The Generalized Riemann Hypothesis and the discriminant of number fields

Under the Generalized Riemann Hypothesis for the Dedekind zeta-function ${\zeta }_{\kappa }$, we obtain a formula for the discriminant $D_{\kappa /\mathbb{Q}}$ of the algebraic number field κ in terms of an integral of ${\zeta }_{\kappa }$ on the critical line.     

Probabilistic properties of number fields

In general a bound on number theoretic invariants under the Generalized Riemann Hypothesis (GRH) for the Dedekind zeta function of a number field K   is much stronger than an unconditional one. In this article, we consider three invariants; the residue of _ζK(s)${\zeta }_K(s)$ at s=1$s=1$, the logarithmic derivative of Artin L-function attached to K   at s=1$s=1$, and the smallest prime which does not split completely in K. We obtain bounds on them just as good as the bounds under GRH except for a density zero set of number fields.     

On the effectiveness of a generalization of Miller’s primality theorem

Berrizbeitia and Olivieri showed in a recent paper that, for any integer r$r$, the notion of ω$\omega$-prime to base a$a$ leads to a primality test for numbers n≡1$n\equiv 1$ mod r$r$, that under the Extended Riemann Hypothesis (ERH) runs in polynomial time. They showed that the complexity of their test is at most the complexity of the Miller primality test (MPT), which is O((logn)^4+o(1))$O\left({(logn)}^{4+o\left(1\right)}\right)$. They conjectured that their test is more effective than the MPT if r$r$ is large.     

Nonclassical symmetry analysis for hyperbolic partial differential equation

We consider the nonclassical symmetry of one-dimensional hyperbolic differential equations of the form u_t + M(u)u_x = 0. For the infinitesimal generator $\mathbf{V}=\tau {\mathrm{\partial }}_t+\xi {\mathrm{\partial }}_x+{\sum }_{i=1}^n{\varphi }_i{\mathrm{\partial }}_{u_i}$, it is shown that ξ is an eigenvalue of the matrix M when ϕ_i = 0 [Souichi M. Nonclassical symmetry and Riemann invariants. Int J Nonlinear Mech, [in press]]. In this paper, we prove a sufficient condition of a lemma.     

Weak convergence of the Stratonovich integral with respect to a class of Gaussian processes

For a Gaussian process X$X$ and smooth function f$f$, we consider a Stratonovich integral of f(X)$f\left(X\right)$, defined as the weak limit, if it exists, of a sequence of Riemann sums. We give covariance conditions on X$X$ such that the sequence converges in law. This gives a change-of-variable formula in law with a correction term which is an Itô integral of f^?$f^{?}$ with respect to a Gaussian martingale independent of X$X$. The proof uses Malliavin calculus and a central limit theorem from Nourdin and Nualart (2010) [8]. This formula was known for fBm with H=1/6$H=1/6$ Nourdin et al. (2010) [9]. We extend this to a larger class of Gaussian processes.     

Transversality family of expanding rational semigroups

We study finitely generated expanding semigroups of rational maps with overlaps on the Riemann sphere. We show that if a d$d$-parameter family of such semigroups satisfies the transversality condition, then for almost every parameter value the Hausdorff dimension of the Julia set is the minimum of 2 and the zero of the pressure function. Moreover, the Hausdorff dimension of the exceptional set of parameters is estimated. We also show that if the zero of the pressure function is greater than 2$2$, then typically the 2-dimensional Lebesgue measure of the Julia set is positive. Some sufficient conditions for a family to satisfy the transversality conditions are given. We give non-trivial examples of families of semigroups of non-linear polynomials with the transversality condition for which the Hausdorff dimension of the Julia set is typically equal to the zero of the pressure function and is less than 2$2$. We also show that a family of small perturbations of the Sierpinski gasket system satisfies that for a typical parameter value, the Hausdorff dimension of the Julia set (limit set) is equal to the zero of the pressure function, which is equal to the similarity dimension. Combining the arguments on the transversality condition, thermodynamical formalisms and potential theory, we show that for each a∈C$a\in \mathbb{C}$ with |a|≠0,1$\left|a\right|\ne 0,1$, the family of small perturbations of the semigroup generated by {z²,az²}$\{z^2,a{z}^2\}$ satisfies that for a typical parameter value, the 2-dimensional Lebesgue measure of the Julia set is positive.     

An Itô–Stratonovich formula for Gaussian processes: A Riemann sums approach

The aim of this paper is to establish a change of variable formula for general Gaussian processes whose covariance function satisfies some technical conditions. The stochastic integral is defined in the Stratonovich sense using an approximation by middle point Riemann sums. The change of variable formula is proved by means of a Taylor expansion up to the sixth order, and applying the techniques of Malliavin calculus to show the convergence to zero of the residual terms. The conditions on the covariance function are weak enough to include processes with infinite quadratic variation, and we show that they are satisfied by the bifractional Brownian motion with parameters (H,K)$(H,K)$ such that 1/6<HK<1$1/6<HK<1$, and, in particular, by the fractional Brownian motion with Hurst parameter H∈(1/6,1)$H\in (1/6,1)$.     

Test ideals of non-principal ideals: Computations,jumping numbers,alterations and division theorems

Given an ideal a⊆R$\mathfrak{a}\subseteq R$ in a (log) Q$\mathbb{Q}$-Gorenstein F  -finite ring of characteristic p>0$p>0$, we study and provide a new perspective on the test ideal τ(R,a^t)$\tau (R,{\mathfrak{a}}^t)$ for a real number t>0$t>0$. Generalizing a number of known results from the principal case, we show how to effectively compute the test ideal and also describe τ(R,a^t)$\tau (R,{\mathfrak{a}}^t)$ using (regular) alterations with a formula analogous to that of multiplier ideals in characteristic zero. We further prove that the F  -jumping numbers of τ(R,a^t)$\tau (R,{\mathfrak{a}}^t)$ as t varies are rational and have no limit points, including the important case where R is a formal power series ring. Additionally, we obtain a global division theorem for test ideals related to results of Ein and Lazarsfeld from characteristic zero, and also recover a new proof of Skoda's theorem for test ideals which directly mimics the proof for multiplier ideals.     

Global solutions with shock waves to the generalized Riemann problem for a class of quasilinear hyperbolic systems of balance laws

It is proven that the generalized Riemann problem for a class of quasilinear hyperbolic systems of balance laws admits a unique global piecewise C¹$C^1$ solution u=u(t,x)$u=u(t,x)$ containing only n$n$ shock waves with small amplitude on t?0$t?0$ and this solution possesses a global structure similar to that of the similarity solution u=U(x/t)$u=U(x/t)$ of the corresponding homogeneous Riemann problem. As an application of our result, we prove the existence of global shock solutions, piecewise continuous and piecewise smooth solution with shock discontinuities, of the flow equations of a model class of fluids with viscosity induced by fading memory with a single jump initial data.     

Positive semidefinite zero forcing

The positive semidefinite zero forcing number _Z+(G)${\mathrm{Z}}_{+}(G)$ of a graph G was introduced in Barioli et al. (2010) [4]. We establish a variety of properties of _Z+(G)${\mathrm{Z}}_{+}(G)$: Any vertex of G   can be in a minimum positive semidefinite zero forcing set (this is not true for standard zero forcing). The graph parameters tw(G)$\mathrm{tw}(G)$ (tree-width), _Z+(G)${\mathrm{Z}}_{+}(G)$, and Z(G)$\mathrm{Z}(G)$ (standard zero forcing number) all satisfy the Graph Complement Conjecture (see Barioli et al. (2012) [3]). Graphs having extreme values of the positive semidefinite zero forcing number are characterized. The effect of various graph operations on positive semidefinite zero forcing number and connections with other graph parameters are studied.     

Lifespan of classical discontinuous solutions to general quasilinear hyperbolic systems of conservation laws with small BV initial data: Rarefaction waves

In the present paper the author investigates the global structure stability of Riemann solutions for general quasilinear hyperbolic systems of conservation laws under small BV perturbations of the initial data, where the Riemann solution contains rarefaction waves, while the perturbations are in BV but they are assumed to be C¹$C^1$-smooth, with bounded and possibly large C¹$C^1$-norms. Combining the techniques employed by Li–Kong with the modified Glimm’s functional, the author obtains a lower bound of the lifespan of the piecewise C¹$C^1$ solution to a class of generalized Riemann problems, which can be regarded as a small BV perturbation of the corresponding Riemann problem. This result is also applied to the system of traffic flow on a road network using the Aw–Rascle model.