Let $k$ be a totally real number field and $p$ prime. We show that the ``complexity'' of Greenberg's conjecture ($\lambda = \mu 0$) is $p$-adic nature governed (under Leopoldt's conjecture) by finite torsion group ${\mathcal T}_k$ Galois maximal abelian $p$-ramified pro-$p$-extension $k$, means images in ideal norms from layers $k_n$ cyclotomic tower (Theorem (5.2)). These are obtained via form...

Permutations of the form \(F(x)=L_1(x^{-1})+L_2(x)\) with linear functions \(L_1,L_2\) are closely related to several interesting questions regarding CCZ-equivalence and EA-equivalence inverse function. In this paper, we show that F cannot be a permutation on binary fields if kernel \(L_1\) or \(L_2\) is large. A key step our proof an observation maximal size subspace V \(\mathbb {F}_{2^n}\) co...

Abstract Let $$\mu $$ μ be a log-concave probability measure on $${\mathbb R}^n$$ R n and for any $$N>n$$ N > consider the random polytope $$K_N=\textrm{conv}\{X_1,\ldots ,X_N\}$$ ...

An edge-ordered graph is a with total ordering of its edges. A path $$P=v_1v_2\ldots v_k$$ in an called increasing if $$(v_iv_{i+1}) < (v_{i+1}v_{i+2})$$ for all $$i = 1,\ldots,k-2$$ ; and it decreasing > . We say that P monotone or decreasing. rooted tree T either every from the root to leaf Let G be graph. In straight-line drawing D G, vertices are drawn as different points plane edges straig...

We study a natural question in the Iwasawa theory of algebraic curves genus $$>1$$ . Fix prime number p. Let X be smooth, projective, geometrically irreducible curve defined over field K $$g>1$$ , such that Jacobian has good ordinary reduction at primes above an odd p and for any integer $$n>1$$ let $$K_n^{(p)}$$ denote degree- $$p^n$$ extension contained $$K(\mu _{p^{\infty }})$$ prove explici...

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Completely independent spanning trees in a graph G are of such that for any two distinct vertices G, the paths between them pairwise edge-disjoint and internally vertex-disjoint. In this paper, we present tight lower bound on maximum number completely L(G), where L(G) denotes line G. Based new characterization with k trees, also show complete $$K_n$$ order $$n \ge 4$$ , there $$\lfloor \frac{n+...

Let \(2&lt;n&lt;\omega\). Then \({\sf CA}_n\) denotes the class of cylindric algebras dimension \(n\), RCA}_n\) representable \(\sf CA_n\)s, CRCA}_n\) completely CA}_n\)s, and Nr}_n{\sf CA}_{\omega}(\subseteq {\sf CA}_n\)) \(n\)-neat reducts CA}_{\omega}\)s. The elementary closure CRCA}_n\)s (\(\mathbf{K_n}\)) non-elementary At}({\sf CA}_{\omega})\) are characterized using two-player zero-sum g...

The Harary-Hill Conjecture states that the number of crossings in any drawing of the complete graph Kn in the plane is at least Z(n) := 1 4 ⌊ n 2 ⌋ ⌊ n−1 2 ⌋ ⌊ n−2 2 ⌋ ⌊ n−3 2 ⌋ . In this paper, we settle the Harary-Hill conjecture for shellable drawings. We say that a drawing D of Kn is s-shellable if there exist a subset S = {v1, v2, . . . , vs} of the vertices and a region R of D with the fo...

Abstract In this work we analyse bucket increasing tree families. We introduce two simple stochastic growth processes, generating random trees of size n , complementing the earlier result Mahmoud and Smythe (1995, Theoret. Comput. Sci. 144 221–249.) for recursive trees. On combinatorial side, define multilabelled generalisations families d -ary generalised plane-oriented Additionally, a cluster...