Invariant Algebraic Curves of Generalized Liénard Polynomial Differential Systems

In this study, we focus on invariant algebraic curves of generalized Liénard polynomial differential systems x′=y, y′=−fm(x)y−gn(x), where the degrees of the polynomials f and g are m and n, respectively, and we correct some results previously stated.

Tau Function and Moduli of Meromorphic Quadratic Differential

The Bergman tau functions are applied to the study of the Picard group of moduli spaces of quadratic differentials with at most n simple poles on genus g complex algebraic curves. This generalizes our previous results on moduli spaces of holomorphic quadratic differentials.

Algebraic surfaces determine analyticity of functions

Let $$f :X \rightarrow \mathbb {R}$$ f : X → R be a function defined on a nonsingular real algebraic set X of dimension at least 3. We prove that f is an analytic (resp. a Nash) function whenever the restriction $$f|_{S}$$ f | S is an analytic (resp. a Nash) function for every nonsingular algebraic surface $$S \subset X$$ S ⊂ X whose each connected component is homeomorphic to the unit 2-sphere. Furthermore, the surfaces S can be replaced by compact nonsingular algebraic curves in X, provided that dim$$X \ge 2$$ X ≥ 2 and f is of class $$\mathcal {C}^{\infty }$$ C ∞ .

Regulous Functions over Real Closed Fields

Let $X$ be a quasi-projective algebraic variety over a real closed field $R$, and let $f \colon U \to R$ be a function defined on an open subset $U$ of the set $X(R)$ of $R$-rational points of $X$. Assume that either the function $f$ is locally semialgebraic or the field $R$ is uncountable. If for every irreducible algebraic curve $C \subset X$ the restriction $f|_{U \cap C}$ is continuous and admits a rational representation, then $f$ is continuous and admits a rational representation. There are also suitable versions of this theorem with algebraic curves replaced by algebraic arcs. Heretofore, results of such a type have been known only for $R={\mathbb{R}}$. The transition from ${\mathbb{R}}$ to $R$ is not automatic at all and requires new methods.

Rational solutions of the defocusing non-local nonlinear Schrödinger equation: asymptotic analysis and soliton interactions

In this paper, we obtain the N th-order rational solutions for the defocusing non-local nonlinear Schrödinger equation by the Darboux transformation and some limit technique. Then, via an improved asymptotic analysis method relying on the balance between different algebraic terms, we derive the explicit expressions of all asymptotic solitons of the rational solutions with the order 1 ≤ N ≤ 4 . It turns out that the asymptotic solitons are localized in the straight lines or algebraic curves, and the exact solutions approach the curved asymptotic solitons with a slower rate than the straight ones. Moreover, we find that all the rational solutions exhibit just five different types of soliton interactions, and the interacting solitons are divided into two halves with each having the same amplitudes. Particularly for the curved asymptotic solitons, there may exist a slight difference for their velocities between at t and − t with certain parametric conditions. In addition, we reveal that the soliton interactions in the rational solutions with N ≥ 2 are stronger than those in the exponential and exponential-and-rational solutions.

GEOMETRIC MODELING OF ADAPTIVE ALGEBRAIC CURVES PASSING THROUGH PREDETERMINED POINTS

In this paper, the geometric theory of multidimensional interpolation was further developed in terms of modeling and using adaptive curves passing through predetermined points. A feature of the proposed approach to modeling curved lines is the ability to adapt to any initial data for high-quality interpolation, which excludes unplanned oscillations, due to the uneven distribution of parameter values, the source of which are the initial data. This is the improvement of the previously proposed method for constructing and analytically describing arcs of algebraic curves passing through predetermined points, obtained on the basis of Bezier curves, which are compiled taking into account the expansion coefficients of the Newton binomial. The paper gives an example of using adaptive algebraic curves passing through predetermined points for geometric modeling of the stress-strain state of membrane coatings cylindrical shells using two-dimensional interpolation. The given example an illustrative showed the advantages of the proposed adaptation of algebraic curves passing through predetermined points and obtained on the basis of Bezier curves for geometric modeling of multifactor processes and phenomena. The use of such adaptation allows not only to avoid unplanned oscillations, but also self-intersection of geometric objects when generalized to a multidimensional space. Adaptive algebraic curves can also be effectively used as formative elements for constructing geometric objects of multidimensional space, both as guide lines and as generatrix’s.

Automorphism Groups of Symmetric and Pseudo-real Riemann Surfaces

The category of smooth, irreducible, projective, complex algebraic curves is equivalent to the category of compact Riemann surfaces. We study automorphism groups of Riemann surfaces which are equivalent to complex algebraic curves with real moduli. A complex algebraic curve C has real moduli when the corresponding surface $$X_C$$ X C admits an anti-conformal automorphism. If no such an automorphism is an involution (symmetry), then the surface $$X_C$$ X C is called pseudo-real and the curve C is isomorphic to its conjugate, but is not definable over reals. Otherwise, the surface $$X_C$$ X C is called symmetric and the curve C is real.