Small Order Research Articles

Bounding the expectation of the supremum of an empirical process over a (weak) VC-major class

Given a bounded class of functions $\mathscr{G}$ and independent random variables $X_{1},\ldots,X_{n}$, we provide an upper bound for the expectation of the supremum of the empirical process over elements of $\mathscr{G}$ having a small variance. Our bound applies when $\mathscr{G}$ is a VC-subgraph or a VC-major class and it is of smaller order than those one could get by using a universal entropy bound over the whole class $\mathscr{G}$. It also involves explicit constants and does not require the knowledge of the entropy of $\mathscr{G}$.

Eigenstrain based reduced order homogenization for polycrystalline materials

Abstract In this manuscript, an eigenstrain based reduced order homogenization method is developed for polycrystalline materials. A two-scale asymptotic analysis is used to decompose the original equations of polycrystal plasticity into micro- and macroscale problems. Eigenstrain based representation of the inelastic response field is employed to approximate the microscale boundary value problem using an approximation basis of much smaller order. The reduced order model takes into account the grain-to-grain interactions through influence functions that are numerically computed over the polycrystalline microstructure. The proposed approach is also endowed with a hierarchical model improvement capability that allows accurate representation of stress and deformation state within subgrains. The proposed approach was implemented and its performance was assessed against crystal plasticity finite element simulations. Numerical studies point to the capability to efficiently compute the mechanical response of the polycrystal RVEs with good accuracy and the ability to capture stress risers near grain boundaries.

Small Regular Graphs of Girth 7

In this paper, we construct new infinite families of regular graphs of girth 7 of smallest order known so far. Our constructions are based on combinatorial and geometric properties of $(q+1,8)$-cages, for $q$ a prime power. We remove vertices from such cages and add matchings among the vertices of minimum degree to achieve regularity in the new graphs. We obtain $(q+1)$-regular graphs of girth 7 and order $2q^3+q^2+2q$ for each even prime power $q \ge 4$, and of order $2q^3+2q^2-q+1$ for each odd prime power $q\ge 5$. A corrigendum was added to this paper on 21 June 2016.

Cyclopsomyces plurioperculatus: a new genus and species of Lobulomycetales (Chytridiomycota, Chytridiomycetes) from Japan

Lobulomycetales is one of the smallest orders of Chytridiomycota, containing only four genera and five species. In a survey in Japan we isolated a chytrid from a soil sample collected in a broadleaf forest, which grouped in Lobulomycetales by BLAST query. To identify this chytrid and determine its taxonomic position, thallus development and morphology were observed by light microscopy and zoospore ultrastructure was examined using a transmission electron microscopy. We conducted a phylogenetic analysis using nuc 28S rDNA sequences. Thallus morphology was characterized by a spherical zoosporangium with multiple operculate discharge papillae, which is different from that of any other species in Lobulomycetales. This chytrid is similar to Chytriomyces multioperculatus in having multiple operculate discharge papillae, but these are distinguished by characters of the discharge papillae and rhizoidal systems. Zoospores of this chytrid had electron-dense material in the kinetosome, a unique character in the order. Our 28S phylogeny placed it in a distinct clade, sister to all described species in Lobulomycetaceae. Based on these results, we propose a new genus and species of Lobulomycetales, Cyclopsomyces plurioperculatus.

Biregular Cages of Odd Girth

Biregular {r,m};g-cages are graphs of girth g that contain vertices of degrees r and m and are of the smallest order among all such graphs. We show that for every ri¾?3 and every odd g=2t+1i¾?3, there exists an integer m0 such that for every evenmi¾?m0, the biregular {r,m},g-cage is of order equal to a natural lower bound analogous to the well-known Moore bound. In addition, when r is odd, the restriction on the parity of m can be removed, and there exists an integer m0 such that a biregular {r,m},g-cage of order equal to this lower bound exists for allmi¾?m0. This is in stark contrast to the result classifying all cages of degree k and girth g whose order is equal to the Moore bound.

Packing Order Splitting Optimization Based on Cloud Model Culture Algorithm

In online-shopping, the size and quantity of most orders are different. So how to pick and pack the objects of orders becomes a critical problem. To solve this problem, a picking and packing integrated model in automated warehouse system is proposed. The objects of one order can be split into smaller orders according to the volume and loading weight of container. Then the objects can be packaged in the suitable containers. A dispatching algorithm based on cloud model and culture algorithm is proposed. The steps of the proposed algorithm are discussed in detail. Finally, the example result shows that it can effectively optimize picking and packing operations and reduce the waste of containers.

Theoretical Studies on Multiphoton Absorption of Ultrashort Laser Pulses in Sapphire

We discuss in detail the multiphoton absorption theory for wide bandgap dielectrics that we have implemented in the framework of the time-dependent Nth order perturbation theory. In particular, we have carried out calculations on the N-photon absorption coefficient K N and interband transition rate W N for a perfect sapphire crystal (α-Al 2 O 3 ). Furthermore, we derive an expression on the laser-pulse induced seed electron density n 0 . Application of this theory on sapphire shows that n 0 , induced by the ultrashort laser pulse with pulse duration of 30 ps, wavelength of 1 μm, and peak intensity of 5 × 10 11 W/cm 2 at the focus point, cannot lead to multiphoton-based ablation. Indeed, our calculations show that the laser pulse in the N = 7 order absorption process generates a conduction electron density n 0 ≈ 6 × 10 13 cm -3 that is far too small compared with the required critical density n cr ≈ 10 19 - 10 21 cm -3 . This is in agreement with the extremely small value of the probability (P ind ≈ 2 × 10 -9 ) that we have estimated for a valence electron to be transferred to the conduction band under the influence of the abovementioned laser-pulse conditions. Therefore, we need to seek other mechanisms than multiphoton absorption, to explain ablation in sapphire. However, we show that by using larger photon energies and smaller order N values, it is possible to induce large enough multiphoton absorption excited electron densities, which will lead to ablation in sapphire, even for the abovementioned laser beam intensity. Finally, we discuss the Keldysh adiabatic parameter y and its interpretation in the case of our experimental pulse-laser setup.

Longest Partial Transversals in Plexes

A k-protoplex of order n is a partial latin square of order n such that each row and column contains precisely k entries and each symbol occurs precisely k times. If a k-protoplex is completable to a latin square, then it is a k-plex. A 1-protoplex is a transversal. Let \({\phi_k}\) denote the smallest order for which there exists a k-protoplex that contains no transversal, and let \({{{\phi_k}^{*}}}\) denote the smallest order for which there exists a k-plex that contains no transversal. We show that \({k \leqslant \phi_k = {{\phi_k}^{*}} \leqslant k+1}\) for all \({k \geqslant 6}\) . Given a k-protoplex P of order n, we define T(P) to be the size of the largest partial transversal in P. We explore upper and lower bounds for T(P). Aharoni et al. have conjectured that \({T(P) \geqslant (k-1)n/k}\) . We find that $$T(P) > max \{ k(1-n^{-1/2}), k-n/(n-k), n-O (nk^{-1/2}log^{3/2} k)\}$$ . In the special case of 3-protoplexes, we improve the lower bound for T(P) to 3n/5.

Generation of a finite group with Hall maximal subgroups by a pair of conjugate elements

For a finite group G, the set of all prime divisors of |G| is denoted by π(G). P. Shumyatsky introduced the following conjecture, which was included in the “Kourovka Notebook” as Question 17.125: a finite group G always contains a pair of conjugate elements a and b such that π(G) = π(〈a, b〉). Denote by \(\mathfrak{Y}\) the class of all finite groups G such that π(H) ≠ π(G) for every maximal subgroup H in G. Shumyatsky’s conjecture is equivalent to the following conjecture: every group from \(\mathfrak{Y}\) is generated by two conjugate elements. Let \(\mathfrak{V}\) be the class of all finite groups in which every maximal subgroup is a Hall subgroup. It is clear that \(\mathfrak{V} \subseteq \mathfrak{Y}\). We prove that every group from \(\mathfrak{V}\) is generated by two conjugate elements. Thus, Shumyatsky’s conjecture is partially supported. In addition, we study some properties of a smallest order counterexample to Shumyatsky’s conjecture.

Synthesis of mesoporous silica materials from municipal solid waste incinerator bottom ash

Incinerator bottom ash contains a large amount of silica and can hence be used as a silica source for the synthesis of mesoporous silica materials. In this study, the conditions for alkaline fusion to extract silica from incinerator bottom ash were investigated, and the resulting supernatant solution was used as the silica source for synthesizing mesoporous silica materials. The physical and chemical characteristics of the mesoporous silica materials were analyzed using BET, XRD, FTIR, SEM, and solid-state NMR. The results indicated that the BET surface area and pore size distribution of the synthesized silica materials were 992m2/g and 2–3.8nm, respectively. The XRD patterns showed that the synthesized materials exhibited a hexagonal pore structure with a smaller order. The NMR spectra of the synthesized materials exhibited three peaks, corresponding to Q2 [Si(OSi)2(OH)2], Q3 [Si(OSi)3(OH)], and Q4 [Si(OSi)4]. The FTIR spectra confirmed the existence of a surface hydroxyl group and the occurrence of symmetric Si–O stretching. Thus, mesoporous silica was successfully synthesized from incinerator bottom ash. Finally, the effectiveness of the synthesized silica in removing heavy metals (Pb2+, Cu2+, Cd2+, and Cr2+) from aqueous solutions was also determined. The results showed that the silica materials synthesized from incinerator bottom ash have potential for use as an adsorbent for the removal of heavy metals from aqueous solutions.

Entire functions sharing an entire function of smaller order with their difference operators

We study a uniqueness question of entire functions sharing an entire function of smaller order with their difference operators, and deal with a question posed by Liu and Yang. The results in this paper extend the corresponding results obtained by Liu-Yang and by Liu-Laine respectively. Examples are provided to show that the results in this paper, in a sense, are the best possible.

Ice Crawlers (Grylloblattodea) – the history of the investigation of a highly unusual group of insects

Grylloblattodea are one of the most unusual groups of insects and the second smallest order. All known extant species are wingless and exhibit a remarkable preference for cold temperatures. Although their morphology was intensively investigated shortly after their discovery, the systematic position has been disputed for a long time. The placement of Grylloblattodea as sister-group to the recently described Mantophasamtodea is supported by morphological and molecular evidence. However, the relationships of this clade, Xenonomia, among the polyneopteran lineages is not clear. Transcriptome analyses, in addition to the study of winged grylloblattodean fossils, may help to clarify the position of Xenonomia and aid in the reconstruction of the “phylogenetic backbone” of Polyneoptera.

Analysis on China's 'Reform and Opening-Up': Based on the Features of Institutional System

Institutional reform needs to recognize the institutional rules from the perspective of the institutional system. Based on the system theory, this paper analyzes two features of the institutional system — the order and the irreversibility. Institutional goal is the steady state of institutional system. The closer to steady state institutional system is, the smaller order is, and the greater disorder is. Meanwhile the order moving to the goal of the institutional system is an irreversible process, so the only way that can change the disorder state is to change institutional goal. In this paper, above conclusions were verified from China’s “Reform and Opening-up”, and the evolutional path was summarized and two suggestions for reform were given.

DIVERSE STRUCTURE SYNCHRONIZATION OF FRACTIONAL ORDER HYPER-CHAOTIC SYSTEMS

This paper studied the dynamic behavior of the fractional order hyper-chaotic Lorenz system and the fractional order hyper-chaotic Rössler system, then numerical analysis of the different fractional orders hyper-chaotic systems are carried out under the predictor–corrector method. We proved the two systems are in hyper-chaos when the maximum and the second largest Lyapunov exponential are calculated. Also the smallest orders of the systems are proved when they are in hyper-chaos. The diverse structure synchronization of the fractional order hyper-chaotic Lorenz system and the fractional order hyper-chaotic Rössler system is realized using active control method. Numerical simulations indicated that the scheme was always effective and efficient.

Representation of cyclotomic fields and their subfields

Let $$\mathbb{K}$$ be a finite extension of a characteristic zero field $$\mathbb{F}$$ . We say that a pair of n × n matrices (A,B) over $$\mathbb{F}$$ represents $$\mathbb{K}$$ if $$\mathbb{K} \cong {{\mathbb{F}\left[ A \right]} \mathord{\left/ {\vphantom {{\mathbb{F}\left[ A \right]} {\left\langle B \right\rangle }}} \right. \kern-\nulldelimiterspace} {\left\langle B \right\rangle }}$$ , where $$\mathbb{F}\left[ A \right]$$ denotes the subalgebra of $$\mathbb{M}_n \left( \mathbb{F} \right)$$ containing A and 〈B〉 is an ideal in $$\mathbb{F}\left[ A \right]$$ , generated by B. In particular, A is said to represent the field $$\mathbb{K}$$ if there exists an irreducible polynomial $$q\left( x \right) \in \mathbb{F}\left[ x \right]$$ which divides the minimal polynomial of A and $$\mathbb{K} \cong {{\mathbb{F}\left[ A \right]} \mathord{\left/ {\vphantom {{\mathbb{F}\left[ A \right]} {\left\langle {q\left( A \right)} \right\rangle }}} \right. \kern-\nulldelimiterspace} {\left\langle {q\left( A \right)} \right\rangle }}$$ . In this paper, we identify the smallest order circulant matrix representation for any subfield of a cyclotomic field. Furthermore, if p is a prime and $$\mathbb{K}$$ is a subfield of the p-th cyclotomic field, then we obtain a zero-one circulant matrix A of size p × p such that (A, J) represents $$\mathbb{K}$$ , where J is the matrix with all entries 1. In case, the integer n has at most two distinct prime factors, we find the smallest order 0, 1-companion matrix that represents the n-th cyclotomic field. We also find bounds on the size of such companion matrices when n has more than two prime factors.

Iterative On-Surface Discretized Boundary Equation Method for 2-D Scattering Problems

The recently developed on-surface discretized boundary equation (OS-DBE) method has low memory requirement and is very suitable for parallel computing because the current at each point can be independently evaluated with a matrix of much smaller order than that in the method of moments (MoM) for electrically large objects. However, repeated solutions of the matrix equation in generating the whole current distribution are still the major computational burden when the scatterer size becomes large. In this paper, an iterative OS-DBE (IT-OS-DBE) method is presented for 2-D scattering problems. It further reduces the OS-DBE matrix order significantly and solves the matrix equation only once. The fast multipole algorithm (FMA) or multilevel FMA (MLFMA) can be incorporated into the present method to reduce the computational cost for concerned matrix vector multiplications. Three optional forms regarding memory usage of the IT-OS-DBE method are given. All the three options have advantage of less CPU time consumption than the MoM-based MLFMA. Two of the three options prevail not only in CPU time consumed but also in memory cost.

Critical behavior in inhomogeneous random graphs

AbstractWe study the critical behavior of inhomogeneous random graphs in the so‐called rank‐1 case, where edges are present independently but with unequal edge occupation probabilities. The edge occupation probabilities are moderated by vertex weights, and are such that the degree of vertex i is close in distribution to a Poisson random variable with parameter wi, where wi denotes the weight of vertex i. We choose the weights such that the weight of a uniformly chosen vertex converges in distribution to a limiting random variable W. In this case, the proportion of vertices with degree k is close to the probability that a Poisson random variable with random parameter W takes the value k. We pay special attention to the power‐law case, i.e., the case where \documentclass{article}\usepackage{mathrsfs}\usepackage{amsmath, amssymb}\pagestyle{empty}\begin{document}\begin{align*}{\mathbb{P}}(W\geq k)\end{align*} \end{document} is proportional to k‐(τ‐1) for some power‐law exponent τ > 3, a property which is then inherited by the asymptotic degree distribution.We show that the critical behavior depends sensitively on the properties of the asymptotic degree distribution moderated by the asymptotic weight distribution W. Indeed, when \documentclass{article}\usepackage{mathrsfs}\usepackage{amsmath, amssymb}\pagestyle{empty}\begin{document}\begin{align*}{\mathbb{P}}(W > k) \leq ck^{-(\tau-1)}\end{align*} \end{document} for all k ≥ 1 and some τ > 4 and c > 0, the largest critical connected component in a graph of size n is of order n2/3, as it is for the critical Erdős‐Rényi random graph. When, instead, \documentclass{article}\usepackage{mathrsfs}\usepackage{amsmath, amssymb}\pagestyle{empty}\begin{document}\begin{align*}{\mathbb{P}}(W > k)=ck^{-(\tau-1)}(1+o(1))\end{align*} \end{document} for k large and some τ∈(3,4) and c > 0, the largest critical connected component is of the much smaller order n(τ‐2)/(τ‐1). © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 42, 480–508, 2013

Identification of parameters with different orders of magnitude in chaotic systems

The synchronization and parameter identification of six unknown parameters in a chaotic neuron model, which one parameter (about 0.006) is 3 orders of magnitude smaller than the others (about 1–5), is investigated by using Lyapunov stability theory and adaptive synchronization in detail. Two gain coefficients (δ1, δ2) are introduced into the Lyapunov function to obtain certain optimized controllers and parameter observers. A selectable amplification factor k 0 is presented using scale conversion and it is used to improve the accuracy of parameter estimation with the smallest order. The parameter space for gain coefficient (δ) versus amplification factor k 0, and the parameter space δ1 versus δ2 at certain fixed amplification factor k 0 are calculated numerically. It is found that the selection values of optimized gain coefficients and amplification factor are critical to estimate the six unknown parameters, particularly for the smallest unknown parameters with an order 0.001. The extensive numerical results show that it is more effective to estimate the smallest unknown parameter r when the two gain coefficients δ1 and δ2 are given the same value and a higher amplification factor k 0 is used. It could be useful to estimate the unknown parameters with large deviation of order magnitude, such as a single chaotic Josephson junction coupled to a Resonant tank and other chaotic systems with potential application [Z.Y. Wang, H.Y. Liao, and S.P. Zhou, Study of the DC biased Josephson junction coupled to a Resonant tank, Acta. Phys. Sin. 50(10) (2001), pp. 1996–2000 (in Chinese)].

Performance Analysis of Finite Element and Energy Based Analytical Methods for Modeling of PCLD Treated Beams

For active/passive vibration control applications, low order models of flexible structures are always preferable. Mathematical models of passive constrained layer damping (PCLD)/active constrained layer damping (ACLD) treatment generated by energy based analytical methods are of much smaller orders as compared to finite element methods (FEM). Using these analytical methods, one can get rid of complex model reduction techniques, since these model reduction techniques are subjected to errors if applied directly to PCLD or ACLD systems. However, analytical techniques cannot be applied blindly to any PCLD system. There is significant error in loss factor prediction for certain boundary conditions. This error is also dependent on the relative thicknesses of Viscoelastic material layer, constraining layer and base beam. For certain combination of the above thicknesses and under certain boundary conditions, the models generated are useless for controller design purposes. On the other hand, FEM are highly robust and can be applied easily to any set of boundary conditions with guaranteed accuracy. Also, results predicted by this method are of high accuracy for any combinations of thicknesses of different layers. The only disadvantage of this technique is the high order of the developed model. Detailed performance comparison of both the techniques to predict the modal parameters of the PCLD treated beam is presented.

Representations for the Drazin inverse of block cyclic matrices

A formula for the Drazin inverse of a block k-cyclic (k � 2) matrix A with nonzeros only in blocks Ai,i+1, for i = 1,...,k (mod k) is presented in terms of the Drazin inverse of a smaller order product of the nonzero blocks of A, namely Bi = Ai,i+1 ···Ai 1,i for some i. Bounds on the index of A in terms of the minimum and maximum indices of these Bi are derived. Illustrative examples and special cases are given.