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The article offers information on dynamical systems and irrational angle construction by paper-folding. Construction of rational angle to any desired accuracy by folding a strip of paper requires approximating method. The construction uses a periodic sequence composed of two types of folding moves, guided by number-theoretic properties of the angle's fractional representation. The number-theoretic scheme must be replaced by something new, the Folds and Switchfold prescription, in the construction of irrational angles.

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]

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 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 paper we consider a family F of 2n-dimensional F q -linear rank metric codes in F q n × n arising from polynomials of the form x q s + δ x q n 2 + s ∈ F q n [ x ] . The family F was introduced by Csajbók et al. (JAMA 548:203–220) as a potential source for maximum rank distance (MRD) codes. Indeed, they showed that F contains MRD codes for n = 8 , and other subsequent partial results have been provided in the literature towards the classification of MRD codes in F for any n. In particular, the classification has been reached when n is smaller than 8, and also for n greater than 8 provided that s is small enough with respect to n. In this paper we deal with the open case n = 8 , providing a classification for any large enough odd prime power q. The techniques are from algebraic geometry over finite fields, since our strategy requires the analysis of certain 3-dimensional F q -rational algebraic varieties in a 7-dimensional projective space. We also show that the MRD codes in F are not equivalent to any other MRD codes known so far. [ABSTRACT FROM AUTHOR]

The paper summarizes some results on algebraic geometry presence in the homotopy theory. For the homotopy group πm(Sn ), denote by πmalg(Sn ) its subset of homotopy classes represented by R-entire rational maps Sm → Sn of spheres. The main result of this paper concerns to the study of π n+k alg(Sn) for k = 0, . . ., 7. [ABSTRACT FROM AUTHOR]

What is striking about de Broglie's foundational work on wave–particle dualism is the role played by pseudo-Riemannian geometry in his early thinking. While exploring a fully covariant description of the Klein–Gordon equation, he was led to the revolutionary idea that a variable rest mass was essential. DeWitt later explained that in order to obtain a covariant quantum Hamiltonian, one must supplement the classical Hamiltonian with an additional energy ℏ 2 Q from which the quantum potential emerges, a potential that Berry has recently shown also arises in classical wave optics. In this paper, we show how these ideas emerge from an essentially geometric structure in which the information normally carried by the wave function is contained within the algebraic description of the geometry itself, within an element of a minimal left ideal. We establish the fundamental importance of conformal symmetry, in which rescaling of the rest mass plays a vital role. Thus, we have the basis for a radically new theory of quantum phenomena based on the process of mass-energy flow. [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]

In this paper, we first give a relationship between generalized algebraic geometry codes (GAG codes) and algebraic geometry codes (AG codes). More precisely, we show that a GAG code is contained (up to isomorphism) in a suitable AG code. Next we recall the concept of an N1N2-automorphism group, a subgroup of the automorphism group of a GAG code. With the use of the relation we obtained between these two classes of codes, we show that the N1N2-automorphism group is a subgroup of the automorphism group of an AG code. [ABSTRACT FROM AUTHOR]

The interplays between topology and algebraic geometry present a set of interesting properties. In this paper, we comprehensively revisit the Bézout theorem in terms of topology, and we present a topological proof of the theorem considering n-dimensional space. We show the role of topology in understanding the complete and finite intersections of algebraic curves within a topological space. Moreover, we introduce the concept of symmetrically complex translations of roots in a zero-set of a real algebraic curve, which is called a fundamental polynomial, and we show that the resulting complex algebraic curve is additively decomposable into multiple components with varying degrees in a sequence. Interestingly, the symmetrically complex translations of roots in a zero-set of a fundamental polynomial result in the formation of isomorphic topological manifolds if one of the complex translations is kept fixed, and it induces repeated real roots in the fundamental polynomial as a component. A set of numerically simulated examples is included in the paper to illustrate the resulting manifold structures and the associated properties. [ABSTRACT FROM AUTHOR]

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]

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]

An MDS code is a code which achieves equality in the singleton bound. The defect of a code measures how far it is from an MDS code. Amplifying on the relationship between the weight distribution of a code and its dual code as in the well-known MacWilliams identities, we show in this paper that there are indeed strong lower bounds on the defects of codes or the dual codes. [ABSTRACT FROM AUTHOR]

We study the coherent cohomology of generalized flag supervarieties. Our main observation is that these groups are closely related to the free resolutions of (certain generalizations of) determinantal ideals. In the case of super Grassmannians, we completely compute the cohomology of the structure sheaf: it is composed of the singular cohomology of a Grassmannian and the syzygies of a determinantal variety. The majority of the work involves studying the geometry of an analog of the Grothendieck-Springer resolution associated with the super Grassmannian; this takes place in the world of ordinary (non-super) algebraic geometry. Our work gives a conceptual explanation of the result of Pragacz-Weyman that the syzygies of determinantal ideals admit an action of the general linear supergroup. In a subsequent paper, we will treat other flag supervarieties in detail. [ABSTRACT FROM AUTHOR]

In this paper, we study an algebraic fiber space in positive characteristic whose generic fiber F has finitely generated canonical ring and sufficiently large Frobenius stable canonical ring. An example of such a case is when F is F-pure and its dualizing sheaf is invertible and ample. We treat a Fujita-type conjecture due to Popa and Schnell concerning direct images of pluricanonical bundles, and prove it under some additional hypotheses. As an application, we show the subadditivity of Kodaira dimensions in some new cases. We also prove an analog of Fujino's result regarding his Fujita-type conjecture. [ABSTRACT FROM AUTHOR]

The formulations of polynomials over a topological simplex combine the elements of topology and algebraic geometry. This paper proposes the formulation of simplicial polynomials and the properties of resulting topological manifolds in two classes, non-degenerate forms and degenerate forms, without imposing the conditions of affine topological spaces. The non-degenerate class maintains the degree preservation principle of the atoms of the polynomials of a topological simplex, which is relaxed in the degenerate class. The concept of hybrid decomposition of a simplicial polynomial in the non-degenerate class is introduced. The decompositions of simplicial polynomial for a large set of simplex vertices generate ideal components from the radical, and the components preserve the topologically isolated origin in all cases within the topological manifolds. Interestingly, the topological manifolds generated by a non-degenerate class of simplicial polynomials do not retain the homeomorphism property under polynomial extension by atom addition if the simplicial condition is violated. However, the topological manifolds generated by the degenerate class always preserve isomorphism with varying rotational orientations. The hybrid decompositions of the non-degenerate class of simplicial polynomials give rise to the formation of simplicial chains. The proposed formulations do not impose strict positivity on simplicial polynomials as a precondition. [ABSTRACT FROM AUTHOR]

There are two main problems in finding the higher genus superstring measure. The first one is that for g ≥ 5 the super moduli space is not projected. Furthermore, the supermeasure is regular for g ≤ 11, a bound related to the source of singularities due to the divisor in the moduli space of Riemann surfaces with even spin structure having holomorphic sections, such a divisor is called the θ-null divisor. A result of this paper is the characterization of such a divisor. This is done by first extending the Dirac propagator, that is the Szegö kernel, to the case of an arbitrary number of zero modes, that leads to a modification of the Fay trisecant identity, where the determinant of the Dirac propagators is replaced by the product of two determinants of the Dirac zero modes. By taking suitable limits of points on the Riemann surface, this holomorphic Fay trisecant identity leads to identities that include points dependent rank 3 quadrics in ℙg−1. Furthermore, integrating over the homological cycles gives relations for the Riemann period matrix which are satisfied in the presence of Dirac zero modes. Such identities characterize the θ-null divisor. Finally, we provide the geometrical interpretation of the above points dependent quadrics and show, via a new θ-identity, its relation with the Andreotti-Mayer quadric. [ABSTRACT FROM AUTHOR]

As a key method to deal with loop integrals, Integration-By-Parts (IBP) method can be used to do reduction as well as establish the differential equations for master integrals. However, when talking about tensor reduction, the Passarino-Veltman (PV) reduction method is also widely used for one-loop integrals. Recently, we have proposed an improved PV reduction method, i.e., the PV reduction method with auxiliary vector R, which can easily give analytical reduction results for any tensor rank. However, our results are only for integrals with propagators with power one. In this paper, we generalize our method to one-loop integrals with general tensor structures and propagators with general powers. Our ideas are simple. We solve the generalised reduction problem by combining differentiation over masses and proper limit of reduction with power-one propagators. Finally, we demonstrate our method with several examples. With the result in this paper, we have shown that our improved PV-reduction method with auxiliary vector is a self-completed reduction method for one-loop integrals. [ABSTRACT FROM AUTHOR]

Variable-DOF (or kinematotropic) mechanisms are a class of reconfigurable mechanisms that have varying degrees of freedom (DOF) in different motion modes and can be reconfigured without disassembly. However, the number of proposed variable-DOF multi-loop planar mechanisms is currently limited. This paper introduces a new 8-link variable-DOF planar mechanism that has five motion modes. Firstly, the 8-link variable-DOF planar mechanism is described. Then, reconfiguration analysis of the mechanism is performed using a hybrid approach that combines elimination and computer algebraic geometry methods. The analysis reveals that the 8-link mechanism has one 2-DOF motion mode and four 1-DOF motion modes. It can switch among three motion modes at four transition configurations and between two motion modes at the remaining four transition configurations. The paper also highlights the geometric characteristics of the mechanism in different motion modes. In contrast to variable-DOF planar mechanisms presented in the literature, the proposed 8-link mechanism has two inactive joints in one of its 1-DOF motion modes. Moreover, both closed-loop 4R kinematic sub-chains of the mechanism must appear as either a pair of parallelograms or a pair of anti-parallelograms in the same motion mode. As a by-product of this research, a method for factoring trigonometric functions in two angles is also proposed. [ABSTRACT FROM AUTHOR]

Transcription factors (TFs) and microRNAs (miRNAs) are co-actors in genome-scale decoding and regulatory networks, often targeting common genes. To discover the symmetries and invariants of the transcription and regulation at the scale of the genome, in this paper, we introduce tools of infinite group theory and of algebraic geometry to describe both TFs and miRNAs. In TFs, the generator of the group is a DNA-binding domain while, in miRNAs, the generator is the seed of the sequence. For such a generated (infinite) group π , we compute the S L (2 , C) character variety, where S L (2 , C) is simultaneously a 'space-time' (a Lorentz group) and a 'quantum' (a spin) group. A noteworthy result of our approach is to recognize that optimal regulation occurs when π looks similar to a free group F r ( r = 1 to 3) in the cardinality sequence of its subgroups, a result obtained in our previous papers. A non-free group structure features a potential disease. A second noteworthy result is about the structure of the Groebner basis G of the variety. A surface with simple singularities (such as the well known Cayley cubic) within G is a signature of a potential disease even when π looks similar to a free group F r in its structure of subgroups. Our methods apply to groups with a generating sequence made of two to four distinct DNA/RNA bases in { A , T / U , G , C } . We produce a few tables of human TFs and miRNAs showing that a disease may occur when either π is away from a free group or G contains surfaces with isolated singularities. [ABSTRACT FROM AUTHOR]

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Inflationary spacetimes have been argued to be past geodesically incomplete in many situations. However, whether the geodesic incompleteness implies the existence of an initial spacetime curvature singularity or whether the spacetime may be extended (potentially into another phase of the universe) is generally unknown. Both questions have important physical implications. In this paper, we take a closer look at the geometrical structure of inflationary spacetimes and investigate these very questions. We first classify which past inflationary histories have a scalar curvature singularity and which might be extendible and/or non-singular in homogeneous and isotropic cosmology with flat spatial sections. Then, we derive rigorous extendibility criteria of various regularity classes for quasi-de Sitter spacetimes that evolve from infinite proper time in the past. Finally, we show that beyond homogeneity and isotropy, special continuous extensions respecting the Einstein field equations with a perfect fluid must have the equation of state of a de Sitter universe asymptotically. An interpretation of our results is that past-eternal inflationary scenarios are most likely physically singular, except in situations with very special initial conditions. [ABSTRACT FROM AUTHOR]

In the Grassmannian formulation of the S-matrix for planar N = 4 Super Yang-Mills, Nk−2MHV scattering amplitudes for k negative and n − k positive helicity gluons can be expressed, by an application of the global residue theorem, as a signed sum over a collection of (k − 2)(n − k − 2)-dimensional residues. These residues are supported on certain positroid subvarieties of the Grassmannian G(k, n). In this paper, we replace the Grassmannian G(3, n) with its torus quotient, the moduli space of n points in the projective plane in general position, and planar N = 4 SYM with generalized biadjoint scalar amplitudes m n 3 as introduced by Cachazo-Early-Guevara-Mizera (CEGM) [1]. Whereas in the Grassmannian formulation residues of the Parke-Taylor form correspond to individual BCFW, or on-shell diagrams, we show that each such (n − 5)-dimensional residue of m n 3 an entire biadjoint scalar partial amplitude m n 2 , that is a sum over all tree-level Feynman diagrams for a fixed planar order. We make a proposal which would do the same for k ≥ 4. [ABSTRACT FROM AUTHOR]

The purpose of the paper is to show that, in low dimensions, the WDVV equations are bi-Hamiltonian. The invariance of the bi-Hamiltonian formalism is proved for N = 3. More examples in higher dimensions show that the result might hold in general. The invariance group of the bi-Hamiltonian pairs that we find for WDVV equations is the group of projective transformations. The significance of projective invariance of WDVV equations is discussed in detail. The computer algebra programs that were used for calculations throughout the paper are provided in a GitHub repository. [ABSTRACT FROM AUTHOR]

This paper proposes an approach to derive the Jacobian matrix of a hybrid mechanism by applying a velocity operator to the transformation matrix. This Jacobian matrix is capable of deducing hybrid singularities, which cannot be identified by using the screw-based Jacobian or velocity-based Jacobian. The transformation matrix was obtained based on the algebraic geometry approach, and it becomes the key point since it was used to not only formulate the Jacobian matrix, but also to define the motion type of hybrid mechanisms. In this paper, two hybrid mechanisms were investigated, which were composed of two distinct parallel mechanisms mounted in series. Hybrid Mechanisms 1 and 2 were composed of 3-PRP-3-RPS and 3-RPS-3-PRP (the underlined P is an actuated joint), respectively. The motion types of Hybrid Mechanisms 1 and 2 were determined from the product of the transformation matrices of the 3-PRP and 3-RPS parallel mechanisms, and vice versa. The developed method was employed to establish the Jacobian matrix to which the conditioning index was applied. Therefore, the kinematic performances of the two hybrid mechanisms can be compared for a given bone surgery trajectory within the workspace. It turns out that Hybrid Mechanism 1 has superior performance than that of Mechanism 2, which indicates that Mechanism 1 is better at transmitting power to the moving platform. [ABSTRACT FROM AUTHOR]

Originally, the expression 'Teichmüller theory' referred to the theory that Oswald Teichmüller developed on deformations and on moduli spaces of marked Riemann surfaces. This theory is not an isolated field in mathematics. At different stages of its development, it received strong impetuses from analysis, geometry, and algebraic topology, and it had a major impact on other fields, including low-dimensional topology, algebraic topology, hyperbolic geometry, geometric group theory, representations of discrete groups in Lie groups, symplectic geometry, topological quantum field theory, theoretical physics, and there are certainly others. Of course, the impacts on these various fields are not equally important, but in some cases (namely, low-dimensional topology, algebraic geometry, and physics) the impact was crucial. At the same time, Teichmüller theory established important connections between the fields mentioned. This, in part, is a consequence of the diversity and the richness of the structure that Teichmüller space itself carries. From a more subjective point of view, the result of pondering on these connections and applications demonstrates the unity of mathematics. The aim of this paper is to survey the origin of Teichmüller theory and the development of its early major ideas. [ABSTRACT FROM AUTHOR]

The Euclidean hull of a linear code $C$ is the intersection of $C$ with its Euclidean dual $C^\perp $. The hull with low dimensions gets much interest due to its crucial role in determining the complexity of algorithms for computing the automorphism group of a linear code and for checking permutation equivalence of two linear codes. The Euclidean hull of a linear code has been applied to the so-called entanglement-assisted quantum error-correcting codes (EAQECCs) via classical error-correcting codes. In this paper, we firstly consider linear codes with one-dimensional Euclidean hull from algebraic geometry codes, and then present a general method to construct linear codes with arbitrary dimensional Euclidean hull. Some new EAQECCs are presented. [ABSTRACT FROM AUTHOR]

Previous algorithms for decoding AG codes up to the designed distance all assumed existence of an extra rational place on the base algebraic curve. The place is used to solve the decoding problem by linear algebra over the base field of the curve. The rationality of the place is essential, and therefore AG codes supported by all rational places on the curve are excluded from the domain of applicability of the decoding algorithms. This paper presents a decoding algorithm for those AG codes using an extra place of higher degree. Hence finally all AG codes, as Goppa defined 40 years ago, are equipped with a fast decoding algorithm. [ABSTRACT FROM AUTHOR]

Nonlinear estimation problems, such as range-only and bearing-only target tracking, are often addressed using linearized estimators, e.g., the extended Kalman filter (EKF). These estimators generally suffer from linearization errors as well as the inability to track multimodal probability density functions. In this paper, we propose a bank of batch maximum a posteriori (MAP) estimators as a general estimation framework that provides relinearization of the entire state trajectory, multihypothesis tracking, and an efficient hypothesis generation scheme. Each estimator in the bank is initialized using a locally optimal state estimate for the current time step. Every time a new measurement becomes available, we relax the original batch-MAP problem and solve it incrementally. More specifically, we convert the relaxed one-step-ahead cost function into polynomial or rational form and compute all the local minima analytically. These local minima generate highly probable hypotheses for the target's trajectory and hence greatly improve the quality of the overall MAP estimate. Additionally, pruning of least probable hypotheses and marginalization of old states are employed to control the computational cost. Monte Carlo simulation and real-world experimental results show that the proposed approach significantly outperforms the standard EKF, the batch-MAP estimator, and the particle filter. [ABSTRACT FROM PUBLISHER]

We present several classes of constraints on the discontinuities of Feynman integrals that go beyond the Steinmann relations. These constraints follow from a geometric formulation of the Landau equations that was advocated by Pham, in which the singularities of Feynman integrals correspond to critical points of maps between on-shell spaces. To establish our results, we review elements of Picard-Lefschetz theory, which connect the homotopy properties of the space of complexified external momenta to the homology of the combined space of on-shell internal and external momenta. An important concept that emerges from this analysis is the question of whether or not a pair of Landau singularities is compatible — namely, whether or not the Landau equations for the two singularities can be satisfied simultaneously. Under conditions we describe, sequential discontinuities with respect to non-compatible Landau singularities must vanish. Although we only rigorously prove results for Feynman integrals with generic masses in this paper, we expect the geometric and algebraic insights that we gain will also assist in the analysis of more general Feynman integrals. [ABSTRACT FROM AUTHOR]

Over the past few decades, the mathematical community has accumulated a significant amount of pure mathematical data, which has been analyzed through supervised, semi-supervised, and unsupervised machine learning techniques with remarkable results, e.g., artificial neural networks, support vector machines, and principal component analysis. Therefore, we consider as disruptive the use of machine learning algorithms to study mathematical structures, enabling the formulation of conjectures via numerical algorithms. In this paper, we review the latest applications of machine learning in the field of geometry. Artificial intelligence can help in mathematical problem solving, and we predict a blossoming of machine learning applications during the next years in the field of geometry. As a contribution, we propose a new method for extracting geometric information from the point cloud and reconstruct a 2D or a 3D model, based on the novel concept of generalized asymptotes. [ABSTRACT FROM AUTHOR]

This paper is a companion to our paper [Poisson geometry of the moduli of local systems on smooth varieties, Publ. RIMS 57 (2021), no. 3–4, 959—991]. We study the moduli functor of flat bundles on the smooth, possibly non-proper, algebraic variety X over a field of characteristic zero. For this we introduce the notion of a formal boundary of X, denoted by ∂Xb, which is a formal analog of the boundary at ∞ of the Betti topological space associated with X. We explain how to construct two derived moduli stacks Vect∇(X) and Vect∇ ∂Xb of flat bundles on X and on ∂Xb, respectively, as well as a restriction map R: Vect∇(X) → Vect∇ ∂Xb . This work contains two main results. First we prove that the morphism R comes equipped with a canonical shifted Lagrangian structure. This result can be understood as the de Rham analog of the existence of Poisson structures on moduli of local systems on X. As a second statement, we prove that the geometric fibers of R are representable by quasi-algebraic spaces, a slight weakening of the notion of algebraic space [ABSTRACT FROM AUTHOR]

In this paper, we study the number of different interference alignment (IA) solutions in a $K$ -user multiple-input multiple-output (MIMO) interference channel, when the alignment is performed via beamforming and no symbol extensions are allowed. We focus on the case where the number of IA equations matches the number of variables. In this situation, the number of IA solutions is finite and constant for any channel realization out of a zero-measure set and, as we prove in this paper, it is given by an integral formula that can be numerically approximated using Monte Carlo integration methods. More precisely, the number of alignment solutions is the scaled average of the determinant of a certain Hermitian matrix related to the geometry of the problem. Interestingly, while the value of this determinant at an arbitrary point can be used to check the feasibility of the IA problem, its average (properly scaled) gives the number of solutions. For single-beam systems, the asymptotic growth rate of the number of solutions is analyzed and some connections with classical combinatorial problems are presented. Nonetheless, our results can be applied to arbitrary interference MIMO networks, with any number of users, antennas, and streams per user. [ABSTRACT FROM PUBLISHER]

In this paper, we study the decoding of dual-containing codes from Hermitian tower and applications to quantum codes. The contribution of this paper is threefold. First, we construct the quantum stabilizer codes from the Hermitian tower. Second, we provide a deterministic decoding algorithm with decoding radius that almost achieves the optimal decoding radius, i.e., $(1-R)/4$ , where $R$ is the rate. Last and most importantly, we present a Monte Carlo algorithm with decoding radius roughly equal to $(1-R)/3$ , which is beyond the optimal decoding radius $(1-R)/4$ . There are several features in this paper. First of all, we employ a differential for the Hermitian tower. This differential plays a crucial role for decoding. We also extend our decoding by passing to the constant field extension. This constant field extension makes the decoding work perfectly. [ABSTRACT FROM PUBLISHER]

In this paper we discuss the construction of conserved tensors (currents) involving conformal Killing-Yano tensors (CKYTs), generalising the corresponding constructions for Killing-Yano tensors (KYTs). As a useful preparation for this, but also of intrinsic interest, we derive identities relating CKYTs and geometric quantities. The behaviour of CKYTs under conformal transformations is also given, correcting formulae in the literature. We then use the identities derived to construct covariantly conserved "currents". We find several new CKYT currents and also include a known one by Penrose which shows that "trivial" currents are also useful. We further find that rank-n currents based on rank-n CKYTs k must have a simple form in terms of dk. By construction, these currents are covariant under a general conformal rescaling of the metric. How currents lead to conserved charges is then illustrated using the Kerr-Newman and the C-metric in four dimensions. Separately, we study a rank-1 current, construct its charge and discuss its relation to the recently constructed Cotton current for the Kerr-Newman black hole. [ABSTRACT FROM AUTHOR]

A Teichmüller curve V \subset \mathcal {M}_g is an isometrically immersed algebraic curve in the moduli space of Riemann surfaces. These rare, extremal objects are related to billiards in polygons, Hodge theory, algebraic geometry and surface topology. This paper presents the six known families of primitive Teichmüller curves that have been discovered over the past 30 years, and a selection of open problems. [ABSTRACT FROM AUTHOR]