and algebraic geometry (fundamental polygons and blowups, respectively), we prove that cutting along the center of a Möbius strip yields a cylinder. [ABSTRACT FROM AUTHOR]

Motivated by the study of the Kahan–Hirota–Kimura discretisation of the Euler top, we characterise the growth and integrability properties of a collection of elements in the Cremona group of a complex projective 3-space using techniques from algebraic geometry. This collection consists of maps obtained by composing the standard Cremona transformation c 3 ∈ Bir (P 3) with projectivities that permute the fixed points of c 3 and the points over which c 3 performs a divisorial contraction. Specifically, we show that three behaviour are possible: (A) integrable with quadratic degree growth and two invariants, (B) periodic with two-periodic degree sequences and more than two invariants, and (C) non-integrable with submaximal degree growth and one invariant. [ABSTRACT FROM AUTHOR]

We define the notion of Y-algebroids, generalising the Lie, Courant, and exceptional algebroids that have been used to capture the local symmetry structure of type II string theory and M-theory compactifications to D ≥ 5 dimensions. Instead of an invariant inner product, or its generalisation arising in exceptional algebroids, Y-algebroids are built around a specific type of tensor, denoted Y , that provides exactly the necessary properties to also describe compactifications to D = 4 dimensions. We classify "M-exact" E7-algebroids and show that this precisely matches the form of the generalised tangent space of E7(7) × ℝ+-generalised geometry, with possible twists due to 1-, 4- and 7-form fluxes, corresponding physically to the derivative of the warp factor and the M-theory fluxes. We translate the notion of generalised Leibniz parallelisable spaces, relevant to consistent truncations, into this language, where they are mapped to so-called exceptional Manin pairs. We also show how to understand Poisson-Lie U-duality and exceptional complex structures using Y-algebroids. [ABSTRACT FROM AUTHOR]