We study some properties of positive solutions to the higher order conformally invariant equation with a singular set $$\begin{aligned} (-\Delta )^m u = u^{\frac{n+2m}{n-2m}} ~~~~~~ \text {in} ~ \Omega \backslash \Lambda , \end{aligned}$$ where $$\Omega \subset {\mathbb {R}}^n$$ is an open domain, $$\Lambda $$ closed subset $${\mathbb $$1 \le m < n/2$$ and integer. first establish local estimat...

We apply the Gromov–Hausdorff metric $$d_G$$ for characterization of certain generalized manifolds. Previously, we have proven that with respect to $$d_G,$$ n-manifolds are limits spaces which obtained by gluing two topological a controlled homotopy equivalence (the so-called 2-patch spaces). In present paper, consider manifold-like $$X^{n},$$ introduced in 1966 Mardeić and Segal, characterized...

There are many different types of lenses, but largely they fall into the three classes of the title: set-based, delta-based and edit-based lenses. This paper develops some of the general relationships between those classes. The main results are that a category of set-based lenses is a full subcategory of a category of delta-based lenses determined by sending sets to codiscrete categories; that ...

If a and b are positive integers with a ≤ b and a ≡ a mod b, then the set Ma,b = {x ∈ N : x ≡ a mod b or x = 1} is a multiplicative monoid known as an arithmetical congruence monoid (or ACM). For any monoid M with units M× and any x ∈ M \M× we say that t ∈ N is a factorization length of x if and only if there exist irreducible elements y1, . . . , yt of M and x = y1 · · · yt. Let L(x) = {t1, . ...

In many applications it is of great importance to handle evolution equations about random closed sets of different (even though integer) Hausdorff dimensions, including local information about initial conditions and growth parameters. Following a standard approach in geometric measure theory such sets may be described in terms of suitable measures. For a random closed set of lower dimension wit...

Clustering has been an active research area of great practical importance for recent years. Most previous clustering models have focused on grouping objects with similar values on a (sub)set of dimensions (e.g., subspace cluster) and assumed that every object has an associated value on every dimension (e.g., bicluster). These existing cluster models may not always be adequate in capturing coher...

Let S = 〈n1, n2, n3〉 be a numerical monoid of embedding dimension 3. We characterize the positive integers m such that m ∈ ∆(S). If n1 = 3, we show that {23 3 − 2} ⊆ ∆(S) ⊆ [1, n2+n3 3 − 2]∩N. We conclude by determining necessary and sufficient conditions on n1, n2 and n3 so that ∆(S) is a singleton.

Discretization of singular functions is an important component in many problems to which level set methods have been applied. We present two methods for constructing consistent approximations to Dirac delta measures concentrated on piecewise smooth curves or surfaces. Both methods are designed to be convenient for level set simulations and are introduced to replace the commonly used but inconsi...

Let $K\Delta$ be the incidence algebra associated with a finite poset $(\Delta,\preceq)$ over algebraically closed field $K$. We present study of algebras that are piecewise hereditary, which we denominate PHI algebras. investigate strong global dimension, simply conectedeness and one-point extension We also give positive answer to so-called Skowronski problem for is not wild quiver type. Tha...