We study the asymptotics of large, simple, labeled graphs constrained by the densities of edges and of k-star subgraphs, k ≥ 2 fixed. We prove that under such constraints graphs are “multipodal”: asymptotically in the number of vertices there is a partition of the vertices into M <∞ subsets V1, V2, . . . , VM , and a set of well-defined probabilities gij of an edge between any vi ∈ Vi and vj ∈ ...

In this paper, we prove the following statements: (1) There exist a Tychonoff space X and a subspace Y of X such that Y is strongly star-Lindelöf in X and e(Y, X) is arbitrarily large, but X is not star-Lindelöf. (2) There exist a Tychonoff space X and a subspace Y of X such that Y is star-Lindelöf in X, but Y is not strongly star-Lindelöf in X.

For a fixed $N$, we analyze the space of all sequences $z=(z_1,\dots,z_N)$, approximating continuous function on circle, with given persistence diagram $P$, and show that typical components this are homotopy equivalent to $S^1$. We also consider functions $Y$-shaped (resp., star-shaped) trees 2-point diagram, is $S^1$ bouquet circles).

In the theory of generalized metric spaces, the notion of knetworks has played an important role. Every locally separable metric space or CW-complex, more generally, every space dominated by locally separable metric spaces has a star-countable k-network. Every LaSnev space, as well as, every space dominated by LaSnev spaces has a a-compact-finite knetwork. We recall that every space has a compa...

We explore galaxy assembly over the last seven billion years by characterizing “normal” galaxies along the Hubble sequence, against strongly disturbed merging/interacting galaxies with the widely used CAS system of concentration (C), asymmetry (A), and ‘clumpiness’ (S) parameters, as well as visual classification. We analyze Hubble Space Telescope (HST) ACS images of ∼4000 intermediate and high...

In this paper, we have dened and studied a generalized form of topological vector spaces called s-topological vector spaces. s-topological vector spaces are dened by using semi-open sets and semi-continuity in the sense of Levine. Along with other results, it is proved that every s-topological vector space is generalized homogeneous space. Every open subspace of an s-topological vector space is...

A common measure for the uniformity of point distributions is the star discrepancy. Let λ s denote the s-dimensional Lebesgue measure. Then the star discrepancy of a mul-tiset P = {p 1 ,. .. , p N } ⊂ [0, 1] s is given by D * N (P) := sup α∈[0,1] s λ s ([0, α)) − 1 N N k=1 1 [0,α) (p k) ; here [0, α) denotes the s-dimensional axis-parallel box [0, α 1) × · · · × [0, α s) and 1 [0,α) its charact...

Let S be a topological semigroup acting on a topological space X. We develop the theory of (weakly) almost periodic functions on X, with respect to S, and form the (weakly) almost periodic compactifications of X and S, with respect to each other. We then consider the notion of an action of Son a Banach space, and on its dual, and after defining S-invariant means for such a space, we give a...