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Algebraic independence of values of certain infinite products is proved, where the transcendence of such numbers was already established by Tachiya. As applications explicit examples of algebraically independent numbers are also given.

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.

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

The main purpose of this paper is to study the problems on the existence of algebraic solutions for some second-order complex differential equations with entire algebraic function element coefficients. Several theorems on the existence of solutions are obtained, which perfect the solution theory of linear complex differential equations.

Altinel and Cherlin proved that any perfect central extension of a simple algebraic group over an algebraically closed field which happens to be of finite Morley rank is actually a finite central extension and is itself an algebraic group. We will prove an infinite rank version of their result with an additional hypothesis, while giving an example which shows the necessity of this hypothesis. The inspiration for the work comes from differential algebra; namely, a differential algebraic version of the results here was used by the second author to answer a question of Cassidy and Singer. The work here also provides an alternate path to the same answer.An almost simple superstable group G is one in which every definable normal subgroup H has the property that RU ( G ) RU ( H ) ? n for any n ? N . We prove that any almost simple superstable group which is a central extension of a simple algebraic group is actually a finite central extension and is an algebraic group. We also explain the applications of this result to differential algebraic groups. Many of the central ideas of the proof of our main theorem are an adaptation of techniques developed by the second author in the setting of differential algebraic groups.

In this paper we characterize all cubic polynomial differential systems in the plane having two circles as invariant algebraic limit cycles. The first author is partially supported by a MINECO grant number MTM2014-53703-P, and an AGAUR (Generalitat de Catalunya) grant number 2014SGR 1204. The second author is partially supported by a MINECO grant MTM2013-40998-P, an AGAUR grant 2014SGR 568, and two grants FP7-PEOPLE-2012-IRSES numbers 316338 and 318999. The third author is partially supported by FCT/Portugal through the project UID/MAT/04459/2013.

In this paper, we iterate the algebraic etale-Brauer set for any nice variety X over a number field k with π 1 et ( X ‾ ) finite and we show that the iterated set coincides with the original algebraic etale-Brauer set. This provides some evidence towards the conjectures by Colliot-Thelene on the arithmetic of rational points on nice geometrically rationally connected varieties over k and by Skorobogatov on the arithmetic of rational points on K3 surfaces over k; moreover, it gives a partial answer to an “algebraic” analogue of a question by Poonen about iterating the descent set.

Agraïments: The second author is partially supported by FCT/Portugal through UID/MAT/04459/2013. Algebraic limit cycles in quadratic polynomial differential systems started to be studied in 1958, and few years later the following conjecture appeared: Quadratic polynomial differential systems have at most one algebraic limit cycle. We prove that for a quadratic polynomial differential system having two pairs of diametrally opposite equilibrium points at infinity, has at most one algebraic limit cycle. Our result provides a partial positive answer to this conjecture.

A power moments based algebraic method that takes into account the local Taylor’s expansion structure of the wave function is applied to find the spectrum for the two dimensional parity-time symmetric potential V(x, y) = x 2 + y 2 + i g x 2 y. Converging results are presented for a wide range of the strength parameter g.

In this paper, we study various properties of algebraic extension of *-A operator. Specifically, we show that every algebraic extension of *-A operator has SVEP and is isoloid. And if T is an algebraic extension of *-A operator, then Weyl's theorem holds for f(T), where f is an analytic functions on some neighborhood of σ(T) and not constant on each of the components of its domain.

We use a generalization of a construction by Ziegler to show that for any field F and any countable collection of countable subsets A i ⊆ F , i ∈ I ⊂ Z > 0 there exist infinitely many fields K of arbitrary greater than one transcendence degree over F and of infinite algebraic degree such that each A i is first-order definable over K. We also use the construction to show that many infinitely axiomatizable theories of fields which are not compatible with the theory of algebraically closed fields are finitely hereditarily undecidable.

In this paper, two constructions of Boolean functions with optimal algebraic immunity are proposed. They generalize previous ones respectively given by Rizomiliotis (IEEE Trans Inf Theory 56:4014---4024, 2010) and Zeng et al. (IEEE Trans Inf Theory 57:6310---6320, 2011) and some new functions with desired properties are obtained. The functions constructed in this paper can be balanced and have optimal algebraic degree. Further, a new lower bound on the nonlinearity of the proposed functions is established, and as a special case, it gives a new lower bound on the nonlinearity of the Carlet-Feng functions, which is slightly better than the best previously known ones. For $$n\le 19$$n≤19, the numerical results reveal that among the constructed functions in this paper, there always exist some functions with nonlinearity higher than or equal to that of the Carlet-Feng functions. These functions are also checked to have good behavior against fast algebraic attacks at least for small numbers of input variables.

In the present paper, we classify varieties of algebraic systems of the type [Formula: see text], for natural numbers [Formula: see text] and [Formula: see text], which are closed under particular derived algebraic systems. If we replace in an algebraic system the [Formula: see text]-ary operation by an [Formula: see text]-ary term operation and the [Formula: see text]-ary relation by the [Formula: see text]-ary relation generated by an [Formula: see text]-ary formula, we obtain a new algebraic system of the same type, which we call derived algebraic system. We shall restrict the replacement to so-called “linear” terms and atomic “linear” formulas, respectively.

For each element a in the Banach algebra A , we define the resolvent space R a and completely characterize it whenever a is algebraic. In particular, we find elements a with R a ≠ { a } ′ . Then we consider the Banach algebra of operators L ( X ) , and show that R A possesses nontrivial invariant subspaces whenever A is an algebraic element of L ( X ) . This assertion becomes stronger than that of the existence of a hyper-invariant subspace for A whenever R A ≠ { A } ′ .

Various aspects of the work of Blok and Rebagliato on the algebraic semantics for deductive systems are studied in the context of logics formalized as π-institutions. Three kinds of semantics are surveyed: institution, matrix (system) and algebraic (system) semantics, corresponding, respectively, to the generalized matrix, matrix and algebraic semantics of the theory of sentential logics. After some connections between matrix and algebraic semantics are revealed, it is shown that every (finitary) N -rule based extension of an N -rule based π-institution possessing an algebraic semantics also possesses an algebraic semantics. This result abstracts one of the main theorems of Blok and Rebagliato. An attempt at a Blok-Rebagliato-style characterization of those π-institutions with a mono-unary category of natural transformations on their sentence functors having an algebraic semantics is also made. Finally, a necessary condition for a π-institution to possess an algebraic semantics is provided. c

Agraïments: FEDER-UNAB10-4E-378 and Consolider CSD2007-00004 "ES" We prove the following two results. First every planar polynomial vector field of degree S with S invariant circles is Darboux integrable without limit cycles. Second a planar polynomial vector field of degree S admits at most S - 1 invariant circles which are algebraic limit cycles. In particular we solve the 16th Hilbert problem restricted to algebraic limit cycles given by circles, because a planar polynomial vector field of degree S has at most S - 1 algebraic limit cycles given by circles, and this number is reached.

We give a local analytic characterization that a minimal surface in the 3-sphere $\, \ES^3 \subset \R^4$ defined by an irreducible cubic polynomial is one of the Lawson's minimal tori. This provides an alternative proof of the result by Perdomo (\emph{Characterization of order 3 algebraic immersed minimal surfaces of $S^3$},Geom. Dedicata 129 (2007), 23--34)., Comment: 19 pages

We conjecture the existence of special elements in odd degree higher algebraic K-groups of number fields that are related in a precise way to the values at strictly negative integers of the derivatives of Artin L-functions of finite dimensional complex representations. We prove this conjecture in certain important cases and also provide other evidence (both theoretical and numerical) in its support., 25 pages

The algebraic L-groups L∗(A, X) are defined for an additive category A with chain duality and a ∆-set X, and identified with the generalized homology groups H∗(X;L•(A)) of X with coefficients in the algebraic L-spectrum L•(A). Previously such groups had only been defined for simplicial complexes X.

We prove a Darboux–Jouanolou type theorem on the algebraic integrability of polynomial differential r -forms.

In this paper, enlightened by the idea of the weight of a polynomial introduced by Swinnerton-Dyer [2002], we find all the invariant algebraic surfaces of the Chen system x′ = a(y - x), y′ = (c - a)x + cy - xz, z′ = xy - bz.

An (affine) algebraic monoid is an affine variety over an algebraically closed field K endowed with a monoid structure such that the product map is an algebraic variety morphism. Let M be an irreducible algebraic monoid, G its unit group, P a parabolic subgroup of G, and e ∈ M a minimal idempotent. We show that P = C P ( e ) R u ( G ) and that the assignment P ↦ C P ( e ) defines a one-to-one correspondence between parabolic subgroups of G and of C G ( e ) .

Recall that an algebraic module is a K G -module that satisfies a polynomial with integer coefficients, with addition and multiplication given by the direct sum and tensor product. In this article we prove that non-periodic algebraic modules are very rare, and that if the complexity of an algebraic module is at least 3, then it is the only algebraic module on its component of the (stable) Auslander–Reiten quiver. For dihedral 2-groups, we also show that there is at most one algebraic module on each component of the (stable) Auslander–Reiten quiver. We include a strong conjecture on the relationship between periodicity and algebraicity.

We present a new method for computing zeros of polynomial systems using the algebraic solver and the QR-method. It is based on the theory of algebraic solvers. The unstable calculation of the determinant of the large matrix is replaced by a stable technique using the QR-method. Algorithms and numerical results are presented.

In this paper, we explicitly construct a large class of symmetric Boolean functions on 2k variables with algebraic immunity not less than d, where integer k is given arbitrarily and d is a given suffix of k in binary representation. If let d=k, our constructed functions achieve the maximum algebraic immunity. Remarkably, 2⌊log 2k⌋+2 symmetric Boolean functions on 2k variables with maximum algebraic immunity are constructed, which are much more than the previous constructions. Based on our construction, a lower bound of symmetric Boolean functions with algebraic immunity not less than d is derived, which is 2⌊ log 2d⌋+2(k-d+1). As far as we know, this is the first lower bound of this kind.

In this paper, we characterize the set of all ternary algebraic (or polynomial) operations of idempotent algebras that have at least one binary and one ternary algebraic operation depending on every variable, and there exists an integer r > 2 such that there is not any r-ary algebraic operation depending on every variable. We prove that this set forms a finite DeMorgan algebra with a fixed element and then we characterize this DeMorgan algebra.

This paper is concerned with properties of the algebraic degree of the Laplace-Stieltjes transform of phase-type (PH) distributions. The main problem of interest is: given a PH generator, how do we find the maximum and the minimum algebraic degrees of all irreducible PH representations with that PH generator? Based on the matrix exponential (ME) order of ME distributions and the spectral polynomial algorithm, a method for computing the algebraic degree of a PH distribution is developed. The maximum algebraic degree is identified explicitly. Using Perron-Frobenius theory of nonnegative matrices, a lower bound and an upper bound on the minimum algebraic degree are found, subject to some conditions. Explicit results are obtained for special cases.

If G is a connected linear algebraic group over the field k, a Levi factor of G is a reductive complement to the unipotent radical of G. If k has positive characteristic, G may have no Levi factor, or G may have Levi factors which are not geometrically conjugate. We give in this paper some sufficient conditions for the existence and the conjugacy of Levi factors of G. Let A be a Henselian discrete valuation ring with fractions K and with perfect residue field k of characteristic p>0. Let G be a connected and reductive algebraic group over K. Bruhat and Tits have associated to G certain smooth A-group schemes P whose generic fibers P/K coincide with G; these are known as *parahoric group schemes*. The special fiber P/k of a parahoric group scheme is a linear algebraic group over k. If G splits over an unramified extension of K, we show that P/k has a Levi factor, and that any two Levi factors of P/k are geometrically conjugate., To appear, Morozov issue of Transformation Groups

In the context of 2-summable Fredholm modules, we prove that the Connes-Karoubi multiplicative character coincides with Brown's determinant invariant on algebraic K-theory.

This dissertation is concerned with the study of irreducible embeddings of simple algebraic groups of exceptional type. It is motivated by the role of such embeddings in the study of positive dimensional closed subgroups of classical algebraic groups. The classification of the maximal closed connected subgroups of simple algebraic groups was carried out by E. B. Dynkin, G. M. Seitz and D. M. Testerman. Their analysis for the classical groups was based primarily on a striking result: if G is a simple algebraic group and o : G → SL(V ) is a tensor indecomposable irreducible rational representation then, with specified exceptions, the image of G is maximal among closed connected subgroups of one of the classical groups SL(V), Sp(V ) or SO(V ). In the case of closed, not necessarily connected, subgroups of the classical groups, one is interested in considering irreducible embeddings of simple algebraic groups and their automorphism groups: given a simple algebraic group Y defined over an algebraically closed field K, one is led to study the embeddings G 0, G is a closed non-connected positive dimensional subgroup of Y and V is a nontrivial irreducible rational KY -module such that V|G is irreducible. We obtain a precise description of such triples (G, Y, V ).

The sixteen families of q-series containing the Ramanujan functions were listed by I.J. Zucker (SIAM J. Math. Anal. 10:192–206, 1979), which are generated from the Fourier series expansions of the Jacobian elliptic functions or some of their squares. This paper discusses algebraic independence properties for these q-series. We determine all the sets of q-series such that, at each algebraic point, the values of the q-series in the set are algebraically independent over ℚ. We also present several algebraic relations over ℚ for two or three of these q-series.

Algebraic immunity (AI) measures the resistance of a Boolean function f against algebraic attack. Extended algebraic immunity (EAI) extends the concept of algebraic immunity, whose point is that a Boolean function f may be replaced by another Boolean function f c called the algebraic complement of f. In this paper, we study the relation between different properties (such as weight, nonlinearity, etc.) of Boolean function f and its algebraic complement f c . For example, the relation between annihilator sets of f and f c provides a faster way to find their annihilators than previous report. Next, we present a necessary condition for Boolean functions to be of the maximum possible extended algebraic immunity. We also analyze some Boolean functions with maximum possible algebraic immunity constructed by known existing construction methods for their extended algebraic immunity.

In the past few years, algebraic attacks against stream ciphers with linear feedback function have been significantly improved. As a response to the new attacks, the notion of algebraic immunity of a Boolean function f was introduced, defined as the minimum degree of the annihilators of f and f + 1. An annihilator of f is a nonzero Boolean function g, such that fg = 0. There is an increasing interest in construction of Boolean functions that possess optimal algebraic immunity, combined with other characteristics, like balancedness, high nonlinearity, and high algebraic degree. In this paper, we investigate a recently proposed infinite class of balanced Boolean functions with optimal algebraic immunity, optimum algebraic degree and much better nonlinearity than all the previously introduced classes of Boolean functions with maximal algebraic immunity. More precisely, we study the resistance of the functions against one of the new algebraic attacks, namely the fast algebraic attacks (FAAs). Using the special characteristics of the family members, we introduce an efficient method for the evaluation of their behavior against these attacks. The new algorithm is based on the well studied Berlekamp---Massey algorithm.

With the help of Abel differential equations we obtain a new class of Darboux integrable planar polynomial differential systems, which have degenerate infinity. Moreover such integrable systems may have algebraic limit cycles. Also we present the explicit expressions of these algebraic limit cycles for quintic systems.

We prove an analogue of the Lindemann-Weierstrass theorem (that the exponentials of a Q-linearly independent set of algebraic numbers are algebraically independent), replacing Q alg by C(t) alg , and G n by a semiabelian variety over C(t) alg . Both the formulations of our results and the methods are dierential algebraic in nature.

Let G be an affine algebraic group and let X be an affine algebraic variety. An action G × X → X is called observable if for any G-invariant, proper, closed subset Y of X there is a nonzero invariant f ∈ \(\Bbbk\) [X]G such that f|Y = 0. We characterize this condition geometrically as follows. The action G × X → X is observable if and only if

Let α in (0, 1] and β > 1 be algebraic numbers. We study the asymptotic behaviour of the function that counts the number of digit changes in the β-expansion of α.

We investigate how well complex algebraic numbers can be approximated by algebraic numbers of degree at most n. We also investigate how well complex algebraic numbers can be approximated by algebraic integers of degree at most n+1. It follows from our investigations that for every positive integer n there are complex algebraic numbers of degree larger than n that are better approximable by algebraic numbers of degree at most n than almost all complex numbers. As it turns out, these numbers are more badly approximable by algebraic integers of degree at most n+1 than almost all complex numbers., Comment: 34 pages

Algebraic immunity is a new cryptographic criterion proposed against algebraic attacks. In order to resist algebraic attacks, Boolean functions used in many stream ciphers should possess high algebraic immunity. This paper presents two main results to find balanced Boolean functions with maximum algebraic immunity. Through swapping the values of two bits, and then generalizing the result to swap some pairs of bits of the symmetric Boolean function constructed by Dalai, a new class of Boolean functions with maximum algebraic immunity are constructed. Enumeration of such functions is also given. For a given function p(x) with deg(p(x)) < Open image in new window, we give a method to construct functions in the form p(x)+q(x) which achieve the maximum algebraic immunity, where every term with nonzero coefficient in the ANF of q(x) has degree no less than Open image in new window.

We investigate the algebraic independence of some derivatives of certain multiplicative arithmetical functions over the field C \mathbb {C} of complex numbers.

Let [math] be an elliptic curve. An irreducible algebraic curve [math] embedded in [math] is called weak-transverse if it is not contained in any proper algebraic subgroup of [math] , and transverse if it is not contained in any translate of such a subgroup. ¶ Suppose [math] and [math] are defined over the algebraic numbers. First we prove that the algebraic points of a transverse curve [math] that are close to the union of all algebraic subgroups of [math] of codimension [math] translated by points in a subgroup [math] of [math] of finite rank are a set of bounded height. The notion of closeness is defined using a height function. If [math] is trivial, it is sufficient to suppose that [math] is weak-transverse. ¶ The core of the article is the introduction of a method to determine the finiteness of these sets. From a conjectural lower bound for the normalized height of a transverse curve [math] , we deduce that the sets above are finite. Such a lower bound exists for [math] . ¶ Concerning the codimension of the algebraic subgroups, our results are best possible.

We provide conditions guaranteeing that a given element of a simple algebraic group G has a fixed vector in every nonzero G-module, and deduce similar results for finite Chevalley groups.

For an algebraic curve X over the finite field F q , we denote by N ( X ) and g ( X ) the number of F q -rational points and genus of X , respectively. Define the asymptotic quantity A ( q ) by lim sup g ( X ) → ∞ N ( X ) / g ( X ) . In this paper, we show an improvement on the best known lower bound on A ( 2 ) .

In the present notes we provide a new uniform way to compute a canonical p-dimension of a split algebraic group G for a torsion prime p using degrees of basic polynomial invariants described by V. Kac. As an application, we compute the canonical p-dimensions for all split simple algebraic groups. The notion of a canonical dimension of an algebraic structure was introduced by Berhuy and Reichstein [1]. For a split algebraic group G and its torsion prime p the canonical p-dimension of G was studied by Karpenko and Merkurjev in [3] and [4]. In particular, this invariant was shown to be related with the size of the image of the characteristic map φG : S ∗(T ) → CH∗(X), (1) where T is the character group of a maximal split torus T , X = G/B is the variety of complete flags and S stands for the symmetric algebra. Namely, one has the following formula for the canonical p-dimension of a group G cdp(G) = min{i | Chi(X)}, (2) where Chi(X) stands for the image of φG in the Chow ring Chi(X) = CHi(X;Z/pZ) with Z/pZ-coefficients. Using this remarkable fact together with the explicit description of the image Rp = Ch(X) Karpenko and Merkurjev computed the canonical p-dimensions for all classical algebraic groups G. The goal of the present notes is to relate the work by V. Kac [2] devoted to the study of the p-torsion of the Chow ring of an algebraic group G with the canonical p-dimensions of G. In particular we provide a different and uniform approach of computing cdp(G).

The Fibonacci-type numbers in the title look like $R_n=g_1\gamma_1^n+ g_2\gamma_2^n$ and $S_n=h_1\gamma_1^n+h_2\gamma_2^n$ for any $n\in\mathbb{Z}$, where the $g$'s, $h$'s, and $\gamma$'s are given algebraic numbers satisfying certain natural conditions. For fixed $k\in\mathbb{Z}_{>0}$, and for fixed non-zero periodic sequences $(a_h),(b_h),(c_h)$ of algebraic numbers, the algebraic independence of the series \[ \sum_{h=0}^\infty \frac{a_h}{\gamma_1^{kr^h}}\,, \quad {\sum_{h=0}^\infty}\,\strut' \frac{b_h}{(R_{kr^h+\ell})^m}\,, \quad {\sum_{h=0}^\infty}\,\strut' \frac{c_h}{(S_{kr^h+\ell})^m} \qquad \big((\ell,m,r)\in\mathbb{Z}\times \mathbb{Z}_{>0}\times\mathbb{Z}_{>1}\big) \] is studied. Here the main tool is Mahler's method which reduces the investigation of the algebraic independence of numbers (over $\mathbb{Q}$) to that of functions (over the rational function field) if they satisfy certain types of functional equations.

In this note, we show that the algebraic K-theory of generalized archimedean valuation rings occurring in Durov's compactification of the spectrum of a number ring is given by stable homotopy groups of certain classifying spaces. We also show that the "residue field at infinity" is badly behaved from a K-theoretic point of view., Comment: Final version. To appear in Journal of Homotopy and Related Structures

Let G be a connected semisimple linear algebraic group over an algebraically closed field k of positive characteristic and let X denote an equivariant embedding of G. We define a distinguished Steinberg fiber N in G, called the zero-fiber, and prove that the closure of N within X is normal and Cohen-Macaulay. Furthermore, when X is smooth we prove that the closure of N is a local complete intersection., Comment: 18 pages

We examine families of twists by an automorphism of the complex polynomial ring on n generators. The multiplication in the twisted algebra determines a Poisson structure on affine n-space. We demonstrate that if the automorphism has a single eigenvalue, then the primitive ideals in the twist are parameterized by the algebraic symplectic leaves associated to this Poisson structure. Furthermore, in this case all of the leaves are algebraic and can be realized as the orbits of an algebraic group.

The special case of the (p + 1)th degree algebraic equations of the kind xp+1 = xp + 1 (p = 1, 2, 3, …) is researched in the present article. For the case p = 1, the given equation is reduced to the well-known Golden Proportion equation x2 = x + 1. These equations are called the golden algebraic equations because the golden p-proportions τp, special irrational numbers that follow from Pascal’s triangle, are their roots. A research on the general properties of the roots of the golden algebraic equations is carried out in this article. In particular, formulas are derived for the golden algebraic equations that have degree greater than p + 1. There is reason to suppose that algebraic equations derived by the authors in the present article will interest theoretical physicists. For example, these algebraic equations could be found in the research of the energy relationships within the structures of many compounds and physical particles. For the case of butadiene (C4H6), this fact is proved by the famous physicist Richard Feynman.