Indagationes Mathematicae-New Series最新文献

We study non-linear traces of the Choquet type and the Sugeno type on the algebra of compact operators. They have certain partial additivities. We show that these partial additivities characterize non-linear traces of both the Choquet type and the Sugeno type respectively. There exists a close relation between non-linear traces of the Choquet type and majorization theory. We study trace class operators for non-linear traces of the Choquet type. More generally we discuss Schatten–von Neumann $p$-class operators for non-linear traces of the Choquet type. We determine when they form Banach spaces. This is an attempt at non-commutative integration theory for non-linear traces of the Choquet type on the algebra of compact operators. We also consider the triangle inequality for non-linear traces of the Sugeno type.

We prove a version of the Mayer–Vietoris sequence for De Rham differential forms in diffeological spaces. It is based on the notion of a generating family instead of that of a covering by open subsets.

There are many research available on the study of a real-valued fractal interpolation function and fractal dimension of its graph. In this paper, our main focus is to study the dimensional results for a vector-valued fractal interpolation function and its Riemann–Liouville fractional integral. Here, we give some results which ensure that dimensional results for vector-valued functions are quite different from real-valued functions. We determine interesting bounds for the Hausdorff dimension of the graph of a vector-valued fractal interpolation function. We also obtain bounds for the Hausdorff dimension of the associated invariant measure supported on the graph of a vector-valued fractal interpolation function. Next, we discuss more efficient upper bound for the Hausdorff dimension of measure in terms of probability vector and contraction ratios. Furthermore, we determine some dimensional results for the graph of the Riemann–Liouville fractional integral of a vector-valued fractal interpolation function.

In an author’s previous work, analytic 1-submanifolds had been classified w.r.t. their symmetry under a given regular and separately analytic Lie group action on an analytic manifold. It was shown that such an analytic 1-submanifold is either free or (via the exponential map) analytically diffeomorphic to the unit circle or an interval. In this paper, we show that each free analytic 1-submanifold is discretely generated by the symmetry group, i.e., naturally decomposes into countably many symmetry free segments that are mutually and uniquely related by the Lie group action. This is shown under the same assumptions that were used in the author’s previous work to prove analogous decomposition results for analytic immersive curves. Together with the results obtained there, this completely classifies 1-dimensional analytic objects (analytic curves and analytic 1-submanifolds) w.r.t. their symmetry under a given regular and separately analytic Lie group action.

We prove that every non-degenerate toric variety, every homogeneous space of a connected linear algebraic group without non-constant invertible regular functions, and every variety covered by affine spaces admit a surjective morphism from an affine space.

Charts are oriented labeled graphs in a disk. Any simple surface braid (2-dimensional braid) can be described by using a chart. Also, a chart represents an oriented closed surface embedded in 4-space. In this paper, we investigate embedded surfaces in 4-space by using charts. Let $\Gamma$ be a chart, and we denote by ${\Gamma }_m$ the union of all the edges of label $m$. A chart $\Gamma$ is of type $(4,3)$ if there exists a label $m$ such that $w\left(\Gamma \right)=7$, $w({\Gamma }_m\cap {\Gamma }_{m+1})=4$, $w({\Gamma }_{m+1}\cap {\Gamma }_{m+2})=3$ where $w\left(G\right)$ is the number of white vertices in $G$. In this paper, we prove that there is no minimal chart of type $(4,3)$.

In this paper, we provide several characterisations for uniform amenability concerning a family of finitely generated groups. More precisely, we show that the Hulanicki–Reiter condition for uniform amenability can be weakened in several directions, including cardinalities of supports and certain operator norms.

Let $0<\theta ⩽1$. A sequence of positive integers ${\left(b_n\right)}_{n=1}^{\infty }$ is called a weak greedy approximation of $\theta$ if ${\sum }_{n=1}^{\infty }1/b_n=\theta$. We introduce the weak greedy approximation algorithm (WGAA), which, for each $\theta$, produces two sequences of positive integers $\left(a_n\right)$ and $\left(b_n\right)$ such that

(a) ${\sum }_{n=1}^{\infty }1/b_n=\theta$;

(b) $1/a_{n+1}<\theta -{\sum }_{i=1}^{n}1/b_i<1/(a_{n+1}-1)$ for all $n⩾1$;

(c) there exists $t⩾1$ such that $b_n/a_n⩽t$ infinitely often.

We then investigate when a given weak greedy approximation $\left(b_n\right)$ can be produced by the WGAA. Furthermore, we show that for any non-decreasing $\left(a_n\right)$ with $a_1⩾2$ and $a_n\to \infty$, there exist $\theta$ and