Using the skein relations for the Links-Gould (LG) invariant discovered by the first author, we give two constructions to provide a pair of different knots or links sharing the same LG invariant. Using this, we give examples of arbitrarily many 2-bridge knots with the same LG invariant. These knots also share the same HOMFLY and Kauffman polynomials. Further, for 2-bridge links we construct a similar example, which also share the same 2-variable Alexander polynomial. We also give a non-amphicheiral hyperbolic knot whose chirality is not undetected by the LG invariant.

We study coincidence points for maps $f_1,f_2\colon E \to B$ into manifolds such that $f_1$ is homotopic to $f_2$. We analyze the first and higher obstructions to deform $f_1$ away to $f_2$. The main results consist in solving this one problem for the (generalized) Hopf bundles, which are $G$-principal bundles $p_nG \colon E_n G \to B_n G$ (the $n$-th stage of Milnor's construction), with $G= S^1,S^3$. We also consider the question for general maps $f\colon E_n G \to B_n G$ with $G= S^1,S^3$.

On the basis of Brylinski's work, we introduce a notion of equivariant smooth Deligne cohomology group, which is a generalization of both ordinary smooth Deligne cohomology and ordinary equivariant cohomology. Using the cohomology group, we classify equivariant circle bundles with connection, and equivariant gerbes with connection.

Every compact real-analytic Riemannian manifold has a complexification called the Grauert tube. We give an asymptotic expansion of the leading coefficient of the logarithmic term of the Szegö kernel for two-dimensional Grauert tubes.

We give some regularization in order to define rigorously a stochastic W.Z.N.W. model or a stochastic Chern-Simons theory. We show that the Markov property of the random field allows us to satisfy the glueing axiom of field theory (of Segal or Atiyah).

An integral test for some multidimensional Lévy processes is given. A law of the iterated logarithm of Chover type is derived from it as a corollary. This law was pointed out by Chover in the case of one-dimensional symmetric stable processes with discrete time.

Singular means here that the parabolic equation is \textit{not} in normal form neither can it be reduced to such a form. For this class of problems, following the operator approach used in [1], we prove global in time existence and uniqueness theorems related to (spatial) $L^p$-spaces. Various improvements to [2], [3] are given.

We consider the Cauchy problem for the Korteweg-de Vries-Burgers equation. Global existence and smoothing effect for the Cauchy problem to the Korteweg-de Vries-Burgers equation was proved previously for the case of large initial data. Recently the first term of the large time asymptotics was obtained without restriction on the size of the initial data. The aim of the present paper is to obtain the second term of the large time asymptotic behavior of solutions to the Cauchy problem for the Korteweg-de Vries-Burgers equation in the case of the initial data of arbitrary size.

Let $\mathbf{A}_{n-1}^+ \subset \mathbb{Z}^n$ denote the set of positive roots of the root system $\mathbf{A}_{n-1}$ and $I_{\mathbf{A}_{n-1}^+}$ its toric ideal. The purpose of the present paper is to study combinatorics and algebra on $\mathbf{A}_{n-1}^+$ and $I_{\mathbf{A}_{n-1}^+}$. First, it will be proved that $I_{\mathbf{A}_{n-1}^+}$ induces an initial ideal $\mathit{in}_{<}\left(I_{\mathbf{A}_{n-1}^+}\right)$ which is generated by quadratic squarefree monomials together with cubic squarefree monomials. Second, we will associate each maximal face $\sigma$ of the unimodular triangulation $\Delta$ arising from $\mathit{in}_{<}\left(I_{\mathbf{A}_{n-1}^+}\right)$ with a certain subgraph $G_\sigma$ on $[n] = \{1,\ldots,n\}$. Third, noting that the number of maximal faces of $\Delta$ is equal to that of anti-standard trees $T$ on $[n]$ with $T \neq \{ (1,2) , (1,3), \ldots , (1,n) \}$, an explicit bijection between the set $\{ G_\sigma \colon \sigma\ \text{is a maximal face of}\ \Delta \}$ and that of anti-standard trees $T$ on $[n]$ with $T \neq \{ (1,2), (1,3), \ldots , (1,n) \}$ will be constructed. In particular, a new combinatorial expression of Catalan numbers arises.

We first show that for each Weyl algebra over a positive characteristic field, we may obtain an affine space with a projectively flat connection on it. We give a set of differential equations which controls the behavior of the connection under endomorphism of the Weyl algebra. The key is the theory of $p$-curvatures.

Next we introduce a field $\mathbb{Q}_{\mathcal{U}}^{(\infty)}$ of characteristic zero as a limit of fields of positive characteristics. We need to fix an ultrafilter on the set of prime numbers to do this. The field is actually isomorphic to the field $\mathbb{C}$ of complex numbers.

Then we show that we may associate with a Weyl algebra over the field $\mathbb{Q}_{\mathcal{U}}^{(\infty)}$ an affine space with a symplectic form in a functorial way. That means, the association is done in such a way that an endomorphism of the Weyl algebra induces a symplectic map of the affine space.

As a result, we show that a solution of the Jacobian conjecture is sufficient for an affirmative answer to the Dixmier conjecture.

We construct derived equivalences between group graded symmetric algebras, starting from equivalences between their $1$-components obtained via a construction of J. Rickard. This applies to the verification of various cases of Broué's abelian defect group conjecture.

We give integration formulas in order to calculate Igusa local zeta functions of prehomogeneous vector spaces. By the integration formula, we determine explicit forms of Igusa local zeta functions of four prehomogeneous vector spaces.

In Theorem 1 of this paper, we establish the necessary and sufficient condition for the values of a power series, a Lambert series, and an infinite product generated by a linear recurrence at the same set of algebraic points to be algebraically dependent. In Theorem 4, from which Theorems 1--3 are deduced, we obtain an easily confirmable condition under which the values more general than those considered in Theorem 1 are algebraically independent, improving the method of [5].