Non-primitive number fields of degree eight and of signature with small discriminant

We give the lists of all non-primitive number fields of degree eight having two, four and six real places of discriminant less than 6688609, 24363884 and 92810082, respectively, in absolute value. For each field in the lists, we give its discriminant, the discriminant of its subfields, a relative polynomial generating the field over one of its subfields and its discriminant, the corresponding polynomial over , and the Galois group of its Galois closure.

     

A new approach to the theory of functional integral equations of fractional order

The aim of the paper is to present a new approach to the theory of functional integral equations of fractional order. That approach depends on converting of the mentioned equations to the form of functional integral equations of Volterra-Stieltjes type. It turns out that the study of functional integral equations of Volterra-Stieltjes type is more convenient and effective than the study of functional integral equations of fractional order. An example illustrating our approach is also discussed.     

Approximate additive mappings in 2-Banach spaces and related topics

In this paper, we investigate approximate additive mappings, approximate Jensen mappings and approximate quadratic mappings in 2-Banach spaces. That is, we prove the generalized Hyers-Ulam stability of the Cauchy functional equation, the Jensen functional equation and the quadratic functional equation in 2-Banach spaces.     

Hyperelliptic curves and homomorphisms to ideal class groups of quadratic number fields

For the function field K of hyperelliptic curves over Q we define a subgroup of the ideal class group called the group of Z-primitive ideals. We then show that there are homomorphisms from this subgroup to ideal class groups of certain quadratic number fields.     

Uniform Hölder exponent of a stationary increments Gaussian process: Estimation starting from average values

Let {X(t)}_t∈R be a stationary increments Gaussian process satisfying some assumptions. By using the notion of generalized quadratic variation we build a strongly consistent and asymptotically normal estimator of the uniform Hölder exponent of X, over a compact interval. Our estimator is obtained starting from average values of the process over a regular grid.     

Quadratic functional equations in a set of Lebesgue measure zero

Let R$\mathbb{R}$ be the set of real numbers, Y   a Banach space and f:R→Y$f:\mathbb{R}\to Y$. We prove the Hyers–Ulam stability theorem for the quadratic functional inequality

‖f(x+y)+f(x−y)−2f(x)−2f(y)‖≤?$\Vert f(x+y)+f(x-y)-2f(x)-2f(y)\Vert \le ?$

for all (x,y)∈Ω$(x,y)\in \Omega$, where Ω⊂R²$\Omega \subset {\mathbb{R}}^2$ is of Lebesgue measure 0. Using the same method we dealt with the stability of two more functional equations in a set of Lebesgue measure 0.     

Quadratic covariation estimates in non-smooth stochastic calculus

Given a Brownian Motion W$W$, in this paper we study the asymptotic behavior, as ε→0$\epsilon \to 0$, of the quadratic covariation between f(εW)$f\left(\epsilon W\right)$ and W$W$ in the case in which f$f$ is not smooth. Among the main features discovered is that the speed of the decay in the case f∈C^α$f\in C^{\alpha }$ is at least polynomial in ε$\epsilon$ and not exponential as expected. We use a recent representation as a backward–forward Itô integral of [f(εW),W]$[f\left(\epsilon W\right),W]$ to prove an ε$\epsilon$-dependent approximation scheme which is of independent interest. We get the result by providing estimates to this approximation. The results are then adapted and applied to generalize the results of Almada Monter and Bakhtin (2011) and Bakhtin (2011) related to the small noise exit from a domain problem for the saddle case.     

Continuous quadratic programming formulations of optimization problems on graphs

Four NP-hard optimization problems on graphs are studied: The vertex separator problem, the edge separator problem, the maximum clique problem, and the maximum independent set problem. We show that the vertex separator problem is equivalent to a continuous bilinear quadratic program. This continuous formulation is compared to known continuous quadratic programming formulations for the edge separator problem, the maximum clique problem, and the maximum independent set problem. All of these formulations, when expressed as maximization problems, are shown to follow from the convexity properties of the objective function along the edges of the feasible set. An algorithm is given which exploits the continuous formulation of the vertex separator problem to quickly compute approximate separators. Computational results are given.