A Galois-theoretic approach to Kanev’s correspondence

Let G be a finite group, an absolutely irreducible -module and w a weight of . To any Galois covering with group G we associate two correspondences, the Schur and the Kanev correspondence. We work out their relation and compute their invariants. Using this, we give some new examples of Prym–Tyurin varieties. This work was supported by FONDECYT No. 11060468 and No. 1060742.     

A relaxation approach to hencky’s plasticity

Givenμ, κ, c>0, we consider the functional

A new approach to Sheppard’s corrections

A very simple closed-form formula for Sheppard’s corrections is recovered by means of the classical umbral calculus. Using this symbolic method, a more general closed-form formula for discrete parent distributions is provided and the generalization to the multivariate case turns out to be straightforward. All these new formulas are particularly suited to be implemented in any symbolic package.     

A geometric approach to Euler’s addition formula

Plane quartic curves given by equations of the form y ²=P(x) with polynomials P of degree 4 represent singular models of elliptic curves which are directly related to elliptic integrals in the form studied by Euler and for which he developed his famous addition formulas. For cubic curves, the well-known secant and tangent construction establishes an immediate connection of addition formulas for the corresponding elliptic integrals with the structure of an algebraic group. The situation for quartic curves is considerably more complicated due to the presence of the singularity. We present a geometric construction, similar in spirit to the secant method for cubic curves, which defines an addition law on a quartic elliptic curve given by rational functions. Furthermore, we show how this addition on the curve itself corresponds to the addition in the (generalized) Jacobian variety of the curve, and we show how any addition formula for elliptic integrals of the form \(\int (1/\sqrt{P(x)})\,\mathrm{d}x\) with a quartic polynomial P can be derived directly from this addition law.     

Recurrent approach to Blaschke’s problem

We have obtained a recurrence formula $I_{n+3} = \frac{4(n+3)}{\pi(n+4)}VI_{n+1}We have obtained a recurrence formula I_n+3 = \frac4(n+3)p(n+4)VI_n+1I_{n+3} = \frac{4(n+3)}{\pi(n+4)}VI_{n+1} for integro-geometric moments in the case of a circle with the area V, where I_n = ò\nolimits_{K ?G}sⁿd GI_n = \int \nolimits_{K \cap G}\sigma^{n}{\rm d} G, while in the case of a convex domain K with the perimeter S and area V the recurrence formula

In+3 = \frac8(n+3)V2(n+1)(n+4)p[\fracd In+1d V - \fracIn+1S \fracd Sd V ] I_{n+3} = \frac{8(n+3)V^2}{(n+1)(n+4)\pi}\Big[\frac{{\rm d} I_{n+1}}{{\rm d} V} - \frac{I_{n+1}}{S} \frac{{\rm d} S}{{\rm d} V} \Big]      6. Nekhoroshev’s approach to Hamiltonian monodromy      Dmitrií A. Sadovskí 《Regular and Chaotic Dynamics》2016,21(6):720-758 Using the hyperbolic circular billiard, introduced in [31] by Delos et al. as possibly the simplest system with Hamiltonian monodromy, we illustrate the method developed by N. N. Nekhoroshev and coauthors [48] to uncover this phenomenon. Nekhoroshev’s very original geometric approach reflects his profound insight into Hamiltonian monodromy as a general topological property of fibrations. We take advantage of the possibility of having closed form elementary function expressions for all quantities in our system in order to provide the most explicit and detailed explanation of Hamiltonian monodromy and its relation to similar phenomena in other domains.      7. Rota’s umbral calculus and recursions      Heinrich?NiederhausenEmail author 《Algebra Universalis》2003,49(4):435-457 Umbral Calculus can provide exact solutions to a wide range of linear recursions. We summarize the relevant theory and give a variety of examples from combinatorics in one, two and three variables.Dedicated to the Memory of Gian-Carlo Rota      8. A regression approach to ROC surface, with applications to Alzheimer’s disease      FINE Jason P 《中国科学 数学(英文版)》2012,55(8):1583-1595 We consider the estimation of three-dimensional ROC surfaces for continuous tests given covariates.Three way ROC analysis is important in our motivating example where patients with Alzheimer’s disease are usually classified into three categories and should receive different category-specific medical treatment.There has been no discussion on how covariates affect the three way ROC analysis.We propose a regression framework induced from the relationship between test results and covariates.We consider several practical cases and the corresponding inference procedures.Simulations are conducted to validate our methodology.The application on the motivating example illustrates clearly the age and sex effects on the accuracy for Mini-Mental State Examination of Alzheimer’s disease.      9. Global optimization approach to Malfatti’s problem      Rentsen Enkhbat 《Journal of Global Optimization》2016,64(1):33-48       10. An algorithmic approach to Ramanujan’s congruences      Silviu Radu 《The Ramanujan Journal》2009,20(2):215-251 In this paper we present an algorithm that takes as input a generating function of the form $\prod_{\delta|M}\prod_{n=1}^{\infty}(1-q^{\delta n})^{r_{\delta}}=\sum_{n=0}^{\infty}a(n)q^{n}In this paper we present an algorithm that takes as input a generating function of the form ?d|M?n=1￥(1-qdn)rd=?n=0￥a(n)qn\prod_{\delta|M}\prod_{n=1}^{\infty}(1-q^{\delta n})^{r_{\delta}}=\sum_{n=0}^{\infty}a(n)q^{n} and three positive integers m,t,p, and which returns true if a(mn+t) o 0 mod p,n 3 0a(mn+t)\equiv0\pmod{p},n\geq0, or false otherwise. Our method builds on work by Rademacher (Trans. Am. Math. Soc. 51(3):609–636, 1942), Kolberg (Math. Scand. 5:77–92, 1957), Sturm (Lecture Notes in Mathematics, pp. 275–280, Springer, Berlin/Heidelberg, 1987), Eichhorn and Ono (Proceedings for a Conference in Honor of Heini Halberstam, pp. 309–321, 1996).      11. An ultrafilter approach to Jin’s theorem      Mathias Beiglböck 《Israel Journal of Mathematics》2011,185(1):369-374 It is well known and not difficult to prove that if C ⊆ ℤ has positive upper Banach density, the set of differences C − C is syndetic, i.e. the length of gaps is uniformly bounded. More surprisingly, Renling Jin showed that whenever A and B have positive upper Banach density, then A − B is piecewise syndetic.      12. On Bellman’s approach to optimal control theory      M. F. Sukhinin 《Mathematical Notes》1999,66(5):636-641 A theorem published earlier by the author is strengthened and a more informative necessary condition for an extremum in optimal control problems is obtained. The result is illustrated by a simple optimal control problem. Translated fromMatematicheskie Zametki, Vol. 66, No. 5, pp. 770–776, November, 1999.      13. A computational approach to Conwayʼs thrackle conjecture      Radoslav Fulek  János Pach 《Computational Geometry》2011,44(6-7):345-355       14. An online version of Rota’s basis conjecture      Guus P. Bollen  Jan Draisma 《Journal of Algebraic Combinatorics》2015,41(4):1001-1012       15. A counterexample to Valette’s conjecture      A. Voynov 《Proceedings of the Steklov Institute of Mathematics》2011,275(1):290-292 We disprove a well-known conjecture of D. Vallete (1978), which states that every d-dimensional self-affine convex body is a direct product of a polytope with a convex body of lower dimension. It is shown that there are counterexamples for dimension d = 4. Additional assumptions under which the conjecture is true are discussed.      16. A Counter-Example to Hausmann’s Conjecture      Virk  Žiga 《Foundations of Computational Mathematics》2022,22(2):469-475 Foundations of Computational Mathematics - In 1995 Jean-Claude Hausmann proved that a compact Riemannian manifold X is homotopy equivalent to its Rips complex $${text {Rips}}(X,r)$$ for small...      17. A complement to Gladyshev’s theorem      Rimas Norvaiša 《Lithuanian Mathematical Journal》2011,51(1):26-35 We prove the almost sure convergence of a weighted quadratic variation for a class of Gaussian processes. The result is applied to a bifractional Brownian motion and a subfractional Brownian motion.      18. Optimal control approach to stability criteria on Hill’s equations      Jian Zu  Yong Li 《高校应用数学学报(英文版)》2018,33(3):253-273 This paper gives a tutorial on how to prove Lyapunov type criteria by optimal control methods. Firstly, we consider stability criteria on Hill's equations with nonnegative potential. By optimal control methods developed in 1990s, we obtain several stability criteria including Lyapunov's criterion, Neǐgauz and Lidskiǐ's criterion. Secondly, we present stability criteria on Hill's equations with sign-changing potential in which Brog's criterion and Krein's criterion are included.      19. Classical approach to Ramanujan’s modular equations of septic degree      Vasuki  K. R.  Mahadevaswamy 《The Ramanujan Journal》2020,51(3):553-561 The Ramanujan Journal - In this paper, we prove six Ramanujan’s modular equations of septic degree by employing Ramanujan’s $$_1\psi _1$$ summation formula and certain theta function...      20. A variational approach to the construction and Malliavin differentiability of strong solutions of SDE’s      Olivier Menoukeu-Pamen  Thilo Meyer-Brandis  Torstein Nilssen  Frank Proske  Tusheng Zhang 《Mathematische Annalen》2013,357(2):761-799 In this article we develop a new approach to construct solutions of stochastic equations with merely measurable drift coefficients. We aim at demonstrating the principles of our technique by analyzing strong solutions of stochastic differential equations driven by Brownian motion. An important and rather surprising consequence of our method which is based on Malliavin calculus is that the solutions derived by Veretennikov (Theory Probab Appl 24:354–366, 1979) for Brownian motion with bounded and measurable drift in $\mathbb{R }^{d}$ are Malliavin differentiable. Further, a strength of our approach, which does not rely on a pathwise uniqueness argument, is that it can be transferred and applied to the analysis of various other types of (stochastic) equations: We obtain a Bismut–Elworthy–Li formula (Elworthy and Li, J Funct Anal 125:252–286, 1994) for spatial derivatives of solutions to the Kolmogorov equation with bounded and measurable drift coefficients. To derive the formula, we use that our approach can be applied to obtain Sobolev differentiability in the initial condition in addition to Malliavin differentiability of the associated stochastic differential equations. Another application of our technique is the construction of unique solutions of the stochastic transport equation with irregular vector fields. Moreover, our approach is also applicable to the construction of solutions of stochastic evolution equations on Hilbert spaces.

Nekhoroshev’s approach to Hamiltonian monodromy

Using the hyperbolic circular billiard, introduced in [31] by Delos et al. as possibly the simplest system with Hamiltonian monodromy, we illustrate the method developed by N. N. Nekhoroshev and coauthors [48] to uncover this phenomenon. Nekhoroshev’s very original geometric approach reflects his profound insight into Hamiltonian monodromy as a general topological property of fibrations. We take advantage of the possibility of having closed form elementary function expressions for all quantities in our system in order to provide the most explicit and detailed explanation of Hamiltonian monodromy and its relation to similar phenomena in other domains.     

Rota’s umbral calculus and recursions

Umbral Calculus can provide exact solutions to a wide range of linear recursions. We summarize the relevant theory and give a variety of examples from combinatorics in one, two and three variables.Dedicated to the Memory of Gian-Carlo Rota     

A regression approach to ROC surface, with applications to Alzheimer’s disease

We consider the estimation of three-dimensional ROC surfaces for continuous tests given covariates.Three way ROC analysis is important in our motivating example where patients with Alzheimer’s disease are usually classified into three categories and should receive different category-specific medical treatment.There has been no discussion on how covariates affect the three way ROC analysis.We propose a regression framework induced from the relationship between test results and covariates.We consider several practical cases and the corresponding inference procedures.Simulations are conducted to validate our methodology.The application on the motivating example illustrates clearly the age and sex effects on the accuracy for Mini-Mental State Examination of Alzheimer’s disease.     

An algorithmic approach to Ramanujan’s congruences

In this paper we present an algorithm that takes as input a generating function of the form $\prod_{\delta|M}\prod_{n=1}^{\infty}(1-q^{\delta n})^{r_{\delta}}=\sum_{n=0}^{\infty}a(n)q^{n}In this paper we present an algorithm that takes as input a generating function of the form ?_d|M?_n=1^￥(1-q^dn)^r_d=?_n=0^￥a(n)qⁿ\prod_{\delta|M}\prod_{n=1}^{\infty}(1-q^{\delta n})^{r_{\delta}}=\sum_{n=0}^{\infty}a(n)q^{n} and three positive integers m,t,p, and which returns true if a(mn+t) o 0 mod p,n 3 0a(mn+t)\equiv0\pmod{p},n\geq0, or false otherwise. Our method builds on work by Rademacher (Trans. Am. Math. Soc. 51(3):609–636, 1942), Kolberg (Math. Scand. 5:77–92, 1957), Sturm (Lecture Notes in Mathematics, pp. 275–280, Springer, Berlin/Heidelberg, 1987), Eichhorn and Ono (Proceedings for a Conference in Honor of Heini Halberstam, pp. 309–321, 1996).     

An ultrafilter approach to Jin’s theorem

It is well known and not difficult to prove that if C ⊆ ℤ has positive upper Banach density, the set of differences C − C is syndetic, i.e. the length of gaps is uniformly bounded. More surprisingly, Renling Jin showed that whenever A and B have positive upper Banach density, then A − B is piecewise syndetic.     

On Bellman’s approach to optimal control theory

A theorem published earlier by the author is strengthened and a more informative necessary condition for an extremum in optimal control problems is obtained. The result is illustrated by a simple optimal control problem. Translated fromMatematicheskie Zametki, Vol. 66, No. 5, pp. 770–776, November, 1999.     

A counterexample to Valette’s conjecture

We disprove a well-known conjecture of D. Vallete (1978), which states that every d-dimensional self-affine convex body is a direct product of a polytope with a convex body of lower dimension. It is shown that there are counterexamples for dimension d = 4. Additional assumptions under which the conjecture is true are discussed.     

A Counter-Example to Hausmann’s Conjecture

Foundations of Computational Mathematics - In 1995 Jean-Claude Hausmann proved that a compact Riemannian manifold X is homotopy equivalent to its Rips complex $${text {Rips}}(X,r)$$ for small...     

A complement to Gladyshev’s theorem

We prove the almost sure convergence of a weighted quadratic variation for a class of Gaussian processes. The result is applied to a bifractional Brownian motion and a subfractional Brownian motion.     

Optimal control approach to stability criteria on Hill’s equations

This paper gives a tutorial on how to prove Lyapunov type criteria by optimal control methods. Firstly, we consider stability criteria on Hill's equations with nonnegative potential. By optimal control methods developed in 1990s, we obtain several stability criteria including Lyapunov's criterion, Neǐgauz and Lidskiǐ's criterion. Secondly, we present stability criteria on Hill's equations with sign-changing potential in which Brog's criterion and Krein's criterion are included.     

Classical approach to Ramanujan’s modular equations of septic degree

The Ramanujan Journal - In this paper, we prove six Ramanujan’s modular equations of septic degree by employing Ramanujan’s $$_1\psi _1$$ summation formula and certain theta function...     

A variational approach to the construction and Malliavin differentiability of strong solutions of SDE’s

In this article we develop a new approach to construct solutions of stochastic equations with merely measurable drift coefficients. We aim at demonstrating the principles of our technique by analyzing strong solutions of stochastic differential equations driven by Brownian motion. An important and rather surprising consequence of our method which is based on Malliavin calculus is that the solutions derived by Veretennikov (Theory Probab Appl 24:354–366, 1979) for Brownian motion with bounded and measurable drift in $\mathbb{R }^{d}$ are Malliavin differentiable. Further, a strength of our approach, which does not rely on a pathwise uniqueness argument, is that it can be transferred and applied to the analysis of various other types of (stochastic) equations: We obtain a Bismut–Elworthy–Li formula (Elworthy and Li, J Funct Anal 125:252–286, 1994) for spatial derivatives of solutions to the Kolmogorov equation with bounded and measurable drift coefficients. To derive the formula, we use that our approach can be applied to obtain Sobolev differentiability in the initial condition in addition to Malliavin differentiability of the associated stochastic differential equations. Another application of our technique is the construction of unique solutions of the stochastic transport equation with irregular vector fields. Moreover, our approach is also applicable to the construction of solutions of stochastic evolution equations on Hilbert spaces.