Geometric Sequence Research Articles

Multilayer quantum secret sharing based on GHZ state and generalized Bell basis measurement in multiparty agents

A multilayer quantum secret sharing protocol based on GHZ state is proposed. Alice has the secret carried by quantum state and wants to distribute this secret to multiple agent nodes in the network. In this protocol, the secret is transmitted and shared layer by layer from root Alice to layered agents. The number of agents in each layer is a geometric sequence with a specific common ratio. By sharing GHZ maximally entangled states and making generalized Bell basis measurement, one qubit state can be distributed to multiparty agents and the secret is shared. Only when all agents at the last layer cooperate together, the secret can be recovered. Compared with other protocols based on the entangled state, this protocol adopts layered construction so that secret can be distributed to more agents with fewer particles GHZ state. This quantum secret sharing protocol can be used in wireless network to ensure the security of information delivery.

A perturbation of the geometric spectral sequence in Khovanov homology

We study the relationship between Bar-Natan’s perturbation in Khovanov homology and Szabó’s geometric spectral sequence, and construct a link invariant that generalizes both into a common theory. We study a few properties of the new invariant, and introduce a family of s -invariants from the new theory in the same spirit as Rasmussen’s s -invariant.

Differential Topology of Semimetals

The subtle interplay between local and global charges for topological semimetals exactly parallels that for singular vector fields. Part of this story is the relationship between cohomological semimetal invariants, Euler structures, and ambiguities in the torsion of manifolds. Dually, a topological semimetal can be represented by Euler chains from which its surface Fermi arc connectivity can be deduced. These dual pictures, and the link to topological invariants of insulators, are organised using geometric exact sequences. We go beyond Dirac-type Hamiltonians and introduce new classes of semimetals whose local charges are subtle Atiyah-Dupont-Thomas invariants globally constrained by the Kervaire semicharacteristic, leading to the prediction of torsion Fermi arcs.

Self-Organized Micro-Spiral of Single-Walled Carbon Nanotubes

Single-walled carbon nanotubes (SWCNTs) are reported to spontaneously align in a rotational pattern by drying a liquid droplet of toluene containing polyfluorene as a dispersant. By situating a droplet of an SWCNT solution around a glass bead, spiral patterns are generated. The parallel alignment of SWCNTs along one stripe of such a pattern is confirmed using scanning electron microscopy and polarized optical microscopy. The orientation order increases toward the outer edge of a stripe. The stripe width in the pattern is proportional to the solute concentration, and the width and position of the stripes follow geometric sequences. The growth of the rotational pattern is also observed in real time. The process of spiral pattern formation is visualized, indicating the role of the annihilation of counter-traveling accompanied by continuous depinning. The geometric sequences for the stripe width and position are explained by the near-constant traveling speed and solute enrichment at the droplet periphery.

English

English

Homotopy Types of Frobenius Complexes

Let Λ be a submonoid of the additive monoid \({\mathbb{N}}\). There is a natural order on Λ defined by \({\lambda \leq \lambda +\mu}\) for \({\lambda,\mu \in \Lambda}\). A Frobenius complex of Λ is defined to be the order complex of an open interval of Λ. Suppose \({r \geq 2}\) and let \({\rho}\) be a reducible element of Λ. We construct the additive monoid \({\Lambda[\rho/r]}\) obtained from Λ by adjoining a solution to the equation \({r\alpha=\rho}\). We show that any Frobenius complex of \({\Lambda[\rho/r]}\) is homotopy equivalent to a wedge of iterated suspensions of Frobenius complexes of Λ. As a consequence, we derive a formula for the multi-graded Poincare series associated to \({\Lambda[\rho/r]}\). As an application, we determine the homotopy types of the Frobenius complexes of some additive monoids. For example, we show that if Λ is generated by a finite geometric sequence, then any Frobenius complex of Λ is homotopy equivalent to a wedge of spheres.

Large fluctuations and growth rates of linear Volterra summation equations

This paper concerns the asymptotic behaviour of solutions of a linear convolution Volterra summation equation with an unbounded forcing term. In particular, we suppose the kernel is summable and ascribe growth bounds to the exogenous perturbation. If the forcing term grows at a geometric rate asymptotically, or is bounded by a geometric sequence, then the solution (appropriately scaled) omits a convenient asymptotic representation. Moreover, this representation is used to show that additional growth properties of the perturbation are preserved in the solution. If the forcing term fluctuates asymptotically, we prove that fluctuations of the same magnitude will be present in the solution and we also connect the finiteness of time averages of the solution with those of the perturbation. Our results, and corollaries thereof, apply to stochastic as well as deterministic equations, and we demonstrate this by studying some representative classes of examples.

Incorporating an allosteric regulatory site in an antibody through backbone design.

Allosteric regulation underlies living cells' ability to sense changes in nutrient and signaling-molecule concentrations, but the ability to computationally design allosteric regulation into non-allosteric proteins has been elusive. Allosteric-site design is complicated by the requirement to encode the relative stabilities of active and inactive conformations of the same protein in the presence and absence of both ligand and effector. To address this challenge, we used Rosetta to design the backbone of the flexible heavy-chain complementarity-determining region 3 (HCDR3), and used geometric matching and sequence optimization to place a Zn2+ -coordination site in a fluorescein-binding antibody. We predicted that due to HCDR3's flexibility, the fluorescein-binding pocket would configure properly only upon Zn2+ application. We found that regulation by Zn2+ was reversible and sensitive to the divalent ion's identity, and came at the cost of reduced antibody stability and fluorescein-binding affinity. Fluorescein bound at an order of magnitude higher affinity in the presence of Zn2+ than in its absence, and the increase in fluorescein affinity was due almost entirely to faster fluorescein on-rate, suggesting that Zn2+ preorganized the antibody for fluorescein binding. Mutation analysis demonstrated the extreme sensitivity of Zn2+ regulation on the atomic details in and around the metal-coordination site. The designed antibody could serve to study how allosteric regulation evolved from non-allosteric binding proteins, and suggests a way to designing molecular sensors for environmental and biomedical targets.

Adaptive Weighted Sensing With Simultaneous Transmission for Dynamic Primary User Traffic

In practical scenarios with random arrival and departure of primary users (PUs), existing simultaneous sensing and transmission schemes allocated the same weight to each sample, and did not consider low signal-to-noise ratio (SNR) situations. This paper proposes an adaptive weighted sensing scheme with simultaneous transmission for dynamic PU traffic. It uses a power function based on the corresponding sampling sequence and could reveal the actual PU state in near real time. The power exponent is further adjusted to the sensing situations to achieve lowest false alarm probability under a certain detection probability constraint. Then, an analytical model considering all possible PU state transitions is developed to evaluate achievable interference, throughput, and energy efficiency. Furthermore, the optimal frame duration yielding both optimal false alarm probability and throughput is computed. After that, a fast search algorithm is proposed to track the optimal duration at an exponential convergence rate. Simulation results are provided to validate the analytical model and demonstrate the improvement in low SNR. The results indicate that the proposed scheme can achieve lower false alarm probability and higher energy efficiency over a wide SNR range than that of the existing weighting schemes, which are based on probability, geometric sequence, and equal weighting.

Interleaved Sequences of Geometric Sequences Binarized with Legendre Symbol of Two Types

A pseudorandom number generator is widely used in cryptography. A cryptographic pseudorandom number generator is required to generate pseudorandom numbers which have good statistical properties as well as unpredictability. An m-sequence is a linear feedback shift register sequence with maximal period over a finite field. M-sequences have good statistical properties, however we must nonlinearize m-sequences for cryptographic purposes. A geometric sequence is a binary sequence given by applying a nonlinear feedforward function to an m-sequence. Nogami, Tada and Uehara proposed a geometric sequence whose nonlinear feedforward function is given by the Legendre symbol. They showed the geometric sequences have good properties for the period, periodic autocorrelation and linear complexity. However, the geometric sequences do not have the balance property. In this paper, we introduce geometric sequences of two types and show some properties of interleaved sequences of the geometric sequences of two types. These interleaved sequences have the balance property and double the period of the geometric sequences by the interleaved structure. Moreover, we show correlation properties and linear complexity of the interleaved sequences. A key of our observation is that the second type geometric sequence is the complement of the left shift of the first type geometric sequence by half-period positions.

Composed Reed Solomon Sequences Generated by ith Partial Sum of Geometrical Sequences

Reed–Solomon codes are an important group of error-correcting that were introduced by Irving S. Reed and Justine Solomon in 1960. They used in the error coding control, special in systems that have two way communication channels in two externally applications: deep telecommunications and the compact disc. They have many important applications, the most prominent of which include consumer technologies such as CDs, DVDs, Blue-ray Discs, QR Codes, data transmission, technologies such as DSL and Wi MAX, broadcast, systems such as DVB and ATSC, and storage systems such as RAID 6 . They are also used in satellite communication. This research is useful to generate new Reed Solomon Codes and their composed sequences using the ith partial sum of geometrical sequences with the bigger lengths and the bigger minimum distance that assists to increase secrecy of these information and increase the possibility of correcting mistakes resulting in the channels of communication.

Application of geometric calculus in numerical analysis and difference sequence spaces

The main purpose of this paper is to introduce the geometric difference sequence space ℓ∞G(ΔG) and prove that ℓ∞G(ΔG) is a Banach space with respect to the norm ‖.‖ΔGG. Also we compute the α-dual, β-dual and γ-dual spaces. Finally we obtain the Geometric Newton–Gregory interpolation formulae.

Improved results on perturbed T-S fuzzy systems with mixed delays using geometric sequence division related partitioning methods

This paper addresses improved stability results for T-S fuzzy systems with mixed delays and nonlinear perturbations. By introducing the geometric sequence division (GSD) method, the discrete delay interval can be separated into multiple subintervals with unequal lengths based on the common ratio α. Meanwhile integral partitioning method is applied to deal with the distributed delay. A Lyapunov-Krasovskii functional (LKF) is newly established with augmented factors and triple integral terms which are constructed by means of the length of every subintervals. In addition, in order to reduce the enlargement when we deal with the estimation of the LKF derivative, a free-matrix-based integral inequality, an extended reciprocal convex combination, and free weight matrices techniques are employed. A stability analysis of the delayed T-S fuzzy systems is presented with much less conservative criteria. At the end numerical examples are given to demonstrate the significant improvements of this proposed design.

Impact of drying-grinding sequence on loblolly pine chips preprocessing effectiveness

The high amount of energy consumed during the processing of biomass inflates the production cost of bio-fuels and bio-products. This study was therefore carried out to evaluate the influence of drying and grinding sequence on biomass energy consumption and properties (specific grinding energy requirement, specific drying energy requirement, particle size distribution, bulk density, tap density, aspect ratio, and flow index of grinds). Among the six drying-grinding sequences (12-S, 30-S, 50-S, 30-L-12-S, 50-L-12-S, and 50-L-30-S), sequence 50–S had the lowest geometric mean diameter and sequence 30–S had the highest geometric mean diameter. Drying-grinding sequence significantly affected the bulk density and tap density of grinds. The bulk density of grinds varied from 267.1kg/m3 (sequence 12-S) to 98.0kg/m3 (sequence 50-S) while the tap density ranged from 342.6kg/m3 (sequence 12-S) to 153.7kg/m3 (sequence 50-S). This variation in the densities followed the changes the particle morphology and aspect ratio distribution. A trade-off was noted between specific grinding energy requirement and specific drying energy requirement. The grinding of samples at higher moisture content consumed higher grinding energy than the grinding of samples at lower moisture content. Moisture loss during grinding however increased as the moisture content of woodchips increased, hence resulted in lower drying energy requirements. The least specific drying energy consumption was (1581kJ/kg d.b.) for sequence 50-S and the highest specific drying energy consumption was (3933kJ/kg d.b.) for sequence 12-S.

Geometric sequence based multipath routing protocol for multi-hop ad hoc networks

Purpose The purpose of this paper is to propose, a new multi-path routing protocol that distributes packets over the available paths between a sender and a receiver in a multi-hop ad hoc network. We call this protocol Geometric Sequence Based Multipath Routing Protocol (GMRP). Design/methodology/approach GMRP distributes packets according to the geometric sequence. GMRP is evaluated using GloMoSim simulator. The authors use packet delivery ratio and end-to-end delay as the comparison performance metrics. They also vary many network configuration parameters such as number of nodes, transmission rate, mobility speed and network area. Findings The simulation results show that GMRP reduces the average end-to-end delay by up to 49 per cent and increases the delivery ratio by up to 8 per cent. Originality/value This study is the first to propose to use of geometric sequence in the multipath routing approach.

Improved results on nonlinear perturbed T-S fuzzy systems with interval time-varying delays using a geometric sequence division method.

This paper presents improved stability results by introducing a new delay partitioning method based on the theory of geometric progression to deal with T–S fuzzy systems in the appearance of interval time-varying delays and nonlinear perturbations. A common ratio alpha is applied to split the delay interval into multiple unequal subintervals. A modified Lyapunov–Krasovskii functional (LKF) is constructed with triple-integral terms and augmented factors including the length of every subintervals. In addition, the recently developed free-matrix-based integral inequality is employed to combine with the extended reciprocal convex combination and free weight matrices techniques for avoiding the overabundance of the enlargement when deducing the derivative of the LKF. Eventually, this developed research work can efficiently obtain the maximum upper bound of the time-varying delay with much less conservatism. Numerical results are conducted to illustrate the remarkable improvements of this proposed method.

On the GF(p) linear complexity of Hall’s sextic sequences and some cyclotomic-set-based sequences

Klapper (1994) showed that there exists a class of geometric sequences with the maximal possible linear complexity when considered as sequences over GF(2), but these sequences have very low linear complexities when considered as sequences over GF(p) (p is an odd prime). This linear complexity of a binary sequence when considered as a sequence over GF(p) is called GF(p) complexity. This indicates that the binary sequences with high GF(2) linear complexities are inadequate for security in the practical application, while, their GF(p) linear complexities are also equally important, even when the only concern is with attacks using the Berlekamp-Massey algorithm [Massey, J. L., Shift-register synthesis and bch decoding, IEEE Transactions on Information Theory, 15(1), 1969, 122–127]. From this perspective, in this paper the authors study the GF(p) linear complexity of Hall’s sextic residue sequences and some known cyclotomic-set-based sequences.

Superlubricity in quasicrystalline twisted bilayer graphene

The unique atomic positions in quasicrystals lead to peculiar self-similarity and fractal-like structural morphology. Accordingly, many of the material properties are supposed to manifest exceptional characteristics. In this Rapid Communication, we explain through numerical simulations the fundamental and peculiar aspects of quasicrystals wearless friction manifested in a 30\ifmmode^\circ\else\textdegree\fi{} twisted bilayer graphene system. In particular, the sliding force exhibits a fractal structure with distinct area correlations due to the natural mixture between both periodic and aperiodic lateral modulations. In addition, zero power scaling of the sliding force with respect to the contact area is demonstrated for a geometric sequence of dodecagonal elements.

Numerical Semigroups on Compound Sequences

We generalize the geometric sequence {ap, ap−1b, ap−2b2,…, bp} to allow the p copies of a (resp. b) to all be different. We call the sequence {a1a2a3… ap, b1a2a3…ap, b1b2a3…ap,…, b1b2b3…bp} a compound sequence. We consider numerical semigroups whose minimal set of generators form a compound sequence, and compute various semigroup and arithmetical invariants, including the Frobenius number, Apéry sets, Betti elements, and catenary degree. We compute bounds on the delta set and the tame degree.

Beckett's Watt and Monastic Tradition

The enigmatic setting of Beckett's novel Watt has been compared to places as diverse as an insane asylum, a boarding school, a womb, and a concentration camp. Watt's experience at Knott's house does seem suggestive of all of these, and yet it may more readily conform to the setting of a monastery. The novel is filled with chants, meditations, choral arrangements, hierarchical classifications, and even silence, all highly evocative of a monastic lifestyle. Some of Watt's dialogue (such as his requests for forgiveness or reflections on the nature of mankind) further echoes various Catholic liturgies. Watt finds little solace in these activities, however. He feels that they are largely rote and purposeless as they are focused on Knott, a figure who in many ways defies linguistic description and physical know-ability. Watt's meditations and rituals become, then, empty catechisms without answers, something that is reflected in the extreme difficulty that Watt has communicating. In the face of linguistic and liturgical instability, the Watt notebooks present a counter reading that can be found in the thousand plus doodles that line its pages. The drawings reinforce as well as subvert their textual counterpart, and they function in many ways as the images in medieval illuminated manuscripts. The doodles in Watt often take the form of decorative letters, elaborate marginal drawings, and depictions of a variety of people and animals, and many of its doodles offer uncanny resemblances in form or theme to those in illuminated manuscripts like The Book of Kells. Doodles of saints, monks, crosses, and scribes even give an occasional pictorial nod to the monastic setting in which illuminated manuscripts were usually produced (and remind us of the monastic conditions in which Beckett found himself writing much of Watt). Beckett's doodles not only channel this medium of illuminated manuscripts, they also modernize its application. Instead of neat geometric shapes extending down the page, his geometric doodle sequences are often abstracted, fragmented, and nonlinear. Beckett also occasionally modernized the content of illuminated manuscripts: instead of the traditional sacramental communion table filled with candles, bread and wine, Beckett doodles a science lab table where Bunsen burners replaces candles and wine glasses function as beakers. It is through these modernized images that Watt attempts to draw contemporary relevance from a classic art form and to restore (at least partial) meaning to rote traditions.