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A symmetric planar central configuration of the Newtonian six-body problem x is called cross central configuration if there are precisely four bodies on a symmetry line of x. We use complex algebraic geometry and Groebner basis theory to prove that for a generic choice of positive real masses $$m_1,m_2,m_3,m_4,m_5=m_6$$ there is a finite number of cross central configurations. We also show one explicit example of a configuration in this class. A part of our approach is based on relaxing the output of the Groebner basis computations. This procedure allows us to obtain upper bounds for the dimension of an algebraic variety. We get the same results considering cross central configurations of the six-vortex problem.

This thesis is dedicated to the constructions of modular lattices with algebraic methods. The goal is to develop new methods as well as constructing new lattices. There are three methods considered: construction from number fields, construction from totally definite quaternion algebras over number fields and construction from linear codes via generalized Construction A. The construction of Arakelov-modular lattices, which result in modular lattices, was first introduced in [6] for ideal lattices from cyclotomic fields. We generalize this construction to other number fields and also to totally definite quaternion algebras over number fields. We give the characterization of Arakelov-modular lattices over the maximal real subfield of a cyclotomic field with prime power degree and totally real Galois fields with odd degrees. Characterizations of Arakelov-modular lattices of trace type, which are special cases of Arakelov- modular lattices, are given for quadratic fields and maximal real subfields of cyclotomic fields with non-prime power degrees. Furthermore, we give the classification of Arakelov-modular lattices of level l for l a prime over totally definite quaternion algebras with base field the field of rationals. Construction A is a well studied method to obtain lattices from codes via quotient of different rings, such as rings of integers, in which case mostly cyclotomic number fields have been considered. In this thesis, we will study Construction A over all totally real and CM fields. Using Construction A, the intersection between a lattice constructed from a linear complementary dual (LCD) code and its dual lattice is investigated. This is an attempt to find an equivalent definition to LCD codes for lattices. Several new constructions of existing extremal lattices as well as a new extremal lattice are obtained from the above mentioned methods. The mathematical concepts used in this thesis mainly involve algebraic number theory, class field theory, non commutative algebra and coding theory. ​Doctor of Philosophy (SPMS)

This work is devoted to interpretation of concepts of Zariski dimension of an algebraic variety over a field and of Krull dimension of a coordinate ring in algebraic geometry over algebraic structures of an arbitrary signature. Proposed dimensions are ordinal numbers (ordinals).

We present an algorithmic strategy to compute an upper bound for the degree of the algebraic solutions of non-degenerate polynomial differential equations in dimension two.

Constructing algebraic polynomials using computer algebra techniques is believed to be state-of-the-art in analyzing gate-level arithmetic circuits. However, the existing approach applies algebraic rewriting directly to the gate-level netlist, which has potential memory explosion problem. This paper introduces an algebraic rewriting technique based on the and-inverter graph (AIG) representation of gate-level designs. Using AIG-based cut-enumeration and truth table computation, an efficient order of algebraic rewriting is identified, resulting in dramatic simplifications of the polynomial under construction. An automatic approach, which further reduces the complexity of algebraic rewriting by handling redundant polynomials, is also proposed.

In this note we prove the following surprising characterization: if X ⊂ is an (embedded, non-empty, proper) algebraic variety deûned over a field k of characteristic zero, then X is a hypersurface if and only if the module of logarithmic vector fields of X is a reflexive -module. As a consequence of this result, we derive that if is a free -module, which is shown to be equivalent to the freeness of the t-th exterior power of for some (in fact, any) t ≤ n, then necessarily X is a Saito free divisor.

We give positive characteristic analogues of complex entire functions having remarkable property that their values as well as their derivatives of any order at any nonzero algebraic numbers are algebraically independent. These results are obtained by establishing a criterion for the algebraic independence of the values of Mahler functions as well as that of the algebraic independence of the Mahler functions themselves over any function fields of positive characteristic.

This work is shown below, the algebraic sum of the two functions selected from class S of univalent functions which is a subclass of this class A of functions f(z) satisfy the conditions analiytic in the open unit disk U={z∈C:|z

This article is a part of our effort to explain the foundations of algebraic geometry over arbitrary algebraic structures [1–8]. We introduce the concept of universal geometrical equivalence of two algebraic structures A and B of a common language L which strengthens the available concept of geometrical equivalence and expresses the maximal affinity between A and B from the viewpoint of their algebraic geometries. We establish a connection between universal geometrical equivalence and universal equivalence in the sense of equality of universal theories.

The present paper is one in our series of works on algebraic geometry over arbitrary algebraic structures, which focuses on the concept of geometrical equivalence. This concept signifies that for two geometrically equivalent algebraic structures $$ \mathcal{A} $$ and ℬ of a language L, the classification problems for algebraic sets over $$ \mathcal{A} $$ and ℬ are equivalent. We establish a connection between geometrical equivalence and quasiequational equivalence.

Bu makalede proksimal relator uzaylarında yaklaşımlı yarıgruplar ve ideallere giriş yapılmıştır. Tanımsal proksimiti bağıntısı ile birlikte dikkate alınan dijital görüntülerde yaklaşımlı yarıgrup ve ideal örnekleri verilmiştir. Bundan başka, nesne tanımlaması homomorfizması kullanılarak tanımsal yaklaşımların bazı özellikleri incelenmiştir.

This work makes a contribution to algebraic parameter estimation as it proposes a simple alternative to the derivation of the algebraic estimation equations. The idea is based on a system representation in the form of an evaluation functional which does not exhibit any singularities in the neighbourhood of zero. Implied is the fact that algebraic estimation of parameters as well as system states can then truly be performed in arbitrary time and with uniform accuracy over the entire estimation interval. Additionally, the result offers a geometric representation of a linear system as a finite dimensional subspace of a Hilbert space, that readily suggests powerful noise rejection methods in which invariance plays a central role.

The paper is a brief survey of the author’s results connected with the lattices of algebraic sets of universal algebras and with the operator of algebraic closure on the subsets of direct powers of basic sets of algebras.

The division algorithm for ideals of algebraic power series satisfying Hironaka’s box condition is shown to be finite when expressed suitably in terms of the defining polynomial codes of the series. In particular, the codes of the reduced standard basis of the ideal can be constructed effectively.

We examine the tangent groups at the identity, and more generally the formal completions at the identity, of the Chow groups of algebraic cycles on a nonsingular quasiprojective algebraic variety over a field of characteristic zero. We settle a question recently raised by Mark Green and Phillip Griffiths concerning the existence of Bloch–Gersten–Quillen-type resolutions of algebraic K-theory sheaves on infinitesimal thickenings of nonsingular varieties, and the relationships between these sequences and their “tangent sequences,” expressed in terms of absolute Kahler differentials. More generally, we place Green and Griffiths’ concrete geometric approach to the infinitesimal theory of Chow groups in a natural and formally rigorous structural context, expressed in terms of nonconnective K-theory, negative cyclic homology, and the relative algebraic Chern character.

We compare families of functions related to the Darboux property (functions having the 𝒜-Darboux property) with family of strong Świątkowski functions using the notions of strong c-algebrability. We also compare families of functions associated with density topologies.

J.R. Sendra is member of the Research Group ASYNACS (Ref.CT-CE2019/683), The resultant theory plays a crucial role in computational algebra and algebraic geometry. The theoryhas two aspects: algebraic and geometric. In this paper, we focus on the algebraic aspect. One of themost important and well known algebraic properties of the resultant is that it is equal to the determinantof the Sylvester matrix. In 2008, Odagiri proved that a similar property holds over the tropical semiringif one replaces subtraction with addition. The tropical semiring belongs to a large family of algebraicstructures called commutative idempotent semiring. In this paper, we prove that the same property(with subtraction replaced with addition) holds over an arbitrary commutative idempotent semiring., Ministerio de Economía y Competitividad

The foundation of algebraic geometry is the solving of systems of polynomial equations. When the equations to be considered are defined over a subfield of the complex numbers, numerical methods can be used to perform algebraic geometric computations forming the area of numerical algebraic geometry. This article provides a short introduction to numerical algebraic geometry with the subsequent articles in this special issue considering three current research topics: solving structured systems, certifying the results of numerical computations, and performing algebraic computations numerically via Macaulay dual spaces.

A smooth variety is called uniformly rational if every point admits a Zariski open neighborhood isomorphic to a Zariski open subset of the affine space. In this note we show that every smooth and rational affine variety endowed with an algebraic torus action such that the algebraic quotient has dimension 0 or 1 is uniformly rational., Comment: 4 pages

Recent research shows that the class of rotation symmetric Boolean functions is potentially rich in functions of cryptographic significance. In this paper, two classes of rotation symmetric Boolean functions having optimal algebraic immunity on even number of variables are presented. We give a lower bound of the algebraic degree of the functions in the first class, and derive the algebraic degree of the second class of functions. Moreover, the algebraic degree of the second class of functions is high enough. It is shown that both classes of functions have much better nonlinearity than all the previously obtained rotation symmetric Boolean functions with optimal algebraic immunity, and have good behavior against fast algebraic attacks at least for small numbers of input variables.

The solution set of a system of polynomial equations, called an algebraic set, can be decomposed into finitely many irreducible components. In numerical algebraic geometry, irreducible algebraic sets are represented by witness sets, whereas general algebraic sets allow a numerical irreducible decomposition comprising a collection of witness sets, one for each irreducible component. We denote the solution set of any system of polynomials f : C N ź C n as V ( f ) ź C N . Given a witness set for some algebraic set Z ź C N and a system of polynomials f : C N ź C n , the algorithms of this paper compute a numerical irreducible decomposition of the set Z ź V ( f ) . While extending the types of intersection problems that can be solved via numerical algebraic geometry, this approach is also a unification of two existing algorithms: the diagonal intersection algorithm and the homotopy membership test. The new approach includes as a special case the "extension problem" where one wishes to intersect an irreducible component A of V ( g ( x ) ) with V ( f ( x , y ) ) , where f introduces new variables, y. For example, this problem arises in computing the singularities of A when the singularity conditions are expressed in terms of new variables associated to the tangent space of A. Several examples are included to demonstrate the effectiveness of our approach applied in a variety of scenarios.

This paper gives conditions for algebraic independence of a collection of functions satisfying a certain kind of algebraic difference relations. As applications, we show algebraic independence of two collections of special functions: (1) Vigneras' multiple gamma functions and derivatives of the gamma function, (2) the logarithmic function, \(q\)-exponential functions and \(q\)-polylogarithm functions. In a similar way, we give a generalization of Ostrowski's theorem.

In Tang and Bai [Math Nachr. 2012;285:922–935] classified a class of non-graded left-symmetric algebraic structures on the Witt algebra under a certain rational condition. In this note, we show that this rational condition is not necessary. This leads to a more elegant classification of the left-symmetric algebraic structures and Novikov algebraic structures on the Witt algebra.

Algebraic tori occupy a special place among linear algebraic groups. An algebraic torus can be defined over an arbitrary field but if the ground field is of arithmetic type, one can additionally consider schemes over the ring of integers of this field, which are related to the original tori and called their integral models. The Neron and Voskresenskiĭ models are most well known among them. There exists a broad range of problems dealing with the construction of these models and the elucidation of their properties. This paper is devoted to the study of the main integral models of algebraic tori over fields of algebraic numbers, to the comparison of their properties, and to the clarification of links between them. At the end of this paper, a special family of maximal algebraic tori unaffected inside semisimple groups of Bn type is presented as an example for realization of previously investigated constructions.

Differential algebraic geometry seeks to extend the results of its algebraic counterpart to objects defined by differential equations. Many notions, such as that of a projective algebraic variety, have close differential analogues but their behavior can vary in interesting ways. Workers in both differential algebra and model theory have investigated the property of completeness of differential varieties. After reviewing their results, we extend that work by proving several versions of a “differential valuative criterion" and using them to give new examples of complete differential varieties. We conclude by analyzing the first examples of incomplete, finite-rank projective differential varieties, demonstrating a clear difference from projective algebraic varieties.

A group action on the input ring or category induces an action on the algebraic $K$-theory spectrum. However, a shortcoming of this naive approach to equivariant algebraic $K$-theory is, for example, that the map of spectra with $G$-action induced by a $G$-map of $G$-rings is not equivariant. We define a version of equivariant algebraic $K$-theory which encodes a group action on the input in a functorial way to produce a $genuine$ algebraic $K$-theory $G$-spectrum for a finite group $G$. The main technical work lies in studying coherent actions on the input category. A payoff of our approach is that it builds a unifying framework for equivariant topological $K$-theory, Atiyah's Real $K$-theory, and existing statements about algebraic $K$-theory spectra with $G$-action. We recover the map from the Quillen-Lichtenbaum conjecture and the representational assembly map studied by Carlsson and interpret them from the perspective of equivariant stable homotopy theory., Comment: Final version to appear in Mathematische Zeitschrift. The last section about Waldhausen G-categories has been removed from this paper

Let P k : = F 2 [ x 1 , x 2 , … , x k ] be the graded polynomial algebra over the prime field of two elements F 2 , in k generators x 1 , x 2 , … , x k , each of degree 1. Being the mod-2 cohomology of the classifying space B ( Z / 2 ) k , the algebra P k is a module over the mod-2 Steenrod algebra A . In this Note, we extend a result of Hưng on Kameko's homomorphism S q ˜ ⁎ 0 : F 2 ⊗ A P k ⟶ F 2 ⊗ A P k . Using this result, we show that Singer's conjecture for the algebraic transfer is true in the case k = 5 and the degree 7.2 s − 5 with s an arbitrary positive integer.

We prove some results in universal algebraic geometry over algebraic structures of arbitrary functional languages with relation ≠ adjoined.

Let $X$ be a connected affine homogenous space of a linear algebraic group $G$ over $\C$. (1) If $X$ is different from a line or a torus we show that the space of all algebraic vector fields on $X$ coincides with the Lie algebra generated by complete algebraic vector fields on $X$. (2) Suppose that $X$ has a $G$-invariant volume form $\omega$. We prove that the space of all divergence-free (with respect to $\omega$) algebraic vector fields on $X$ coincides with the Lie algebra generated by divergence-free complete algebraic vector fields on $X$ (including the cases when $X$ is a line or a torus). The proof of these results requires new criteria for algebraic (volume) density property based on so called module generating pairs., Comment: 21 pages

Global observability of continuous-time polynomial systems is studied. An algebraic necessary and sufficient condition of global observability is proved. It is expressed with the aid of real radicals of ideals in the ring of polynomials and is based on the real theorem of zeros from real algebraic geometry.

The goals of this paper fall into two closely related areas. First, we develop a formal framework for deterministic unital quantities in which measurement unitization is understood to be a built-in feature of quantities rather than a mere annotation of their numerical values with convenient units. We introduce this idea within the setting of certain ordered semigroups of physical-geometric states of classical physical systems. States are assumed to serve as truth makers of metrological statements about quantity values. A unital quantity is presented as an isomorphism from the target system’s ordered semigroup of states to that of positive reals. This framework allows us to include various derived and variable quantities, encountered in engineering and the natural sciences. For illustration and ease of presentation, we use the classical notions of length, time, electric current and mean velocity as primordial examples. The most important application of the resulting unital quantity calculus is in dimensional analysis. Second, in evaluating measurement uncertainty due to the analog-to-digital conversion of the measured quantity’s value into its measuring instrument’s pointer quantity value, we employ an ordered semigroup framework of pointer states. Pointer states encode the measuring instrument’s indiscernibility relation, manifested by not being able to distinguish the measured system’s topologically proximal states. Once again, we focus mainly on the measurement of length and electric current quantities as our motivating examples. Our approach to quantities and their measurement is strictly state-based and algebraic in flavor, rather than that of a representationalist-style structure-preserving numerical assignment.

Generalized hypergeometric differential equations of arbitrary order are considered. Necessary and sufficient conditions for the algebraic independence of solutions of collections of such equations, as well as of their values at algebraic points, are obtained.

A global generalization of continued fraction that yields the best Diophantine approximations of any dimension is considered. In the algebraic case, this generalization underlies a method for calculating the fundamental units of algebraic rings and the periods of best approximations, as well as the identification of the fundamental domain with respect to these periods. The units of an algebraic field are understood as the units of maximal order of this field.

The authors consider current problems of the modern theory of dynamic systems on manifolds, which are actively developing. A brief review of such trends in the theory of dynamic systems is given. The results of the algebra of dual numbers, quaternionic algebras, biquaternions (dual quaternions), and their application to the analysis of infinitesimal neighborhoods and infinitesimal deformations of manifolds (schemes) are presented. The theory of differential---algebraic equations over the field of real numbers and their dynamics, as well as elements of trajectory optimization of respective dynamic systems, are outlined. On the basis of connection in bundles, the theory of differential---algebraic equations is extended to algebraic manifolds and schemes over arbitrary fields and schemes, respectively.

New multivariable asymmetric public-key encryption schemes based on the NP-complete problem of simultaneous algebraic Riccati equations over finite fields are suggested. We also provide a systematic way to describe any set of quadratic equations over any field, as a set of algebraic Riccati equations. This has the benefit of systematic algebraic crypt-analyzing any encryption scheme based on quadratic equations, to any possible vulnerable hidden structure, in view of the fact that the set of all solutions to any given single algebraic Riccati equation is fully described in terms of all the T -invariant subspaces of some restricted dimension, where T is the matrix of coefficients of the related algebraic Riccati equation.

Under study is the component algebraic immunity of vectorial Boolean functions. We prove a theorem on the correspondence between the maximal component algebraic immunity of a function and its balancedness. Some relationship is obtained between the maximal component algebraic immunity and matrices of a special form. We construct several functions with maximal component algebraic immunity in case of few variables.

We use a geometric technique based on embeddings of graphs to provide an explicit formula for the absolute algebraic connectivity and its eigenvectors of double brooms. Besides, we give a polynomial time combinatorial algorithm that computes the absolute algebraic connectivity of a given tree.

For connected simple algebraic groups defined over an algebraically closed field of characteristic zero, the classifications of irreducible algebraic representations of modalities 0, 1, and 2 are obtained.