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We construct qualitatively new examples of superabundant tropical curves which are non‐realisable in genuses 3 and 4. These curves are in R3${\mathbb {R}}^3$ and R4${\mathbb {R}}^4$, respectively, and have properties resembling canonical embeddings of genus 3 and 4 algebraic curves. In particular, the genus 3 example is a degree 4 planar tropical curve, and the genus 4 example is contained in the product of a tropical line and a tropical conic. They have excess dimension of deformation space equal to 1. Non‐realisability follows by combining this with a dimension calculation for the corresponding space of logarithmic curves. [ABSTRACT FROM AUTHOR]

We prove that any geometrically connected curve X$X$ over a field k$k$ is an algebraic K(π,1)$K(\pi,1)$, as soon as its geometric irreducible components have nonzero genus. This means that the cohomology of any locally constant constructible étale sheaf of Z/nZ$\mathbb {Z}/n\mathbb {Z}$‐modules, with n$n$ invertible in k$k$, is canonically isomorphic to the cohomology of its corresponding π1(X)$\pi _1(X)$‐module. To this end, we explicitly construct some Galois coverings of X$X$ corresponding to Galois coverings of the normalisation of its irreducible components. When k$k$ is finite or algebraically closed, we precisely describe finite quotients of π1(X)$\pi _1(X)$ that allow to compute the cohomology groups of the sheaf, and give explicit descriptions of the cup products H1×H1→H2$\operatorname{H}^1\times \operatorname{H}^1\rightarrow \operatorname{H}^2$ and H1×H2→H3$\operatorname{H}^1\times \operatorname{H}^2\rightarrow \operatorname{H}^3$ in étale cohomology in terms of finite group cohomology. [ABSTRACT FROM AUTHOR]

Consider a pair consisting of an abstract tropical curve and an effective divisor from the linear system associated to k$k$ times the canonical divisor for k∈Z⩾1$k \in \mathbb {Z}_{\geqslant 1}$. In this article, we give a purely combinatorial criterion to determine if such a pair arises as the tropicalization of a pair consisting of a smooth algebraic curve over a non‐Archimedean field with algebraically closed residue field of characteristic 0 together with an effective pluri‐canonical divisor. To do so, we introduce tropical normalized covers as special instances of cyclic tropical Hurwitz covers and reduce the realizability problem for pluri‐canonical divisors to the realizability problem for normalized covers. Our main result generalizes the work of Möller–Ulirsch–Werner on realizability of tropical canonical divisors and incorporates the recent progress on compactifications of strata of k$k$‐differentials in the work of Bainbridge–Chen–Gendron–Grushevsky–Möller. [ABSTRACT FROM AUTHOR]

Rational algebraic curves have been intensively studied in the last decades, both from the theoretical and applied point of view. In applications (e.g. level curves, linear homotopy deformation, geometric constructions in computer aided design, image detection, algebraic differential equations, etc.), there often appear unknown parameters. It is possible to adjoin these parameters to the coefficient field as transcendental elements. In some particular cases, however, the curve has a different behavior than in the generic situation treated in this way. In this paper, we show when the singularities and thus the (geometric) genus of the curves might change. More precisely, we give a partition of the affine space, where the parameters take values, so that in each subset of the partition the specialized curve is either reducible or its genus is invariant. In particular, we give a Zariski-closed set in the space of parameter values where the genus of the curve under specialization might decrease or the specialized curve gets reducible. For the genus zero case, and for a given rational parametrization, a finer partition is possible such that the specialization of the parametrization parametrizes the specialized curve. Moreover, in this case, the set of parameters where Hilbert's irreducibility theorem does not hold can be identified. We conclude the paper by illustrating these results by some concrete applications. [ABSTRACT FROM AUTHOR]

We consider reduced generalized jet schemes L Z (Y) along Z of a infinitesimal deformation Y of a complete intersection variety X over the same fat point Z. By definition, these reduced generalized jet schemes L Z (Y) live over the so-called reduced auto-arc spaces A Z , which were originally introduced by H. Schoutens and later studied by the author. Some well-known results of M. Mustaţǎ on classical jet schemes of locally complete intersections carry over to this case. For example, as a consequence of miracle flatness and some generalizations of the aforementioned results of M. Mustaţǎ, it is shown that the reduced generalized jet space, denoted L n (X n) , of a flat deformation X n → D n along ▪ is usually a global flat deformation over A k n of the classical jet scheme L n (X) provided the base scheme X is a locally complete intersection and the fibers are all of minimal dimension. Similar interesting relativized versions of those aforementioned results are also explored. Also, the initiation of the so-called regulated defect of a formal deformation in analogy to log canonical threshold is introduced. In the end, the situation is studied when the base scheme X is an algebraic curve, and the notions of so-called strong/weak deformations is introduced in this context. Finally, a so-called deformed motivic volume is defined. [ABSTRACT FROM AUTHOR]

In this paper, we introduce new plane curves, i.e. Cassinian focal-directorial curves as curves of a transitory type, of Cassinian multidirectorial curves as counterpart to Cassinian multifocal curves; and demonstrate their geometric genesis, and their form variation for special choices of foci and directrices. The form variation of Cassinian focal-directorial curves (CFDC) for different values of the particular parameter is also considered. Furthermore, it is shown that these algebraic curves also might have applications in architectural space design. [ABSTRACT FROM AUTHOR]

We revisit the seminal Brill–Noether algorithm for plane curves with ordinary singularities. Our new approach takes advantage of fast algorithms for polynomials and structured matrices. We design a new probabilistic algorithm of Las Vegas type that computes a Riemann–Roch space in expected sub-quadratic time. [ABSTRACT FROM AUTHOR]

In this paper, we study the algebraic limit cycle of degree 2 for the system x ˙ = - y + a 30 x 3 + a 21 x 2 y + a 12 x y 2 + a 03 y 3 , y ˙ = x + b 30 x 3 + b 21 x 2 y + b 12 x y 2 + b 03 y 3 , which is called a linear type center plus cubic homogeneous polynomial differential system. We derive the normal form of this system with an algebraic limit cycle of degree 2. It is shown that, if an invariant algebraic curve of degree 2 is a limit cycle of the above mentioned system, then it is a unique limit cycle in the phase plane. [ABSTRACT FROM AUTHOR]

Let C be a smooth projective curve of genus g defined over an algebraically closed field of characteristic p ≠ 2 and let Δ be the diagonal of C × C. We observe the complement X : = (C × C) ∖ Δ. If g = 0 , X is an affine hypersurface x y = z 2 − 1 in A 3 which is the simplest example of Danielewski surfaces. If g = 1 then Δ is a fiber of an elliptic fibration over C so that (Δ 2) = 0 and κ ‾ (X) = 1 , and if g > 1 , (Δ 2) = 2 − 2 g and Δ is contractible. In the case C has genus g > 1 , X is embedded bijectively into the Jacobian variety if C is non-hyperelliptic, though X is generically a double covering of a surface in the Jacobian variety if C is hyperelliptic. Some observations will be made in the case k has characteristic 2. [ABSTRACT FROM AUTHOR]

We study the topology of the moduli space of unramified Z / p -covers of tropical curves of genus g ≥ 2 , where p is a prime number. We use recent techniques by Chan–Galatius–Payne to identify contractible subcomplexes of the moduli space. We then use this contractibility result to show that this moduli space is simply connected. In the case of genus 2, we determine the homotopy type of this moduli space for all primes p. This work is motivated by prospective applications to the top-weight cohomology of the space of prime cyclic étale covers of smooth algebraic curves. [ABSTRACT FROM AUTHOR]

In this paper, we represent vector bundles over a regular algebraic curve as pairs of lattices over the maximal orders of its function field and we give polynomial time algorithms for several tasks: computing determinants of vector bundles, kernels and images of global homomorphisms, isomorphisms between vector bundles, cohomology groups, extensions, and splitting into a direct sum of indecomposables. Most algorithms are deterministic except for computing isomorphisms when the base field is infinite. Some algorithms are only polynomial time if we may compute Hermite forms of pseudo-matrices in polynomial time. All algorithms rely exclusively on algebraic operations in function fields. For applications, we give an algorithm enumerating isomorphism classes of vector bundles on an elliptic curve, and to construct algebraic geometry codes over vector bundles. We implement all our algorithms into a SageMath package. [ABSTRACT FROM AUTHOR]

We prove in this paper that a hybrid van der Pol-Rayleigh oscillator x ˙ = y , y ˙ = − x − α y − β y 3 − x 2 y with (α , β) ∈ R 2 has no irreducible invariant algebraic curves in C [ x , y ] except for (α , β) = (0 , − 1) , or α 2 = − β − 1 / β + 2 , β ≠ 0 or α ≠ 0 , β = 1 , and that the oscillator has only one irreducible invariant algebraic curve in R [ x , y ] when (α , β) = (0 , − 1) , or α 2 = − β − 1 / β + 2 , β < 0 , or α ∈ R , β = 1. These together with that in Chen et al. (2021) [7] ensure that the hybrid van der Pol-Rayleigh oscillator has an algebraic limit cycle if and only if β = 1 and α < 0. This result solves the conjecture posed in that paper. [ABSTRACT FROM AUTHOR]

Elliptic curves are considered highly secure due to their complex mathematical structure and are widely used in cryptographic key exchange algorithms. This study aimed to construct a confusion component for a symmetric block cipher using a hybrid approach that involves elliptic curve points and an algebraic group structure. Generally, elliptic curve points do not generate much randomness, so they are not recommended for constructing a highly nonlinear S-box. In this study, we first randomized the elliptic curve points to the maximum possible level and generate initial S-boxes. These initial S-boxes were then evolved through algebraic group action to enhance their nonlinearity. The strength of the proposed substitution box was assessed using various tests, including nonlinearity, bit independence criterion, strict avalanche criterion, and differential and linear probabilities. Using this approach, we designed an S-box with a nonlinearity value of 114, which is higher than the standard value presented by AES. We compared the results of other cryptographic tests with well-known S-boxes to validate the effectiveness of the suggested confusion component. [ABSTRACT FROM AUTHOR]

Classical invariant theory is mainly concerned with form invariance. This involves finding some function of the coefficients of a homogeneous polynomial which, under a linear transformation of the variables in the polynomial, has its value changed by a factor which only depends upon the transformation. In this article a wider class of invariants is considered. The polynomials now involves only a single variable. If a function of the coefficients can be found which is unchanged by a simple change of origin then it is said to be polynomial invariant. It is shown that there is an intimate connection between such an invariant and pencils of lines associated with an algebraic curve and this is extended to cover a collection of polynomials. [ABSTRACT FROM AUTHOR]

We study constructing an algebraic curve from a Riemann surface given via a translation surface, which is a collection of finitely many polygons in the plane with sides identified by translation. We use the theory of discrete Riemann surfaces to give an algorithm for approximating the Jacobian variety of a translation surface whose polygon can be decomposed into squares. We first implement the algorithm in the case of L shaped polygons where the algebraic curve is already known. The algorithm is also implemented in any genus for specific examples of Jenkins–Strebel representatives, a dense family of translation surfaces that, until now, lived squarely on the analytic side of the transcendental divide between Riemann surfaces and algebraic curves. Using Riemann theta functions, we give numerical experiments and resulting conjectures up to genus 5. [ABSTRACT FROM AUTHOR]

Several applications of function fields over finite fields, or equivalently, algebraic curves over finite fields, require computing bases for Riemann–Roch spaces. In this paper, we determine explicit bases for Riemann–Roch spaces of linearized function fields, and we give a lower bound for the minimum distance of generalized algebraic geometry codes. As a consequence, we construct generalized algebraic geometry codes with good parameters. [ABSTRACT FROM AUTHOR]

In this paper, as a first step, we show that the elementary addition law on the circle can be extended to any non singular algebraic curve of degree three. In a second step, we introduce the Edwards curves recently discovered by Harold M. Edwards. These curves are elliptic and they are deformations of the circle. Finally, we obtain the addition laws on the circle and on the elliptic curves are the samething. For simplicity, we write the abbreviation [ABSTRACT FROM AUTHOR]

Let G be a reductive group, and let X be an algebraic curve over an algebraically closed field k with positive characteristic. We prove a version of nonabelian Hodge correspondence for tame G-local systems over X and logarithmic G-Higgs bundles over the Frobenius twist X ′ . To obtain a full description of the correspondence for the noncompact case, we introduce the language of parahoric group schemes to establish the correspondence. [ABSTRACT FROM AUTHOR]

For any almost-simple group G over an algebraically closed field k of characteristic zero, we describe the automorphism group of the moduli space of semistable G -bundles over a connected smooth projective curve C of genus at least 4. The result is achieved by studying the singular fibers of the Hitchin fibration. As a byproduct, we provide a description of the irreducible components of two natural closed subsets in the Hitchin basis: the divisor of singular cameral curves and the divisor of singular Hitchin fibers. [ABSTRACT FROM AUTHOR]

The resonance varieties are cohomological invariants that are studied in a variety of topological, combinatorial, and geometric contexts. We discuss their scheme structure in a general algebraic setting and introduce various properties that ensure the reducedness of the associated projective resonance scheme. We prove an asymptotic formula for the Hilbert series of the associated Koszul module, then discuss applications to vector bundles on algebraic curves and to Chen ranks formulas for finitely generated groups, with special emphasis on Kähler and right-angled Artin groups. [ABSTRACT FROM AUTHOR]

In this paper, we show that ∞-adic multiple zeta values associated to the function field of an algebraic curve of higher genus over a finite field are not zero, under certain assumption on the gap sequence associated to the rational point ∞ on the given curve. Using arguments and results of Sheats and Thakur for the case of the projective line, we calculate the absolute values of power sums in the series defining multiple zeta values, and show that the calculation implies the non-vanishing result. [ABSTRACT FROM AUTHOR]

We study the multiviews of algebraic space curves X from n pin-hole cameras of a real or complex projective space. We assume the pin-hole centers to be known, i.e., we do not reconstruct them. Our tools are algebro-geometric. We give some general theorems, e.g., we prove that a projective curve (over complex or real numbers) may be reconstructed using four general cameras. Several examples show that no number of badly placed cameras can make a reconstruction possible. The tools are powerful, but we warn the reader (with examples) that over real numbers, just using them correctly, but in a bad way, may give ghosts: real curves which are images of the emptyset. We prove that ghosts do not occur if the cameras are general. Most of this paper is devoted to three important cases of space curves: unions of a prescribed number of lines (using the Grassmannian of all lines in a 3-dimensional projective space), plane curves, and curves of low degree. In these cases, we also see when two cameras may reconstruct the curve, but different curves need different pairs of cameras. [ABSTRACT FROM AUTHOR]

We prove a rigidity criterion for period maps of admissible variations of graded-polarizable mixed Hodge structure, and establish rigidity in a number of cases, including families of quasi-projective curves, projective curves with ordinary double points, the complement of the canonical curve in families of Kynev-Todorov surfaces, period maps attached to the fundamental groups of smooth varieties and normal functions. [ABSTRACT FROM AUTHOR]

We consider for structure groups SU (n) ⊂ SL (n , ℂ) a densely defined toric structure on the moduli space of framed parabolic sheaves on a three-punctured sphere, which degenerates to an actual toric structure. In combination with previous degeneration results, these extend to similar moduli for arbitrary Riemann surfaces. [ABSTRACT FROM AUTHOR]

In this paper, we study the bivariate truncated moment problem (TMP) on curves of the form $ y=q(x) $ y = q (x) , $ q(x)\in \mathbb R[x] $ q (x) ∈ R [ x ] , $ \deg q\geq ~3 $ deg ⁡ q ≥ 3 and $ yx^\ell =1 $ y x ℓ = 1 , $ \ell \in \mathbb N\setminus \{1\} $ ℓ ∈ N ∖ { 1 }. For even degree sequences, the solution based on the size of moment matrix extensions was first given by Fialkow [Fialkow L. Solution of the truncated moment problem with variety $ y=x^3 $ y = x 3 . Trans Amer Math Soc. 2011;363:3133–3165.] using the truncated Riesz–Haviland theorem [Curto R, Fialkow L. An analogue of the Riesz–Haviland theorem for the truncated moment problem. J Funct Anal. 2008;255:2709–2731.] and a sum-of-squares representations for polynomials, strictly positive on such curves [Fialkow L. Solution of the truncated moment problem with variety $ y=x^3 $ y = x 3 . Trans Amer Math Soc. 2011;363:3133–3165.; Stochel J. Solving the truncated moment problem solves the moment problem. Glasgow J Math. 2001;43:335–341.]. Namely, the upper bound on this size is quadratic in the degrees of the sequence and the polynomial determining a curve. We use a reduction to the univariate setting technique, introduced in [Zalar A. The truncated Hamburger moment problem with gaps in the index set. Integral Equ Oper Theory. 2021;93:36.doi: .; Zalar A. The truncated moment problem on the union of parallel lines. Linear Algebra Appl. 2022;649:186–239. .; Zalar A. The strong truncated Hamburger moment problem with and without gaps. J Math Anal Appl. 2022;516:126563. doi: .], and improve Fialkow's bound to $ \deg q-1 $ deg ⁡ q − 1 (resp. $ \ell +1 $ ℓ + 1) for curves $ y=q(x) $ y = q (x) (resp. $ yx^\ell =1 $ y x ℓ = 1). This in turn gives analogous improvements of the degrees in the sum-of-squares representations referred to above. Moreover, we get the upper bounds on the number of atoms in the minimal representing measure, which are $ k\deg q $ kdeg ⁡ q (resp. $ k(\ell +1) $ k (ℓ + 1)) for curves $ y=q(x) $ y = q (x) (resp. $ yx^\ell =1 $ y x ℓ = 1) for even degree sequences, while for odd ones they are $ k\deg q-\big \lceil \frac {\deg q}{2} \big \rceil $ kdeg ⁡ q − ⌈ deg ⁡ q 2 ⌉ (resp. $ k(\ell +1)-\big \lfloor \frac {\ell }{2} \big \rfloor +1 $ k (ℓ + 1) − ⌊ ℓ 2 ⌋ + 1) for curves $ y=q(x) $ y = q (x) (resp. $ yx^\ell =1 $ y x ℓ = 1). In the even case, these are counterparts to the result by Riener and Schweighofer [Riener C, Schweighofer M. Optimization approaches to quadrature:a new characterization of Gaussian quadrature on the line and quadrature with few nodes on plane algebraic curves, on the plane and in higher dimensions. J Complex. 2018;45:22–54., Corollary 7.8], which gives the same bound for odd degree sequences on all plane curves. In the odd case, their bound is slightly improved on the curves we study. Further on, we give another solution to the TMP on the curves studied based on the feasibility of a linear matrix inequality, corresponding to the univariate sequence obtained, and finally we solve concretely odd degree cases to the TMP on curves $ y=x^\ell $ y = x ℓ , $ \ell =2,3 $ ℓ = 2 , 3 , and add a new solvability condition to the even degree case on the curve $ y=x^2 $ y = x 2 . [ABSTRACT FROM AUTHOR]

We give a complete classification of Dembowski-Ostrom polynomials from the composition of Dickson polynomials of arbitrary kind and monomials over finite fields. Moreover, by using a variant of the Weil bound for the number of points of affine algebraic curves over finite fields, we discuss the planarity of the obtained Dembowski-Ostrom polynomials. [ABSTRACT FROM AUTHOR]

Let X \subset \mathbb {C}^{2n} be an n-dimensional algebraic variety. We define the algebraic version of the generic symmetry defect set (Wigner caustic) of X. Moreover, we compute its singularities as well as degree, genus and Euler characteristic for X_d being a generic (smooth and transversal to the line at infinity) curve of degree d in \mathbb {C}^2. [ABSTRACT FROM AUTHOR]

In [V. I. Arnold, Simple singularities of curves, Proc. Steklov Inst. Math. 226(3) (1999) 20–28, Sec. 5, p. 32], Arnold writes: 'Classification of singularities of curves can be interpreted in dual terms as a description of "co-artin" subalgebras of finite co-dimension in the algebra of formal series in a single variable (up to isomorphism of the algebra of formal series)'. In the paper, such a description is obtained but up to isomorphism of algebraic curves (i.e. this description is finer). Let K be an algebraically closed field of arbitrary characteristic. The aim of the paper is to give a classification (up to isomorphism) of the set of subalgebras of the polynomial algebra K [ x ] that contains the ideal x m K [ x ] for some m ≥ 1. It is proven that the set = ∐ m , Γ (m , Γ) is a disjoint union of affine algebraic varieties (where Γ ∐ { 0 , m , m + 1 , ... } is the semigroup of the singularity and m − 1 is the Frobenius number). It is proven that each set (m , Γ) is an affine algebraic variety and explicit generators and defining relations are given for the algebra of regular functions on (m , Γ). An isomorphism criterion is given for the algebras in . For each algebra A ∈ (m , Γ) , explicit sets of generators and defining relations are given and the automorphism group Aut K (A) is explicitly described. The automorphism group of the algebra A is finite if and only if the algebra A is not isomorphic to a monomial algebra, and in this case | Aut K (A) | < dim K (A / A) where A is the conductor of A. The set of orders of the automorphism groups of the algebras in (m , Γ) is explicitly described. [ABSTRACT FROM AUTHOR]

For a , b , c , d , e ∈ Q × , denote by C a , b , c , d , e the nonsingular projective curve over Q given by the affine equation C a , b , c , d , e : a x 2 + b y 2 = c + d x y + e x 2 y 2. In this paper, we find a relation between the number of points on C a , b , c , d , e over F q and the finite field Appell series F 4 ∗ . As a consequence, in view of certain transformations for finite field Appell series F 4 ∗ , we prove the existence of isogenies between C a , b , c , d , ab c and the twisted Edward family of elliptic curves E α , 16 a b α d 2 . [ABSTRACT FROM AUTHOR]

A number field K is primitive if K and Q are the only subextensions of K. Let C be a curve defined over Q . We call an algebraic point P ∈ C (Q ¯) primitive if the number field Q (P) is primitive. We present several sets of sufficient conditions for a curve C to have finitely many primitive points of a given degree d. For example, let C / Q be a hyperelliptic curve of genus g, and let 3 ≤ d ≤ g - 1 . Suppose that the Jacobian J of C is simple. We show that C has only finitely many primitive degree d points, and in particular it has only finitely many degree d points with Galois group S d or A d . However, for any even d ≥ 4 , a hyperelliptic curve C / Q has infinitely many imprimitive degree d points whose Galois group is a subgroup of S 2 ≀ S d / 2 . [ABSTRACT FROM AUTHOR]

Bounding the number of rational points of height at most |$H$| on irreducible algebraic plane curves of degree |$d$| has been an intense topic of investigation since the work by Bombieri and Pila. In this paper we establish optimal dependence on |$d$| by showing the upper bound |$C d^{2} H^{2/d} (\log H)^{\kappa }$| with some absolute constants |$C$| and |$\kappa $|⁠. This bound is optimal with respect to both |$d$| and |$H$|⁠ , except for the constants |$C$| and |$\kappa $|⁠. This answers a question raised by Salberger, leading to a simplified proof of his results on the uniform dimension growth conjectures of Heath-Brown and Serre, and where at the same time we replace the |$H^{\varepsilon }$| factor by a power of |$\log H$|⁠. The main strength of our approach comes from the combination of a new, efficient form of smooth parametrizations of algebraic curves with a century-old criterion of Pólya, which allows us to save one extra power of |$d$| compared with the standard approach using Bézout's theorem. [ABSTRACT FROM AUTHOR]

Over the moduli space of pointed smooth algebraic curves, the projectivized k th Hodge bundle is the space of k -canonical divisors. The incidence loci are defined by requiring the k -canonical divisors to have prescribed multiplicities at the marked points. We compute the classes of the closure of the incidence loci in the projectivized k th Hodge bundle over the moduli space of curves with rational tails. The classes are expressed as a linear combination of tautological classes indexed by decorated stable graphs with coefficients enumerating appropriate weightings. As a consequence, we obtain an explicit expression for some relations in tautological rings of moduli of curves with rational tails. [ABSTRACT FROM AUTHOR]

The control problem of the rotary double inverted pendulum (double Furuta pendulum) is nontrivial because of underactuation and strong nonlinearities in the associated state‐space model. The system has three degrees of freedom (one actuated and two unactuated joints) while receiving only one control input. In this article, a novel nonlinear optimal (H‐infinity) control approach is developed for the dynamic model of the rotary double inverted pendulum. First, the dynamic model of the double pendulum undergoes approximate linearization with the use of first‐order Taylor series expansion and through the computation of the associated Jacobian matrices. The linearization process takes place at each sampling instance around a temporary operating point which is defined by the present value of the system's state vector and by the last sampled value of the control inputs vector. At a next stage a stabilizing H‐infinity feedback controller is designed. To compute the controller's feedback gains an algebraic Riccati equation has to be solved at each time‐step of the control algorithm. The global stability properties of the control scheme are proven through Lyapunov analysis. To implement state estimation‐based control without the need to measure the entire state vector of the rotary double‐pendulum the H‐infinity Kalman filter is used as a robust state observer. The nonlinear optimal control method achieves fast and accurate tracking of setpoints by all state variables of the rotary double inverted pendulum under moderate variations of the control input. [ABSTRACT FROM AUTHOR]

We study moduli stacks of principal C ∗ -bundles over nodal complex algebraic curves and determine their rational cohomology algebras in terms of Chern classes. [ABSTRACT FROM AUTHOR]

We show that on a metric graph of genus g, a divisor of degree n generically has g (n - g + 1) Weierstrass points. For a sequence of generic divisors on a metric graph whose degrees grow to infinity, we show that the associated Weierstrass points become distributed according to the Zhang canonical measure. In other words, the limiting distribution is determined by effective resistances on the metric graph. This distribution result has an analogue for complex algebraic curves, due to Neeman, and for curves over non-Archimedean fields, due to Amini. [ABSTRACT FROM AUTHOR]

Spectral curves are algebraic curves associated to commutative subalgebras of rings of ordinary differential operators (ODOs). Their origin is linked to the Korteweg–de Vries equation and to seminal works on commuting ODOs by I. Schur and Burchnall and Chaundy. They allow the solvability of the spectral problem L y = λ y , for an algebraic parameter λ and an algebro-geometric ODO L, whose centralizer is known to be the affine ring of an abstract spectral curve Γ. In this work, we use differential resultants to effectively compute the defining ideal of the spectral curve Γ , defined by the centralizer of a third-order differential operator L, with coefficients in an arbitrary differential field of zero characteristic. For this purpose, defining ideals of planar spectral curves associated to commuting pairs are described as radicals of differential elimination ideals. In general, Γ is a non-planar space curve and we provide the first explicit example. As a consequence, the computation of a first-order right factor of L − λ becomes explicit over a new coefficient field containing Γ. Our results establish a new framework appropriate to develop a Picard–Vessiot theory for spectral problems. [ABSTRACT FROM AUTHOR]

The Riccati polynomial differential systems are differential systems of the form x ′ = c 0 (x) , y ′ = b 0 (x) + b 1 (x) y + b 2 (x) y 2 , where c 0 and b i for i = 0 , 1 , 2 are polynomial functions. We characterize all the Riccati polynomial differential systems having an invariant algebraic curve. We show that the coefficients of the first four highest degree terms of the polynomial in the variable y defining the invariant algebraic curve determine completely the Riccati differential system. A similar result is obtained for any Abel polynomial differential system. [ABSTRACT FROM AUTHOR]

Let X be a compact algebraic curve of genus g ≥ 2. The nontrivial outer automorphism σ of the complex Lie group E6 acts on the moduli space M(E6) of principal E6-bundles over X, and this action defines an automorphism fσ of M(E6). The group H¹ (X, Z(E6)) of principal Z(E6)-bundles over X also acts on M(E6) by tensor product, Z(E6) being the center of E6, so each choice of an element L ∈ H¹ (X, Z(E6)) defines an automorphism fL of M(E6). In this paper two theorems describing the simple fixed points of the automorphism fL of M(E6) and the composition fL o fσ are proved. [ABSTRACT FROM AUTHOR]

We study the existence of some irreducible projective plane curves of degree 8 with some prescribed topological type of singularities in the algebraic and symplectic worlds. [ABSTRACT FROM AUTHOR]

We show that for each algebraic space that is separated and of finite type over a field, and whose affinization is connected and reduced, there is a universal morphism to a para-abelian variety. The latter are schemes that acquire the structure of an abelian variety after some ground field extension. This generalizes classical results of Serre on universal morphisms from algebraic varieties to abelian varieties. Our proof relies on corresponding facts for the proper case, together with the structural properties of group schemes, removal of singularities by alterations, and ind-objects. It turns out that the formation of the Albanese variety commutes with base-change up to universal homeomorphisms. We also give a detailed analysis of Albanese maps for algebraic curves and algebraic groups, with special emphasis on imperfect ground fields. [ABSTRACT FROM AUTHOR]

We consider the following Kirchhoff problem - (a+b ∫R³ |▽u|²) Δu + (1+εV(x))u = |u|p-2u, where a,b > 0, and 2 < p < 6. Under suitable assumptions on V, by using the Lyapunov-Schmidt reduction method, we obtain the existence of multi-bump solutions. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to investigate the upper bound and H¨older continuity for a general class of parametric elliptical variational-hemivariational inequalities via regularized gap functions. More precisely, we deliver a formulation of the elliptical variational-hemivariational inequalities in the case of the perturbed parameters governed by both the set of constraints and the mappings (for brevity, PEVHI (CM)). Based on the arguments of monotonicity and properties of the Clarke's generalized directional derivative, we establish an upper bound result for the PEVHI (CM) and provide the H¨older continuity of the solution mapping for the PEVHI (CM) under suitable assumptions on the data. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the notion of surjective radical parametrization and we prove sufficient conditions for a radical curve parametrization to be surjective. [ABSTRACT FROM AUTHOR]

According to the polynomial recursion formalism, the modified Jaulent–Miodek hierarchy is derived in a standard way. The first two nontrivial members in the modified Jaulent–Miodek hierarchy are listed correspondingly. Based on the squared eigenfunctions, an algebraic curve κ n and a Riemann surface S with arithmetic genus n are introduced, then the Dubrovin-type equations are obtained naturally. With the help of the conservation laws, the Baker–Akhiezer functions are defined. Finally, the asymptotic properties of the Baker–Akhiezer functions are analyzed, from which the algebro-geometric solutions of the modified Jaulent–Miodek hierarchy are constructed in term of the Riemann theta function. [ABSTRACT FROM AUTHOR]

(N , γ) -hyperelliptic semigroups were introduced by Fernando Torres to encapsulate the most salient properties of Weierstrass semigroups associated with totally ramified points of N-fold covers of curves of genus γ . Torres characterized (2 , γ) -hyperelliptic semigroups of maximal weight whenever their genus is large relative to γ . Here we do the same for (3 , γ) -hyperelliptic semigroups, and we formulate a conjecture about the general case whenever N ≥ 3 is prime. [ABSTRACT FROM AUTHOR]

In this paper we systematically consider various ways of generating integrable and separable Hamiltonian systems in canonical and in non-canonical representations from algebraic curves on the plane. In particular, we consider Stäckel transformbetween two pairs of Stäckel systems, generated by 2n-parameter algebraic curves on the plane, as well as Miura maps between Stäckel systems generated by (n+N)-parameter algebraic curves, leading to multi-Hamiltonian representation of these systems. [ABSTRACT FROM AUTHOR]

Propagation is a standard way of producing certain new conformal blocks from old ones that corresponds to the geometric procedure of adding new distinct points to a pointed compact Riemann surface. On the other hand, sewing conformal blocks corresponds to sewing compact Riemann surfaces. In this article, we clarify the relationships between these two procedures. Most importantly, we show that, "sewing and propagation are commuting procedures." More precisely: letϕbe a conformal block associated to a vertex operator algebra 𝕍 and a compact Riemann surface to be sewn, and let ≀𝑛ϕ be its 𝑛-times propagation. If the sewing ˜ 𝒮ϕ converges, then ˜ 𝒮 ≀𝑛 ϕ (the sewing of ≀𝑛ϕ) automatically converges, and it equals ≀𝑛 ˜ 𝒮ϕ (the 𝑛-times propagation of the sewing ˜ 𝒮ϕ). The proof of this result relies on establishing the propagation of conformal blocks associated to holomorphic families of compact Riemann surfaces. We prove this in our paper using the idea that, "propagation is itself a sewing followed by an analytic continuation." This result generalizes previous ones on single Riemann surfaces [Zhu94, FB04], and supplements those on algebraic families of complex algebraic curves [Cod19, DGT19a]. The results in this paper will be used in [Gui21] as the main technical tools to relate the (genus-0) permutation-twisted V⊗k-conformal blocks (i.e. intertwining operators) and the untwisted V-conformal blocks (of possibly higher genera). [ABSTRACT FROM AUTHOR]