Irreducibility of Hypersurfaces

Given a polynomial P in several variables over an algebraically closed field, we show that except in some special cases that we fully describe, if one coefficient is allowed to vary, then the polynomial is irreducible for all but at most deg(P)² ? 1 values of the coefficient. We more generally handle the situation where several specified coefficients vary.     

Infinite systems of noncolliding generalized meanders and Riemann–Liouville differintegrals

Yor’s generalized meander is a temporally inhomogeneous modification of the 2(ν + 1)-dimensional Bessel process with ν  >   ? 1, in which the inhomogeneity is indexed by $\kappa \in [0, 2(\nu+1))$ . We introduce the noncolliding particle systems of the generalized meanders and prove that they are Pfaffian processes, in the sense that any multitime correlation function is given by a Pfaffian. In the infinite particle limit, we show that the elements of matrix kernels of the obtained infinite Pfaffian processes are generally expressed by the Riemann–Liouville differintegrals of functions comprising the Bessel functions J _ν used in the fractional calculus, where orders of differintegration are determined by ν ? κ. As special cases of the two parameters (ν, κ), the present infinite systems include the quaternion determinantal processes studied by Forrester, Nagao and Honner and by Nagao, which exhibit the temporal transitions between the universality classes of random matrix theory.     

Eta products of weight one,and decomposition of primes in non-abelian extensions

This article is a slightly extended and revised version of a conference talk at “Arithmetik an der A7” in Würzburg, June 23rd, 2017. We present a conjecture on the coincidence of Hecke theta series of weight 1 on three distinct quadratic fields. Then we discuss a special instance of the Deligne–Serre Theorem, implying that the decomposition of prime numbers in a certain extension of the rationals is governed by the coefficients of the eta product \(\eta^{2}(z)\).     

Polynomial behavior of special values of partial zeta functions of real quadratic fields at s = 0

We compute the special values of partial zeta functions at s = 0 for family of real quadratic fields K _n and ray class ideals ${\mathfrak{b}_n}$ such that ${\mathfrak{b}_n^{-1} = [1, \delta(n)]}$ where the continued fraction expansion of δ(n) ? 1 is purely periodic and terms are polynomials in n of degree bounded by d. With additional assumptions, we prove that the special values of the partial zeta functions at s = 0 are given by a quasi-polynomial of degree less than or equal to d as a function of n. We apply this to conclude that the special values of the Hecke’s L-functions at s = 0 for the family ${(K_n, \mathfrak{b}_n, \chi_n:= \chi \circ N_{K_n/\mathbb{Q}})}$ for any Dirichlet character χ behave like quasi-polynomial as well. We compute explicitly the coefficients of the quasi-polynomials. Two examples satisfying the condition are presented, and for these two families, the special values of the partial zeta functions at s = 0 are given.     

Mutation Classes of Skew-Symmetric 3 × 3-Matrices

In this article, we establish a bijection between the set of mutation classes of mutation-cyclic skew-symmetric integral 3 × 3-matrices and the set of triples of integers (a, b, c) such that 2 ≤ a ≤ b ≤ c and ab ≥ c. We also give an algorithm allowing to verify whether a matrix is mutation-cyclic or not. We prove that given a, b, the two cases depend on whether c is large enough or not.     

Projective Normality of Special Scrolls

We study the projective normality of a linearly normal special scroll R of degree d and speciality i over a smooth curve X of genus g. We relate it with the Clifford index of the base curve X. If d ≥ 4g ? 2i ? Cliff(X) + 1, i ≥ 3 and R is smooth, we prove that the projective normality of the scroll is equivalent to the projective normality of its directrix curve of minimum degree.     

Global Asymptotics of the Second Painlevé Equation in Okamoto’s Space

We study the solutions of the second Painlevé equation (P II) in the space of initial conditions first constructed by Okamoto, in the limit as the independent variable, x, goes to infinity. Simultaneously, we study solutions of the related equation known as the thirty-fourth Painlevé equation (P 34). By considering degenerate cases of the autonomous flow, we recover the known special solutions, which are either rational functions or expressible in terms of Airy functions. We show that the solutions that do not vanish at infinity possess an infinite number of poles. An essential element of our construction is the proof that the union of exceptional lines is a repeller for the dynamics in Okamoto’s space. Moreover, we show that the limit set of the solutions exists and is compact and connected.     

The checkpoint ordering problem

We suggest a new variant of a row layout problem: Find an ordering of n departments with given lengths such that the total weighted sum of their distances to a given checkpoint is minimized. The Checkpoint Ordering Problem (COP) is both of theoretical and practical interest. It has several applications and is conceptually related to some well-studied combinatorial optimization problems, namely the Single-Row Facility Layout Problem, the Linear Ordering Problem and a variant of parallel machine scheduling. In this paper we study the complexity of the (COP) and its special cases. The general version of the (COP) with an arbitrary but fixed number of checkpoints is NP-hard in the weak sense. We propose both a dynamic programming algorithm and an integer linear programming approach for the (COP) . Our computational experiments indicate that the (COP) is hard to solve in practice. While the run time of the dynamic programming algorithm strongly depends on the length of the departments, the integer linear programming approach is able to solve instances with up to 25 departments to optimality.     

Rotational integral geometry of tensor valuations

We derive a new rotational Crofton formula for Minkowski tensors. In special cases, this formula gives (1) the rotational average of Minkowski tensors defined on linear subspaces and (2) the functional defined on linear subspaces with rotational average equal to a Minkowski tensor. Earlier results obtained for intrinsic volumes appear now as special cases.     

The two-point spectral Nevanlinna-Pick problem

We present an analytic solution of certain special cases of the spectral Nevanlinna-Pick problem for two interpolation points and 2×2 matrices.     

On a problem of Croft on optimally nested regular polygons

We present a solution for the largest regular m-gon contained in a regular n-gon. We find that the answer depends critically on the coprimality of m and n. We show that the optimal polygons are concentric if and only if gcd(m, n) > 1. Our principal result is a complete solution for the case where m and n share a common divisor. For the case of coprime m and n, we present partial results and a conjecture for the general solution. Our findings subsume some special cases which have previously been published on this problem.     

On projectively flat Finsler spaces

First we present a short overview of the long history of projectively flat Finsler spaces. We give a simple and quite elementary proof of the already known condition for the projective flatness, and we give a criterion for the projective flatness of a special Lagrange space (Theorem 1). After this we obtain a second-order PDE system, whose solvability is necessary and sufficient for a Finsler space to be projectively flat (Theorem 2). We also derive a condition in order that an infinitesimal transformation takes geodesics of a Finsler space into geodesics. This yields a Killing type vector field (Theorem 3). In the last section we present a characterization of the Finsler spaces which are projectively flat in a parameter-preserving manner (Theorem 4), and we show that these spaces over ${\mathbb {R}}^{n}$ are exactly the Minkowski spaces (Theorems 5 and 6).     

When is the Diagonal Functor Frobenius?

Given a complete, cocomplete category 𝒞, we investigate the problem of describing those small categories I such that the diagonal functor Δ: 𝒞 → Functors(I, 𝒞) is a Frobenius functor. This condition can be rephrased by saying that the limits and the colimits of functors I → 𝒞 are naturally isomorphic. We find necessary conditions on I for a certain class of categories 𝒞, and, as an application, we give both necessary and sufficient conditions in the two special cases 𝒞 =Set or _R ?, the category of left modules over a ring R.     

The Module Decomposition of Super-Symmetric Powers of Matrices

ABSTRACT

Let M be the k  ×  m matrices over ?. The GL ( k ) ×  GL ( m ) decompositions of the symmetric and of the exterior powers of M are described by two classical theorems. We describe a theorem for Lie superalgebras, which implies both of these classical theorems as special cases. The constructions of both the exterior and the symmetric algebras are generalized to a class of algebras defined by partitions. That superalgebra theorem is further generalized to these algebras.     

Amalgamations of the Painlevé Equations

We present new hierarchies of nonlinear ordinary differential equations (ODEs) that are generalizations of the Painlevé equations. These hierarchies contain the Painlevé equations as special cases. We emphasize the sixth-order ODEs. Special solutions for one of them are expressed via the general solutions of the P ₁ and P ₂ equations and special cases of the P ₃ and P ₅ equations. Four of the six Painlevé equations can be considered special cases of these sixth-order ODEs. We give linear representations for solving the Cauchy problems for the hierarchy equations using the inverse monodromy transform.     

Hybrid multi-objective Bayesian estimation of distribution algorithm: a comparative analysis for the multi-objective knapsack problem

Nowadays, a number of metaheuristics have been developed for efficiently solving multi-objective optimization problems. Estimation of distribution algorithms are a special class of metaheuristic that intensively apply probabilistic modeling and, as well as local search methods, are widely used to make the search more efficient. In this paper, we apply a Hybrid Multi-objective Bayesian Estimation of Distribution Algorithm (HMOBEDA) in multi and many objective scenarios by modeling the joint probability of decision variables, objectives, and the configuration parameters of an embedded local search (LS). We analyze the benefits of the online configuration of LS parameters by comparing the proposed approach with LS off-line versions using instances of the multi-objective knapsack problem with two to five and eight objectives. HMOBEDA is also compared with five advanced evolutionary methods using the same instances. Results show that HMOBEDA outperforms the other approaches including those with off-line configuration. HMOBEDA not only provides the best value for hypervolume indicator and IGD metric in most of the cases, but it also computes a very diverse solutions set close to the estimated Pareto front.     

Maximal Subgroups of Finite Orthogonal Groups Stabilizing Spreads of Lines

We prove that, for q odd and n ≥ 3, the group G = O _n (q ²) · 2 is maximal in either the orthogonal group O _2n (q) or the special orthogonal group SO _2n (q). The group G corresponds to the stabilizer of a spread of lines of PG(2n ? 1, q) in which some lines lie on a quadric, some are secant to the quadric, and others are external to the quadric.     

Jacobian Conditions For p-Bases

We discuss various sufficient conditions, involving derivations and jacobians, for elements of a domain of characteristic p > 0, to form a p-basis of a ring of constants of a derivation. We also prove that in a special case of eigenvectors these conditions are both sufficient and necessary.     

Simply sequentially additive labelings of 2-regular graphs

We conjecture that any 2-regular simple graph has an SSA labeling. We provide several special cases to support our conjecture. Most of our constructions are based on Skolem sequences and on an extension of it. We establish a connection between simply sequentially additive labelings of 2-regular graphs and ordered graceful labelings of spiders.     

More on the Hochschild and Cyclic Homologies of Crossed Modules of Algebras

We investigate the Hochschild and cyclic homologies of crossed modules of algebras in some special cases. We prove that the cotriple cyclic homology of a crossed module of algebras (I, A, ρ) is isomorphic to HC _*(ρ): HC _*(I) → HC _*(A), provided I is H-unital and the ground ring is a field with characteristic zero. We also calculate the Hochschild and cyclic homologies of a crossed module of algebras (R, 0, 0) for each algebra R with trivial multiplication. At the end, we give some applications proving a new five term exact sequence.