Graph Realizations Constrained by Connected Local Dimensions and Connected Local Bases

For an ordered set W = {w1,w2, ...,wk} of k distinct vertices in a connected graph G, the representation of a vertex v of G with respect to W is the k-vector r(v|W) = (d(v,w1), d(v,w2), ..., d(v,wk)), where d(v,wi) is the distance from v to wi for 1 ≤ i ≤ k. The setW is called a connected local resolving set of G if the representations of every two adjacent vertices of G with respect to W are distinct and the subgraph ⟨W⟩ induced by W is connected. A connected local resolving set of G of minimum cardinality is a connected local basis of G. The connected local dimension cld(G) of G is the cardinality of a connected local basis of G. In this paper, the connected local dimensions of some well-known graphs are determined. We study the relationship between connected local bases and local bases in a connected graph, and also present some realization results.

The Partition Dimension of a Path Graph

Resolving partition is part of graph theory. This article, explains about resolving partition of the path graph, with. Given a connected graph and is a subset of writen . Suppose there is , then the distance between and is denoted in the form . There is an ordered set of -partitions of, writen then the representation of with respect tois the The set of partitions ofis called a resolving partition if the representation of each to is different. The minimum cardinality of the solving-partition to is called the partition dimension of G which is denoted by . Before getting the partition dimension of a path graph, the first step is to look for resolving partition of the graph. Some resolving partitions of path graph, with , and are obtained. Then, the partition dimension of the path graph which is the minimum cardinality of resolving partition, namely pd (Pn)=2Resolving partition is part of graph theory. This article, explains about resolving partition of the path graph, with. Given a connected graph and is a subset of writen . Suppose there is , then the distance between and is denoted in the form . There is an ordered set of -partitions of, writen then the representation of with respect tois the The set of partitions ofis called a resolving partition if the representation of each to is different. The minimum cardinality of the solving-partition to is called the partition dimension of G which is denoted by . Before getting the partition dimension of a path graph, the first step is to look for resolving partition of the graph. Some resolving partitionsof path graph, with , and are obtained. Then, the partition dimension of the path graph which is the minimum cardinality of resolving partition, namely.

A variation of the prime k-tuples conjecture with applications to quantum limits

Let $$\mathcal {H}^{*}=\{h_1,h_2,\ldots \}$$ H ∗ = { h 1 , h 2 , … } be an ordered set of integers. We give sufficient conditions for the existence of increasing sequences of natural numbers $$a_j$$ a j and $$n_k$$ n k such that $$n_k+h_{a_j}$$ n k + h a j is a sum of two squares for every $$k\ge 1$$ k ≥ 1 and $$1\le j\le k.$$ 1 ≤ j ≤ k . Our method uses a novel modification of the Maynard–Tao sieve together with a second moment estimate. As a special case of our result, we deduce a conjecture due to D. Jakobson which has several implications for quantum limits on flat tori.

Probability Models of Distributed Proof Generation for zk-SNARK-Based Blockchains

The paper is devoted to the investigation of the distributed proof generation process, which makes use of recursive zk-SNARKs. Such distributed proof generation, where recursive zk-SNARK-proofs are organized in perfect Mercle trees, was for the first time proposed in Latus consensus protocol for zk-SNARKs-based sidechains. We consider two models of a such proof generation process: the simplified one, where all proofs are independent (like one level of tree), and its natural generation, where proofs are organized in partially ordered set (poset), according to tree structure. Using discrete Markov chains for modeling of corresponding proof generation process, we obtained the recurrent formulas for the expectation and variance of the number of steps needed to generate a certain number of independent proofs by a given number of provers. We asymptotically represent the expectation as a function of the one variable n/m, where n is the number of provers m is the number of proofs (leaves of tree). Using results obtained, we give numerical recommendation about the number of transactions, which should be included in the current block, idepending on the network parameters, such as time slot duration, number of provers, time needed for proof generation, etc.

Application To Lipschitzian and Integral Systems Via Quadruple Coincidence Point in Fuzzy Metric Spaces

Without a partially ordered set, in this manuscript, we investigate quadruple coincidence point (QCP) results for commuting mapping in the setting of fuzzy metric spaces (FMSs). Furthermore, some relevant ndings are presented to generalize some of the previous results in this direction. Ultimately, non-trivial examples and applications to nd a unique solution for Lipschitzian and integral quadruple systems are provided to support and strengthen our theoretical results.

Symmetry-resolved entanglement detection using partial transpose moments

We propose an ordered set of experimentally accessible conditions for detecting entanglement in mixed states. The k-th condition involves comparing moments of the partially transposed density operator up to order k. Remarkably, the union of all moment inequalities reproduces the Peres-Horodecki criterion for detecting entanglement. Our empirical studies highlight that the first four conditions already detect mixed state entanglement reliably in a variety of quantum architectures. Exploiting symmetries can help to further improve their detection capabilities. We also show how to estimate moment inequalities based on local random measurements of single state copies (classical shadows) and derive statistically sound confidence intervals as a function of the number of performed measurements. Our analysis includes the experimentally relevant situation of drifting sources, i.e. non-identical, but independent, state copies.

Mal’tsev bases for partially commutative nilpotent groups

We construct an ordered set of commutators in a partially commutative nilpotent group [Formula: see text]. This set allows us to define a canonical form for each element of [Formula: see text]. Namely, we construct a Mal’tsev basis for the group [Formula: see text]

On a stronger reconstruction notion for monoids and clones

Motivated by reconstruction results by Rubin, we introduce a new reconstruction notion for permutation groups, transformation monoids and clones, called automatic action compatibility, which entails automatic homeomorphicity. We further give a characterization of automatic homeomorphicity for transformation monoids on arbitrary carriers with a dense group of invertibles having automatic homeomorphicity. We then show how to lift automatic action compatibility from groups to monoids and from monoids to clones under fairly weak assumptions. We finally employ these theorems to get automatic action compatibility results for monoids and clones over several well-known countable structures, including the strictly ordered rationals, the directed and undirected version of the random graph, the random tournament and bipartite graph, the generic strictly ordered set, and the directed and undirected versions of the universal homogeneous Henson graphs.

The organizational and economic mechanism of the determinant’s activation of the regional development intensification in the conditions of digitalization

The purpose of the study is to substantiate the organizational and economic mechanism of the determinant’s activation of the regional economic systems development intensification in the context of digitalization. The study uses general and special methods of scientific research, including scientific knowledge, analysis, synthesis, dialectical, monographic, generalizations, functional, process, system and other methods. It is determined that the organizational and economic mechanism of the determinants activation of the development intensification of regional economic systems is an ordered set of objects, subjects, set goals, objectives, priorities, principles and specific management tools in terms of digitalization of economic development. As a result of the research, the conceptual identification of key elements of instrumental support of the determinants activation of the modern economic systems development intensification in the conditions of digitalization was carried out. The normative-legal mechanisms of the regional development intensification are offered.