Terms Of High Degree Research Articles

Nonsmooth cryptanalysis, with an application to the stream cipher MICKEY

A new approach to the cryptanalysis of symmetric algorithms based on non- smooth optimisation is presented. We develop this technique as a novel way of dealing with nonlinearity over F2 by modeling the equations corresponding to the algorithm as a continuous optimisation problem that avoids terms of higher degree. The resulting prob- lems are not continuously differentiable, but can be approached with techniques from non- smooth analysis. Applied to the stream cipher MICKEY, which is part of the eSTREAM final portfolio, this method can solve instances corresponding to the full cipher, although with time complexity greater than brute force. Finally, we compare this approach to clas- sical pseudo-Boolean programming.

Lower bounds for a polynomial in terms of its coefficients

We determine new sufficient conditions in terms of the coefficients for a polynomial \({f\in \mathbb{R}[\underline{X}]}\) of degree 2d (d ≥ 1) in n ≥ 1 variables to be a sum of squares of polynomials, thereby strengthening results of Fidalgo and Kovacec (Math. Zeitschrift, to appear) and of Lasserre (Arch. Math. 89 (2007) 390–398). Exploiting these results, we determine, for any polynomial \({f\in \mathbb{R}[\underline{X}]}\) of degree 2d whose highest degree term is an interior point in the cone of sums of squares of forms of degree d, a real number r such that f − r is a sum of squares of polynomials. The existence of such a number r was proved earlier by Marshall (Canad. J. Math. 61 (2009) 205–221), but no estimates for r were given. We also determine a lower bound for any polynomial f whose highest degree term is positive definite.

Non-abelian Reciprocity Laws on a Riemann Surface

On a Riemann surface, there are relations among the periods of holomorphic differential forms called Riemann’s relations. We prove them, using iterated integrals. In this paper, we generalize these relations to relations among generating series of iterated integrals. The generating series gives infinitely many relations—one for each coefficient of the generating series. The lower-order terms give the well-known classical relations. The new result is reciprocity for the higher degree terms, which give non-trivial relations among iterated integrals on a Riemann surface. As an application we refine the definition of Manin’s non-commutative modular symbol in order to include Eisenstein series. Finally, this paper contains some constructions needed for multi-dimensional reciprocity laws like a refinement of one of the Kato–Parshin reciprocity laws (Horozov, I. “A refinement of the Parshin symbol for surfaces”. Preprint, arXiv.org, AG, 1002.2698].

Parametric normal forms of vector fields and their further simplification

Given a family of vector fields parametrized by ξ in its linear part, one usually obtains its versal unfolding by the so-called two-step approach, i.e. find a Poincaré normal form for the system with fixed ξ0 and then apply the obtained near-identity transformation to the system with general ξ. It is also a common practice to treat ξ as a component together with those state components and calculate normal forms of the extended system. In this paper we reformulate normal forms on modules of homogeneous polynomials over the ring of all continuous functions of ξ and give a direct computation of versal unfolding. Our procedure enables us to determine coefficients of all terms of a certain degree in the normal form before we give a near-identity transformation of this degree. We can give all available near-identity transformations and choose an appropriate one to eliminate more terms of higher degree for a simpler normal form. We prove that the normal form reduced in our procedure is the simplest and unique. We illustrate our method with systems of linear centre and nilpotent linear parts separately.

Modeling single-particle energy levels and resonance currents in a coherent electronic quantum dot mixer

We present model calculations based on a coherent tunneling picture, which reproduce well both the single-particle energy level position and the resonant current strength at two typical anticrossings, one involving two levels and the other three levels in a coherent mixer composed of two weakly coupled vertical quantum dots. An essential ingredient is the inclusion of higher degree terms to account for deviations from an ideal elliptical parabolic confining potential in realistic dots. We also calculate density plots of the mixed states for the modified potential.

Counting quiver representations over finite fields via graph enumeration

Let Γ be a quiver on n vertices v 1 , v 2 , … , v n with g i j edges between v i and v j , and let α ∈ N n . Hua gave a formula for A Γ ( α , q ) , the number of isomorphism classes of absolutely indecomposable representations of Γ over the finite field F q with dimension vector α . Kac showed that A Γ ( α , q ) is a polynomial in q with integer coefficients. Using Hua's formula, we show that for each integer s ⩾ 0 , the sth derivative of A Γ ( α , q ) with respect to q, when evaluated at q = 1 , is a polynomial in the variables g i j , and we compute the highest degree terms in this polynomial. Our formulas for these coefficients depend on the enumeration of certain families of connected graphs.

Hierarchical reconstruction for spectral volume method on unstructured grids

The hierarchical reconstruction (HR) [Y.-J. Liu, C.-W. Shu, E. Tadmor, M.-P. Zhang, Central discontinuous Galerkin methods on overlapping cells with a non-oscillatory hierarchical reconstruction, SIAM J. Numer. Anal. 45 (2007) 2442–2467; Z.-L. Xu, Y.-J. Liu, C.-W. Shu, Hierarchical reconstruction for discontinuous Galerkin methods on unstructured grids with a WENO type linear reconstruction and partial neighboring cells, J. Comput. Phys. 228 (2009) 2194–2212] is applied to a piecewise quadratic spectral volume method on two-dimensional unstructured grids as a limiting procedure to prevent spurious oscillations in numerical solutions. The key features of this HR are that the reconstruction on each control volume only uses adjacent control volumes, which forms a compact stencil set, and there is no truncation of higher degree terms of the polynomial. We explore a WENO-type linear reconstruction on each hierarchical level for the reconstruction of high degree polynomials. Numerical computations for scalar and system of nonlinear hyperbolic equations are performed. We demonstrate that the hierarchical reconstruction can generate essentially non-oscillatory solutions while keeping the resolution and desired order of accuracy for smooth solutions.

Counting Quiver Representations over Finite Fields Via Graph Enumeration

Let $\Gamma$ be a quiver on $n$ vertices $v_1, v_2, \ldots , v_n$ with $g_{ij}$ edges between $v_i$ and $v_j$, and let $\boldsymbol{\alpha} \in \mathbb{N}^n$. Hua gave a formula for $A_{\Gamma}(\boldsymbol{\alpha}, q)$, the number of isomorphism classes of absolutely indecomposable representations of $\Gamma$ over the finite field $\mathbb{F}_q$ with dimension vector $\boldsymbol{\alpha}$. We use Hua's formula to show that the derivatives of $A_{\Gamma}(\boldsymbol{\alpha}, q)$ with respect to $q$, when evaluated at $q = 1$, are polynomials in the variables $g_{ij}$, and we can compute the highest degree terms in these polynomials. The formulas for these coefficients depend on the enumeration of certain families of connected graphs. This note simply gives an overview of these results; a complete account of this research is available on the arXiv and has been submitted for publication. Soit $\Gamma$ un carquois sur $n$ sommets $ v_1, v_2, \ldots , v_n$ avec $g_{ij}$ arêtes entre $v_i$ et $v_j$, et soit $\boldsymbol{\alpha} \in \mathbb{N}^n$. Hua a donné une formule pour $A_{\Gamma}(\boldsymbol{\alpha}, q)$, le nombre de classes d'isomorphisme absolument indécomposables de représentations de $\Gamma$ sur le corps fini $\mathbb{F}_q$ avec vecteur de dimension $\boldsymbol{\alpha}$. Nous utilisons la formule de Hua pour montrer que les dérivées de $A_{\Gamma}(\boldsymbol{\alpha}, q)$ par rapport à $q$, alors évaluée à $q=1$, sont des polynômes dans les variables $g_{ij}$, et on peut calculer les termes de plus haut degré de ces polynômes. Les formules pour ces coefficients dépendent de l'énumération de certaines familles de graphes connectés. Cette note donne simplement un aperçu de ces résultats, un compte rendu complet de cette recherche est disponible sur arXiv et a été soumis pour publication.

Stability of discrete-time systems in the simplest critical case

The paper considers nonlinear time-invariant discrete-time systems of arbitrary orders. The right-hand sides of the equations are sums of polynomials of arbitrary degrees and continuous functions which are infinitely small with respect to the highest-degree terms of the polynomials. It is assumed that the characteristic polynomial of the linearized system has one root equal to unity, while its other roots lie inside the unit disk.

An improved EZ-GCD algorithm for multivariate polynomials

The EZ-GCD algorithm often has the bad-zero problem, which has a remarkable influence on polynomials with higher-degree terms. In this paper, by applying special ideals, the EZ-GCD algorithm for sparse polynomials is improved. This improved algorithm greatly reduces computational complexity because of the sparseness of polynomials. The author expects that the use of these ideals will be useful as a resolution for obtaining a GCD of sparse multivariate polynomials with higher-degree terms.

Twisted Alexander polynomials of twist knots for nonabelian representations

In this paper, we describe the twisted Alexander polynomial of twist knots for nonabelian SL ( 2 , C ) -representations and investigate in detail the coefficient of the highest degree term as a function on the representation space of the knot group. In particular, we introduce the notion of monic representation and discuss its relation to the fiberedness of knots.

Stanley's character polynomials and coloured factorisations in the symmetric group

In Stanley [R.P. Stanley, Irreducible symmetric group characters of rectangular shape, Sém. Lothar. Combin. 50 (2003) B50d, 11 p.] the author introduces polynomials which help evaluate symmetric group characters and conjectures that the coefficients of the polynomials are positive. In [R.P. Stanley, A conjectured combinatorial interpretation of the normalised irreducible character values of the symmetric group, math.CO/0606467, 2006] the same author gives a conjectured combinatorial interpretation for the coefficients of the polynomials. Here, we prove the conjecture for the terms of highest degree.

Some robust designs for polynomial regression models

AbstractThe author identifies static optimal designs for polynomial regression models with or without intercept. His optimality criterion is an average between the D‐optimality criterion for the estimation of low‐degree terms and the D8‐optimality criterion for testing the significance of higher degree terms. His work relies on classical results concerning canonical moments and the theory of continued fractions.

A state space embedding approach to approximate feedback linearization of single input nonlinear control systems

AbstractThis contribution presents a numerical approach to approximate feedback linearization which transforms the Taylor expansion of a single input nonlinear system into an approximately linear system by considering the terms of the Taylor expansion step by step. In the linearization procedure, higher degree terms are taken into account by using a state space embedding such that the corresponding system representation has not to be computed in every linearization step. Linear matrix equations are explicitly derived for determining the nonlinear change of coordinates and the nonlinear feedback that approximately linearize the nonlinear system. If these linear matrix equations are not solvable, a least square solution by applying the Moore–Penrose inverse is proposed. The results of the paper are illustrated by the approximate feedback linearization of an inverted pendulum on a cart. Copyright © 2006 John Wiley & Sons, Ltd.

Alternating formulas for K -theoretic quiver polynomials

The main theorem here is the K -theoretic analogue of the cohomological ``stable double component formula'' for quiver polynomials in [KMS]. This K -theoretic version is still in terms of lacing diagrams, but nonminimal diagrams contribute terms of higher degree. The motivating consequence is a conjecture of Buch [B1] on the sign alternation of the coefficients appearing in his expansion of quiver K -polynomials in terms of stable Grothendieck polynomials for partitions.

To the phenomenological theory of ferroelectric phase transitions

Some insufficiently studied aspects of proper ferroelectric phase transitions are considered. These are the role played by higher-degree terms of the expansion of a thermodynamic potential in polarization, the universal temperature dependences of physical quantities represented in terms of dimensionless variables, the soft mode (which, along with other modes, gives the contribution to polarization) and the polarization considered as an order parameter, and also the phase transition from piezoelectric phases.

Extended version of the van der Waals capillarity theory

An extended version of the van der Waals capillarity theory describing the liquid-vapor interface in the temperature range from the triple to the critical point is suggested. A model functional of thermodynamic potential for a two-phase Lennard-Jones system taking into account the effect of the highest degree terms of gradient expansion has been constructed. The identity of the thermodynamic and the mechanical definition of Tolman's length has been proved in the framework of the adopted form of functional. The properties of nuclei of the liquid and the vapor phase are described. The paper determines: the work of formation of a nucleus, density profiles, size dependences of the surface tension, and the parameter delta in the Gibbs-Tolman-Koenig-Buff equation.

Formal Normal Forms for the Perturbations of a Quasi-Homogeneous Hamiltonian Vector Field

We classify up to a formal change of variable the vector fields of two complex variables together with the foliations that they de.ne which are a perturbation of a quasi-homogeneous Hamiltonian vector field X0 by terms of higher degree of quasi-homogeneity. We do not require anything about the degree δ0 of the initial vector field X0, but we assume that the perturbed vector field still keeps invariant the separatrices of X0. We obtain formal normal forms which extend those obtained in the case of an initial vector field with a semisimple or nilpotent linear part. We give an interpretation of the dual version of these normal forms through the relative cohomology with respect to the dual initial di.erential form ω0.

Expressions for IAU 2000 precession quantities

A new precession-nutation model for the Celestial Intermediate Pole (CIP) was adopted by the IAU in 2000 (Resolution B1.6). The model, designated IAU 2000A, includes a nutation series for a non-rigid Earth and corrections for the precession rates in longitude and obliquity. The model also specifies numerical values for the pole osets at J2000.0 between the mean equatorial frame and the Geocentric Celestial Reference System (GCRS). In this paper, we discuss precession models consistent with IAU 2000A precession-nutation (i.e. MHB 2000, provided by Mathews et al. 2002) and we provide a range of expressions that implement them. The final precession model, designated P03, is a possible replacement for the precession com- ponent of IAU 2000A, oering improved dynamical consistency and a better basis for future improvement. As a preliminary step, we present our expressions for the currently used precession quantities A;A; zA, in agreement with the MHB corrections to the precession rates, that appear in the IERS Conventions 2000. We then discuss a more sophisticated method for improving the precession model of the equator in order that it be compliant with the IAU 2000A model. In contrast to the first method, which is based on corrections to the t terms of the developments for the precession quantities in longitude and obliquity, this method also uses corrections to their higher degree terms. It is essential that this be used in conjunction with an improved model for the ecliptic precession, which is expected, given the known discrepancies in the IAU 1976 expressions, to contribute in a significant way to these higher degree terms. With this aim in view, we have developed new expressions for the motion of the ecliptic with respect to the fixed ecliptic using the developments from Simon et al. (1994) and Williams (1994) and with improved constants fitted to the most recent numerical planetary ephemerides. We have then used these new expressions for the ecliptic together with the MHB corrections to precession rates to solve the precession equations for providing new solution for the precession of the equator that is dynamically consistent and compliant with IAU 2000. A number of perturbing eects have first been removed from the MHB estimates in order to get the physical quantities needed in the equations as integration constants. The equations have then been solved in a similar way to Lieske et al. (1977) and Williams (1994), based on similar theoretical expressions for the contributions to precession rates, revised by using MHB values. Once improved expressions have been obtained for the precession of the ecliptic and the equator, we discuss the most suitable precession quantities to be considered in order to be based on the minimum number of variables and to be the best adapted to the most recent models and observations. Finally we provide developments for these quantities, denoted the P03 solution, including a revised Sidereal Time expression.

Algorithms for computing sparsest shifts of polynomials in power, Chebyshev, and Pochhammer bases

We give a new class of algorithms for computing sparsest shifts of a given polynomial. Our algorithms are based on the early termination version of sparse interpolation algorithms: for a symbolic set of interpolation points, a sparsest shift must be a root of the first possible zero discrepancy that can be used as the early termination test. Through reformulating as multivariate shifts in a designated set, our algorithms can compute the sparsest shifts that simultaneously minimize the terms of a given set of polynomials. Our algorithms can also be applied to the Pochhammer and Chebyshev bases for the polynomials, and potentially to other bases as well. For a given univariate polynomial, we give a lower bound for the optimal sparsity. The efficiency of our algorithms can be further improved by imposing such a bound and pruning the highest degree terms.