This paper studies precise estimates of integral kernels of some integral operators on the boundary $\partial D$ of bounded and strictly convex domains with sufficiently regular boundary. Assume that an integral operator $K_{\mu }$ on $\partial D$ has the integral kernel $K_{\mu }(x,y)$ with estimate $\left|K_{\mu }\right(x,y\left)\right|\le C\mu e^{-\mu |x-y|}$ ($x,y\in \partial D$, $\mu \gg 1$). Then, from the Neumann series, the operator $K_{\mu }(I-K_{\mu }{)}^{-1}$ is also an integral operator. The problem is whether the integral kernel of $K_{\mu }(I-K_{\mu }{)}^{-1}$ can be estimated by the term $\mu e^{-\mu |x-y|}$ up to a constant or not. If the boundary $\partial D$ is strictly convex, such types of estimates hold.

The most important point is that the obtained estimates have the same decaying behavior as $\mu \to \infty$ and the same exponential term as for the original kernel $K_{\mu }(x,y)$. These advantages are essentially needed to handle some inverse initial boundary value problems whose governing equation is the heat equation in three dimensions.

In this paper, we show the so-called weak duality theorem of Tannaka type for an inductive limit–type topological group $G={lim\hspace{0.17em}}_{n\to \infty }{G}_n$ in the case where each $G_n$ is a locally compact group, and $G_n$ is embedded into $G_{n+1}$ homeomorphically as a closed subgroup. First, we explain what a weak duality theorem of Tannaka type is and explain the difference between the case of locally compact groups and the case of nonlocally compact groups. Then we introduce the concept “separating system of unitary representations (SSUR),” which assures the existence of sufficiently many unitary representations. The present $G$ has an SSUR. We prove that $G$ is complete. We give semiregular representations and their extensions for $G$. Using them, we deduce a fundamental formula about “birepresentation” on $G$. Combining these results, we can prove the weak duality theorem of Tannaka type for $G$.

We introduce four different notions of weak Tannaka-type duality theorems, and we define three categories of topological groups, called T-type groups, strongly T-type groups, and NOS-groups.

We call a one-parameter subgroup a nontrivial homomorphic image of the additive group $\mathbf{R}$ of real numbers into a topological group $G$. When $G$ does not contain any one-parameter subgroup, we call $G$ a NOS-group.

The aim of this paper is to show the following relations. In the table below, the symbol $⟺$ means that for a given topological group $G$ the duality theorem on the left-hand side holds if and only if $G$ is of type cited on the right-hand side:

(1) u-duality $⟺$ T-type,

(2) i-duality $⟺$ strongly T-type,

(3) b-duality $⟺$ locally compact,

(4) c-duality $⟺$ locally compact NOS.

We give in the last section some examples which show the actual differences among (1)–(4).

We conjecture that the Joyce–Song wall-crossing formula for Donaldson–Thomas invariants arises naturally from an asymptotic expansion in the field-theoretic work of Gaiotto, Moore, and Neitzke. This would also give a new perspective on how the formulae of Joyce and Song and of Kontsevich and Soibelman are related. We check the conjecture in many examples.

Let $R$ be a commutative Noetherian one-dimensional domain containing $\mathbb{Q}$. In this paper we prove that if an $R$-algebra $A$ is such that $A^{\left[n\right]}{\cong }_R{R}^{[n+2]}$, for some $n\ge 1$, then $A{\cong }_R{R}^{\left[2\right]}$. In terms of affine fibrations this means that every stably trivial ${\mathbb{A}}^2$-fibration over $R$ is actually trivial. On the other hand, it is known that this result does not hold in general if $R$ has dimension at least two or if $R$ does not contain $\mathbb{Q}$.

In this work we deal with vector bundles of rank two on a Fano manifold $X$ with second and fourth Betti numbers equal to one. We study the nef and pseudoeffective cones of the corresponding projectivizations and how these cones are related to the decomposability of the vector bundle. As consequences, we obtain the complete list of ${\mathbb{P}}^1$-bundles over $X$ that have a second ${\mathbb{P}}^1$-bundle structure, classify all the uniform rank two vector bundles on this class of Fano manifolds, and show the stability of indecomposable Fano bundles (with one exception on ${\mathbb{P}}^2$).

For a block $b$ of a normal subgroup of a finite group $G$, E. C. Dade has defined a normal subgroup $G\left[b\right]$ of the inertial group of $b$ in $G$. Let $S_G^0\left(b\right)$ be the subgroup of $G$ consisting of all elements of $G$ fixing all irreducible characters of height 0 in $b$. Under the Alperin–McKay conjecture we show that $S_G^0\left(b\right)/G\left[b\right]$ has a normal Sylow $p$-subgroup. Using this theorem, we show that (under the Alperin–McKay conjecture) the class-preserving outer automorphism group ${Out}_c\left(G\right)$ of a group $G$ has $p$-length at most one for any prime $p$. This rectifies C. H. Sah’s incorrect proof that this group is solvable (under the Schreier conjecture). We obtain also other results on the structures of $S_G^0\left(b\right)/G\left[b\right]$ and ${Out}_c\left(G\right)$ which are derived from the Alperin–McKay conjecture. Main results of the present paper depend on the classification theorem of finite simple groups.