Blockchain Consensus Mechanism Based on Improved Distributed Consistency and Hash Entropy

To improve the performance for distributed blockchain system, a novel and effective consensus algorithm is designed in this paper. It firstly constructs a more random additive constant through the generation matrix of the error correction code and uses the value of the hash entropy to prove that the constructed hash function can meet the requirements of high throughput and fast consensus in performance. In addition, a distributed consensus coordination service system is used in the blockchain system to realize the synchronization of metadata and ensure the consistency of block data, configuration information, and transaction information. The experiment results show that our proposed strategy can reduce the waste of computing resources, increase the block generation speed, and ensure the fairness of nodes participating in the competition, which is an effective solution to ensure the stable operation of the blockchain system.

The two-loop remainder function for eight and nine particles

Two-loop MHV amplitudes in planar $$ \mathcal{N} $$ N = 4 supersymmetric Yang Mills theory are known to exhibit many intriguing forms of cluster-algebraic structure. We leverage this structure to upgrade the symbols of the eight- and nine-particle amplitudes to complete analytic functions. This is done by systematically projecting onto the components of these amplitudes that take different functional forms, and matching each component to an ansatz of multiple polylogarithms with negative cluster-coordinate arguments. The remaining additive constant can be determined analytically by comparing the collinear limit of each amplitude to known lower-multiplicity results. We also observe that the nonclassical part of each of these amplitudes admits a unique decomposition in terms of a specific A3 cluster polylogarithm, and explore the numerical behavior of the remainder function along lines in the positive region.

Characterisation of homogeneous fractional Sobolev spaces

Our aim is to characterize the homogeneous fractional Sobolev–Slobodeckiĭ spaces $$\mathcal {D}^{s,p} (\mathbb {R}^n)$$ D s , p ( R n ) and their embeddings, for $$s \in (0,1]$$ s ∈ ( 0 , 1 ] and $$p\ge 1$$ p ≥ 1 . They are defined as the completion of the set of smooth and compactly supported test functions with respect to the Gagliardo–Slobodeckiĭ seminorms. For $$s\,p < n$$ s p < n or $$s = p = n = 1$$ s = p = n = 1 we show that $$\mathcal {D}^{s,p}(\mathbb {R}^n)$$ D s , p ( R n ) is isomorphic to a suitable function space, whereas for $$s\,p \ge n$$ s p ≥ n it is isomorphic to a space of equivalence classes of functions, differing by an additive constant. As one of our main tools, we present a Morrey–Campanato inequality where the Gagliardo–Slobodeckiĭ seminorm controls from above a suitable Campanato seminorm.

Private Sequential Learning

Can we learn privately and efficiently through sequential interactions? A private learning model is formulated to study an intrinsic tradeoff between privacy and query complexity in sequential learning. The formulation involves a learner who aims to learn a scalar value by sequentially querying an external database and receiving binary responses. In the meantime, an adversary observes the learner’s queries, although not the responses, and tries to infer from them the scalar value of interest. The objective of the learner is to obtain an accurate estimate of the scalar value using only a small number of queries while simultaneously protecting his or her privacy by making the scalar value provably difficult to learn for the adversary. The main results provide tight upper and lower bounds on the learner’s query complexity as a function of desired levels of privacy and estimation accuracy. The authors also construct explicit query strategies whose complexity is optimal up to an additive constant.

Linear Time Additively Exact Algorithm for Transformation of Chain-Cycle Graphs for Arbitrary Costs of Deletions and Insertions

We propose a novel linear time algorithm which, given any directed weighted graphs a and b with vertex degrees 1 or 2, constructs a sequence of operations transforming a into b. The total cost of operations in this sequence is minimal among all possible ones or differs from the minimum by an additive constant that depends only on operation costs but not on the graphs themselves; this difference is small as compared to the operation costs and is explicitly computed. We assume that the double cut and join operations have identical costs, and costs of the deletion and insertion operations are arbitrary strictly positive rational numbers.

Bayesian analysis of running holographic Ricci dark energy

ABSTRACT Holographic Ricci dark energy evolving through its interaction with dark matter is a natural choice for the running vacuum energy model. We have analysed the relative significance of two versions of this model in the light of type Ia supernovae (SN1a), the Cosmic Microwave Background (CMB), the Baryonic Acoustic Oscillations (BAO), and Hubble data sets using the method Bayesian inferences. The first one, model 1, is the running holographic Ricci dark energy (rhrde) having a constant additive term in its density form and the second is one, model 2, having no additive constant, instead the interaction of rhrde with dark matter (ΛCDM) is accounted through a phenomenological coupling term. The Bayes factor of these models in comparison with the standard Lambda cold dark matter have been obtained by calculating the likelihood of each model for four different data combinations, SNIa(307)+CMB+BAO, SNIa(307)+CMB+BAO+Hubble data, SNIa(580)+CMB+BAO, and SNIa(580)+CMB+BAO+Hubble data. Suitable flat priors for the model parameters has been assumed for calculating the likelihood in both cases. Our analysis shows that, according to the Jeffreys scale, the evidence for ΛCDM against both model 1 and model 2 is very strong as the Bayes factor of both models are much less than one for all the data combinations.

Integrals of functions containing parameters

Calculus students are taught that an indefinite integral is defined only up to an additive constant, and as a consequence generations of students have assiduously added ‘+C’ to their calculus homework. Although ubiquitous, these constants rarely garner much attention, and typically loiter without intent around the ends of equations, feeling neglected. There is, however, useful work they can do, work which is particularly relevant in the contexts of integral tables and computer algebra systems. We begin, therefore, with a discussion of the context, before returning to coax the constants out of the shadows and assign them their tasks.

Nonsingular black holes from charged dust collapse: A concrete mechanism to evade interior singularities in general relativity

In this paper, we examine the gravitational collapse of a nonrelativistic charged perfect fluid interacting with a dark energy component. Given a simple factor for the energy transfer, we obtain a nonsingular interior solution which naturally matches the Reissner–Nordström–de Sitter exterior geometry. We also show that the interacting parameter is proportional to the overall charge of the final black hole thus formed. For the case of quasi-extremal configurations, we propose a statistical model for the entropy of the collapsed matter. This entropy extends Bekenstein’s geometrical entropy by an additive constant proportional to the area of the extremal black hole.

A Degree Sequence Version of the Kühn–Osthus Tiling Theorem

A fundamental result of Kühn and Osthus [The minimum degree threshold for perfect graph packings, Combinatorica, 2009] determines up to an additive constant the minimum degree threshold that forces a graph to contain a perfect $H$-tiling. We prove a degree sequence version of this result which allows for a significant number of vertices to have lower degree.

Running vacuum in quantum field theory in curved spacetime: renormalizing $$\rho _{vac}$$ without $$\sim m^4$$ terms

The $$\Lambda $$Λ-term in Einstein’s equations is a fundamental building block of the ‘concordance’ $$\Lambda $$ΛCDM model of cosmology. Even though the model is not free of fundamental problems, they have not been circumvented by any alternative dark energy proposal either. Here we stick to the $$\Lambda $$Λ-term, but we contend that it can be a ‘running quantity’ in quantum field theory (QFT) in curved space time. A plethora of phenomenological works have shown that this option can be highly competitive with the $$\Lambda $$ΛCDM with a rigid cosmological term. The, so-called, ‘running vacuum models’ (RVM’s) are characterized by the vacuum energy density, $$\rho _{vac}$$ρvac, being a series of (even) powers of the Hubble parameter and its time derivatives. Such theoretical form has been motivated by general renormalization group arguments, which look plausible. Here we dwell further upon the origin of the RVM structure within QFT in FLRW spacetime. We compute the renormalized energy-momentum tensor with the help of the adiabatic regularization procedure and find that it leads essentially to the RVM form. This means that $$\rho _{vac}(H)$$ρvac(H) evolves as a constant term plus dynamical components $${{\mathcal {O}}}(H^2)$$O(H2) and $$\mathcal{O}(H^4)$$O(H4), the latter being relevant for the early universe only. However, the renormalized $$\rho _{vac}(H)$$ρvac(H) does not carry dangerous terms proportional to the quartic power of the masses ($$\sim m^4$$∼m4) of the fields, these terms being a well-known source of exceedingly large contributions. At present, $$\rho _{vac}(H)$$ρvac(H) is dominated by the additive constant term accompanied by a mild dynamical component $$\sim \nu H^2$$∼νH2 ($$|\nu |\ll 1$$|ν|≪1), which mimics quintessence.