A generalized Burgers equation with measure data is studied. The existence of a weak solution of an initial boundary-value problem in a bounded cylindrical domain, is established. Time-periodic solutions are shown to exist and an optimization problem related to an inverse problem is considered.

In this paper, we prove that the multiple $L$-value of conductor $4$ can be expressed in terms of lower depth multiple $L$-values under the condition on the parity of its depth and weight. This can be regarded as a character analogue of what is called the "parity result" for multiple zeta values which was proved by Zagier.

We show in this paper that the multiplicities of mixed representations are uniformly infinite or uniformly finite and bounded, in the setting of completely solvable Lie groups extending then the situation of nilpotent Lie groups. Necessary and sufficient conditions for these multiplicities to be finite are provided.

The Kauffman bracket approach is used to give estimates on the size of the determinant (and this way also on the coefficients of the Jones and Alexander polynomial) of a link of given crossing number, or equivalently on the number of spanning trees of planar graphs with given number of edges. Properties of the knots and links with maximal determinant for given crossing number are investigated.

In this paper we study the exponents of 2-multiarrangements. More precisely, we compose a basis for $D(\mathcal{A},k)$ in the case where $\mathcal{A}$ consists of three lines using $\mathbf{Q}$-polynomials ${X \choose \lambda}$. Here ${X \choose \lambda}$ is the generalized binomial coefficient of the partition $\lambda$.

We give a characterization of Möbius transformation by use of Apollonius points introduced by Haruki and Rassias [2]. Our result is stronger than theirs.

In this note we investigate the spectral gaps of the Schrödinger operator $$H=-\frac{d^2}{dx^2}+\sum_{l=-\infty}^{\infty}\big(\beta_1\delta^{\prime}(x-2\pi l)+\beta_2\delta^{\prime}(x-\kappa-2\pi l)\big) \quad \textrm{in} \quad L^2(\mathbf{R})\,,$$ where $\beta_1$, $\beta_2 \in \mathbf{R}\setminus\{0\}$ and $\kappa/\pi \in \mathbf{Q}$. By $G_{j}$ we denote the $j$-th gap of the spectrum of $H$. We provide the asymptotic expansion of the length of $G_{j}$ as $j\rightarrow\infty$.

In this paper we compare topological methods and dynamical methods corresponding to recent developments in the classification of inverse limit spaces of one dimensional maps on graphs. We elucidate the role of shifting the periodic base point in applying Williams' theory. We exploit the Fox calculus to define and apply the Bowen-Franks trace---a shift equivalence invariant of free group homomorphisms. We show that augmented cohomology of certain suspensions associated with wrapping rules in a substitution yields augmented dimension groups that have a relatively simple product structure. We complete the classification of a family of examples of generalized solenoids initiated by R.~F. Williams.

We discuss the asymptotic behavior of the largest amount of winnings in the generalized St. Petersburg games and show that the results on the random variables associated with the amount of winnings can be applied to the random variables associated with the degree of the polynomial digits of the continued fraction and the Oppenheim expansions of formal Laurent series.

In this paper we present a set $\mathcal{T}_f^+$ of rational numbers $s\in\mathbf{Q}$ such that the minimal splitting fields $L_s$ of $X^3-3sX^2-(3s+3)X-1$ are cyclic cubic fields with a given conductor $f$. The set $\mathcal{T}_f^+$ has exactly one $s$ for each field $L$ of conductor $f$. The Weil's height of every number $s\in \mathcal{T}_f^+$ is minimal among all of the rational numbers $s\in\mathbf{Q}$ such that $L_s=L$. If a cyclic cubic field $L$ of conductor $f$ is given, then we can choose the number $s\in \mathcal{T}_f^+$ corresponding to $L$ by sequencing the explicit Artin symbols.

We consider the Schrödinger and Schrödinger type operators $H_{1}=-\Delta+V$ and $H_2=(-\Delta)^2+V^2$ with non-negative potentials $V$ on $\mathbf{R}^n$. We assume that the potential $V$ belongs to the reverse Hölder class which includes non-negative polynomials. We establish estimates of the fundamental solution for $H_{2}$ and show some $L^p$ estimates for Schrödinger type operators. Moreover, we show that the operator $\nabla^4H_{2}^{-1}$ is a Calderón-Zygmund operator.

We shall give some criteria for the validity of Greenberg's generalized conjecture (GGC) for imaginary quadratic fields and the prime 2. Using this, we shall give examples of imaginary quadratic fields satisfying GGC for the prime 2.

The cohomology of the complement of hyperplanes with coefficients in the rank one local system associated to a generic weight vanishes except in the highest dimension. In this paper, we construct matroids or arrangements admitting weights for which the Orlik-Solomon algebra has non-vanishing cohomology, using decomposable relations arising from Latin hypercubes.

We define a Courant bracket on an associative algebra using the theory of Hochschild homology, and we introduce the notion of Dirac algebra. We show that the bracket of an omni-Lie algebra is quite a kind of Courant bracket.

In the stochastic context, an invariant set is decomposed into the union of ergodic basins, and each of basin possesses the fractal structure determined by ergodic measures. This paper is to show that when a hyperbolic SRB measure is mixing, the set of measures with zero entropy and the set of measures with positive entropy but without SRB are both dense on the set of all invariant measures on the closure of the ergodic basin in the Pesin set, and moreover that in the set of invariant measures as above a measure with ergodicity and SRB exists uniquely.