Arbitrary High Degree Research Articles

Invariant Algebraic Curves and Hyperelliptic Limit Cycles of Liénard Systems

In this paper we deal with the Lienard system $$\dot{x}=y, \dot{y}=-f_m(x)y-g_n(x),$$ where $$f_m(x)$$ and $$g_n(x)$$ are real polynomials of degree m and n, respectively. We call this system the Lienard system of type (m, n). For this system, we proved that if $$m+1\le n\le [\frac{4m+2}{3}]$$ , then the maximum number of hyperelliptic limit cycles is $$n-m-1$$ , and this bound is sharp. This result indicates that the Lienard system of type $$(m,m+1)$$ has no hyperelliptic limit cycles. Secondly, we present examples of irreducible algebraic curves of arbitrary high degree for Lienard systems of type $$(m,2m+1)$$ . Moreover, these systems have a rational first integral. Finally, we proved that the Lienard system of type (2, 5) has at most one hyperelliptic limit cycle, and this bound is sharp.

Dynamical Liouville

The aim of this paper is to analyse an SPDE which arises naturally in the context of Liouville quantum gravity. This SPDE shares some common features with the so-called Sine-Gordon equation and is built to preserve the Liouville measure which has been constructed recently on the two-dimensional sphere S2 and the torus T2 in the work by David-Kupiainen-Rhodes-Vargas [14,16]. The SPDE we shall focus on has the following (simplified) form:∂tX=14πΔX−eγX+ξ, where ξ is a space-time white noise on R+×S2 or R+×T2. The main aspect which distinguishes this singular stochastic SPDE with well-known SPDEs studied recently (KPZ, dynamical Φ34, dynamical Sine-Gordon, generalised KPZ, etc.) is the presence of intermittency. One way of picturing this effect is that a naive rescaling argument suggests the above SPDE is sub-critical for all γ>0, while we don't expect solutions to exist when γ>2. In this work, we initiate the study of this intermittent SPDE by analysing what one might call the “classical” or “Da Prato-Debussche” phase which corresponds here to γ∈[0,γdPD=22−6). By exploiting the positivity of the non-linearity eγX, we can push this classical threshold further and obtain this way a weaker notion of solution when γ∈[γdPD,γpos=22−2). Our proof requires an analysis of the Besov regularity of natural space/time Gaussian multiplicative chaos (GMC) measures. Regularity Structures of arbitrary high degree should potentially give strong solutions all the way to the same threshold γpos and should not push this threshold further unless the notion of regularity is suitably adapted to the present intermittent situation. Of independent interest, we prove along the way (using techniques from [32]) a stronger convergence result for approximate GMC measures μϵ→μ which holds in Besov spaces.

Fast computation of sine/cosine series coefficients of associated Legendre function of arbitrary high degree and order

Abstract In order to accelerate the spherical/spheroidal harmonic synthesis of any function, we developed a new recursive method to compute the sine/cosine series coefficient of the 4π fully- and Schmidt quasi-normalized associated Legendre functions. The key of the method is a set of increasing-degree/order mixed-wavenumber two to four-term recurrence formulas to compute the diagonal terms. They are used in preparing the seed values of the decreasing-order fixed-degree, and fixed-wavenumber two- and three-term recurrence formulas, which are obtained by modifying the classic relations. The new method is accurate and capable to deal with an arbitrary high degree/ order/wavenumber. Also, it runs significantly faster than the previous method of ours utilizing the Wigner d function, say around 20 times more when the maximum degree exceeds 1,000.

The Combinatorial k-Deck

We consider colouring reconstruction problems for a finite group G acting on a finite set X. For a given set F of colours the G-action on X induces a G-action on the set $$F^X=\{c:X\longrightarrow F\}$$ of colourings of X as well as on the set $$\bigcup _{K\subseteq X}F^K$$ of partial colourings of X. The combinatorial k-deck of a colouring $$c\in F^X$$ consists of the multiset of G-classes of the restrictions of c to all k-element subsets of X. The problem is to reconstruct the G-class of the full colouring c from the partial information given by the combinatorial k-deck. This new notion of deck generalizes the well-known k-deck for subsets in finite groups and is a natural alternative to another (more analytical) notion of deck for real- or integer-valued functions considered by various authors. We compare these kinds of decks and show that the combinatorial deck is to arbitrary high degree stronger than the traditional analytical one. However, we also show that even for the combinatorial deck there is no global colouring reconstruction number for all finite groups. Additionally, we demonstrate that a global subset reconstruction number (if it exists) has to be at least 6.

Transformation between surface spherical harmonic expansion of arbitrary high degree and order and double Fourier series on sphere

In order to accelerate the spherical harmonic synthesis and/or analysis of arbitrary function on the unit sphere, we developed a pair of procedures to transform between a truncated spherical harmonic expansion and the corresponding two-dimensional Fourier series. First, we obtained an analytic expression of the sine/cosine series coefficient of the $$4 \pi $$ fully normalized associated Legendre function in terms of the rectangle values of the Wigner d function. Then, we elaborated the existing method to transform the coefficients of the surface spherical harmonic expansion to those of the double Fourier series so as to be capable with arbitrary high degree and order. Next, we created a new method to transform inversely a given double Fourier series to the corresponding surface spherical harmonic expansion. The key of the new method is a couple of new recurrence formulas to compute the inverse transformation coefficients: a decreasing-order, fixed-degree, and fixed-wavenumber three-term formula for general terms, and an increasing-degree-and-order and fixed-wavenumber two-term formula for diagonal terms. Meanwhile, the two seed values are analytically prepared. Both of the forward and inverse transformation procedures are confirmed to be sufficiently accurate and applicable to an extremely high degree/order/wavenumber as $$2^{30}\,{\approx }\,10^9$$ . The developed procedures will be useful not only in the synthesis and analysis of the spherical harmonic expansion of arbitrary high degree and order, but also in the evaluation of the derivatives and integrals of the spherical harmonic expansion.

Rectangular rotation of spherical harmonic expansion of arbitrary high degree and order

In order to move the polar singularity of arbitrary spherical harmonic expansion to a point on the equator, we rotate the expansion around the y-axis by $$90^{\circ }$$ such that the x-axis becomes a new pole. The expansion coefficients are transformed by multiplying a special value of Wigner D-matrix and a normalization factor. The transformation matrix is unchanged whether the coefficients are $$4 \pi $$ fully normalized or Schmidt quasi-normalized. The matrix is recursively computed by the so-called X-number formulation (Fukushima in J Geodesy 86: 271–285, 2012a). As an example, we obtained $$2190\times 2190$$ coefficients of the rectangular rotated spherical harmonic expansion of EGM2008. A proper combination of the original and the rotated expansions will be useful in (i) integrating the polar orbits of artificial satellites precisely and (ii) synthesizing/analyzing the gravitational/geomagnetic potentials and their derivatives accurately in the high latitude regions including the arctic and antarctic area.

Center problem for systems with two monomial nonlinearities

We study the center problem for planar systems with a linear center at the origin that in complex coordinates have a nonlinearity formed by the sum of two monomials. Our first result lists several centers inside this family. To the best of our knowledge this list includes a new class of Darboux centers that are also persistent centers. The rest of the paper is dedicated to try to prove that the given list is exhaustive. We get several partial results that seem to indicate that this is the case. In particular, we solve the question for several general families with arbitrary high degree and for all cases of degree less or equal than 19. As a byproduct of our study we also obtain the highest known order for weak-foci of planar polynomial systems of some given degrees.

On the Complexity of Computing the Capacity of Codes That Avoid Forbidden Difference Patterns

We consider questions related to the computation of the capacity of codes that avoid forbidden difference patterns. The maximal number of $n$-bit sequences whose pairwise differences do not contain some given forbidden difference patterns increases exponentially with $n$. The exponent is the capacity of the forbidden patterns, which is given by the logarithm of the joint spectral radius of a set of matrices constructed from the forbidden difference patterns. We provide a new family of bounds that allows for the approximation, in exponential time, of the capacity with arbitrary high degree of accuracy. We also provide a polynomial time algorithm for the problem of determining if the capacity of a set is positive, but we prove that the same problem becomes NP-hard when the sets of forbidden patterns are defined over an extended set of symbols. Finally, we prove the existence of extremal norms for the sets of matrices arising in the capacity computation. This result makes it possible to apply a specific (even though non polynomial) approximation algorithm. We illustrate this fact by computing exactly the capacity of codes that were only known approximately.

There is no Bogomolov type restriction theorem for strong semistability in positive characteristic

We give an example of a strongly semistable vector bundle of rank two on the projective plane such that there exist smooth curves of arbitrary high degree with the property that the restriction of the bundle to the curve is not strongly semistable anymore. This shows that a Bogomolov type restriction theorem does not hold for strong semistability in positive characteristic.

On invariant Gibbs measures for the generalized KdV equations

We consider the defocusing generalized KdV equations on the circle. In particular, we construct global-in-time solutions with initial data distributed according to the Gibbs measure and show that the law of the random solutions, at any time, is again given by the Gibbs measure. In handling a nonlinearity of an arbitrary high degree, we make use of the Hermite polynomials and the white noise functional.