Let χm+1=T(χm) or even χm+1=T(χm,χm?1, …, χm?q), m=1,2,3 … be an iteration method for solving the nonlinear problem F(χ)=0, where F(χ) and its derivatives possess all of the properties required by T(χm). Then if it can be established that for the problem at hand ∥F(χm+1)∥?βm∥F(χm)∥, ? m > M0 (M0<∞) and 0?βm<1 , definitions are established and theorems proven concerning convergence, uniqueness and bounds on the error after ‘m’ successive iterations of a new approach to convergence properties T(χm). These charateristics are referred to as “alternate” (local, global) convergence properties and none of the proofs given are restricted to any specific type of method such as, e.g. contraction mapping types. Application of results obtained are illustrated using Newton's method as well as the general concept of Newton-like methods. 相似文献
The general mth order difference equation X(n+m)+a1X(n+m?1)+…+amX(n) = F[n,X(n),…,X(n+m?1)] is considered. The stability properties of its solutions are studied using the discrete form of Liapunov's direct method. A quadratic form is selected as a possible Liapunov function V(n,X) and a scheme is developed for determining appropriate conditions on this function to insure that its total difference ΔV(n,X) is negative semi-definite or negative definite with respect to the difference equation. The approach is applied to the fourth-order difference equation in full detail to illustrate the method for determining the conditions which imply either uniform stability or uniform asymptotic stability and specific results are obtained. Several comments on, and extensions of, the work done by Puri and Drake for the cases m = 2 and m = 3 are presented.The results of the present approach in the homogeneous case where F[n,X(n),…,X(n+m?1)] = 0 are compared with the usual Schur-Cohn criteria and are shown to be at least as good. 相似文献
Understanding the biological feeding strategy and characteristics of a microorganism as an actuator requires the detailed and quantitative measurement of flow velocity and flow rate induced by the microorganism. Although some velocimetry methods have been applied to examine the flow, the measured dimensions were limited to at most two-dimensional two-component measurements. Here we have developed a method to measure three-dimensional two-component flow velocity fields generated by the microorganism Vorticella picta using a piezoscanner and a confocal microscope. We obtained the two-component velocities of the flow field in a two-dimensional plane denoted as the XY plane, with an observation area of 455×341 μm2 and the resolution of 9.09 μm per each velocity vector by a confocal microparticle image velocimetry technique. The measurement of the flow field at each height took 37.5 ms, and it was repeated in 16 planes with a 2.50 μm separation in the Z direction. We reconstructed the three-dimensional two-component flow velocity field. From the reconstructed data, the flow velocity field [u(x,y,z),v(x,y,z)] in an arbitrary plane can be visualized. The flow rates through YZ and ZX planes were also calculated. During feeding, we examined a suction flow to the mouth of the Vorticella picta and measured it to be to 300 pl∕s. 相似文献
In this paper, necessary and sufficient conditions are derived for the existence of temporally periodic “dissipative structure” solutions in cases of weak diffusion with the reaction rate terms dominant in a generic system of reaction-diffusion equations ?ci/?t = Di?2ci+Qi(c), where the enumerator index i runs 1 to n, ci = ci(x, t) denotes the concentration or density of the ith participating molecular or biological species, Di is the diffusivity constant for the ith species and Qi(c), an algebraic function of the n-tuple c = (c1,\3., cn), expresses the local rate of production of the ith species due to chemical reactions or biological interactions. 相似文献
Fixed point properties of the binomial function are developed. It is shown that for any 1 < L < N, TLNhas a unique fixed point p? in (0, 1), and that for large N, the fixed point is L/N. This has application to signal detection schemes commonly used in communication systems. When detecting the presence or absence of a signal with an initial false alarm probability pFAand an initial detection probability pD, then TLN(pFA) < pFAand TLN(pD) > pDif, and only if, pFA < p? < pD. When this condition is satisfied, as N → ∞, TLN(pFA) → 0 and TLN(pD → 1. 相似文献
The usual model for (Poissonian) linear birth-death processes is extended to multiple birth-death processes with fractional birth probabilities in the form λi(Δt)α+o((Δt)α, 0<α<1. The probability generating function for the time dependent population size is provided by a fractional partial differential equation. The solution of the latter is obtained and comparison with the usual model is made. The probability of ultimate extinction is obtained. One considers the special case of fractional Poissonian processes with individual arrivals only, and then one outlines basic results for continuous processes defined by fractional Poissonian noises. The key is the Taylor’s series of fractional order f(x+h)=Eα(hαDxα)f(x), where Eα(·) is the Mittag-Leffler function, and Dxα is the modified Riemann-Liouville fractional derivative, as previously introduced by the author. 相似文献
If T maps a convex domain DT into itself, and if {ωn} is a real sequence with range in (0, 1] then the recursive averaging process, generates a sequence {x?n}; with range in DT. Under suitable conditions on DT, T and {ωn} the sequence {x?n} will converge in some sense to a fixed point of T. We prove that if DT is a closed convex subset of a complex Hilbert space H, if Tω = (1 ? ω) I + ωT is a strict contraction for some ω ? (0, 1], and if {ωn} satisfies the conditions, and then, for arbitrary ξ ? DT, {x?n} converges strongly to (the unique) fixed point of T. We also prove that if DT and {ωn} satisfy the foregoing conditions, if T has at least one fixed point, and if Tω is non-expansive for some ω ? (0, 1], then for all ξ ? DT, {x?n} converges at least weakly to some fixed point of T. Finally, we apply these results to linear equations involving bounded normal operators and obtain an extension of the classical Neumann operator series. 相似文献
An n-port gyrator, terminated by s1-type capacitors in m1-ports, by s2-type capacitors in the next m2-ports and by resistors in the remaining (n-m1-m2)-ports is considered. The determinant of the admittance matrix of the network can be made to yield VSHP (a two-variable Hurwitz polynomial without non-essential singularities of the second kind) under certain conditions involving the sub-determinants of the gyrator matrix. With the gyrator constants as variables and the above conditions as constraints, some 2-D stable low-pass filters have been designed using a suitable optimization procedure. The method is illustrated by examples. 相似文献
Using data generated by progressive nucleation mechanism on the cumulative fraction of citations of individual papers published successively by a hypothetical author, an expression for the time dependence of the cumulative number Lsum(t) of citations of progressively published papers is proposed. It was found that, for all nonzero values of constant publication rate ΔN, the cumulative citations Lsum(t) of the cumulative N papers published by an author in his/her entire publication career spanning over T years may be represented in distinct regions: (1) in the region 0 < t < Θ0 (where Θ0 ≈ T/3), Lsum(t) slowly increases proportionally to the square of the citation time t, and (2) in the region t > Θ0, Lsum(t) approaches a constant Lsum(max) at T. In the former region, the time dependence of Lsum(t) of an author is associated with three parameters, viz. the citability parameter λ0, the publication rate ΔN and his/her publication career t. Based on the predicted dependence of Lsum(t) on t, a useful scientometric age-independent measure, defined as citation acceleration a = Lsum(t)/t2, is suggested to analyze and compare the scientific activities of different authors. Confrontation of the time dependence of cumulative number Lsum(t) of citations of papers with the theoretical equation reveals one or more citation periods during the publication careers of different authors. 相似文献
In this paper we present a method for testing the Hurwitz property of a segment of polynomials (1−λ)p0(s)+λp1(s), where λ∈[0,1] and p0(s) and p1(s) are nth-degree polynomials with complex coefficients. The method consists in constructing a parametric Routh-like array with polynomial entries and generating Sturm sequences for checking the absence of zeros of two real λ-polynomials of degrees 2 and 2n in the interval (0,1). The presented method is easy to implement. Moreover, it accomplishes the test in a finite number of arithmetic operations because it does not invoke any numerical root-finding procedure. 相似文献
This paper gives a general review of the Theory of Nonlinear Systems. In 1960, the author presented a paper “Theory of Nonlinear Control” at the First IFAC Congress at Moscow. Professor Norbert Wiener, who attended this Congress, drew attention to his work on the synthesis and analysis of nonlinear systems in terms of Hermitian polynomials in the Laguerre coefficients of the past of the input.Wiener's original idea was to use white noise as a probe on any nonlinear system. Applying this input to a Laguerre network gives u1, u2,…, us, and then to a Hermite polynomial generator gives V(α)'s. Applying the same input to the actual nonlinear system gives output c(t). Putting c(t) and V(α)'s through a product averaging device, we get , where the upper bar denotes time average and Aα's can be considered as characteristic coefficients of the nonlinear system. A desired output z(itt) may replace c(itt) to get a new set of Aα's.The Volterra functional method suggested by Wiener in 1942 has been greatlydeveloped from 1955 to the present. The method involves a multi-dimensional convolution integral with multi- dimensional kernels. The associated multi-dimensional transforms are given by Y.H. Ku and A.A. Wolf (J. Franklin Inst., Vol. 281, pp. 9–26, 1966). Wiener extended the Volterra functionals by forming an orthogonal set of functionals known as G-functionals, using Gaussian white noise as input. Volterra kernels and Wiener kernels can be correlated and form the characteristic functions of nonlinear systems.From an extension of the linear system to the nonlinear system, the input-output crosscorrelation φxy can be shown to be equal to the convolution of system impulse response h1 with the autocorrelation φxx. Using the white noise as input, where its power density spectrum is a constant, say, A, the crosscorrelation is given by φxy(σ) = Ah1(σ), while the autocorrelation is φxx(τ) = Au(τ). This extension forms the basis of an optimum method for nonlinear system identification. Measurement of kernels can be made through proper circuitry.Parallel to the Volterra series and the Wiener series, another series based on Taylor-Cauchy transforms developed since 1959 are given for comparison. The Taylor-Cauchy transform method can be applied in the analysis of simultaneous nonlinear systems. It is noted that the Volterra functional method and the Taylor-Cauchy transform method give identical final results.A selected Bibliography is appended not only to include other aspects of nonlinear system theory but also to show the wide application of nonlinear system characterization and identification to problems in biology, ecology, physiology, cybernetics, control theory, socio- economic systems, etc. 相似文献
The problem is to determine the linear graph that has the maximum number of spanning trees, where only the number of nodes N and the number of branches B are prescribed. We deal with connected graphs G(N,B) obtained by deleting D branches from a complete graph KN. Our solution is for D less than or equal to N相似文献
Matrix A with characteristic polynomial Q(z) is defined positive or negative Hurwitz according to whether Q(z) or Q(-z) is a Hurwitz polynomial. Leading principle sections of the Tikhonova growth matrix have associated characteristic polynomials Pn(-z) which satisfy the recursionThat the Tikhonova growth matrix is negative Hurwitz is established through applying the Wall-Stieltjes theory of continued fraction expansions to show the Pn(-z) are Hurwitz polynomials. The Kayeya-Enestrom theorem and a procedure for refinement of the Gerschgorin estimate are used to obtain analytical bounds on spectral radii for the Tikhonova model, which provides estimates of maximal growth rates. The theory allows generalization to more complicated growth models. 相似文献
The selective cell separation is a critical step in fundamental life sciences, translational medicine, biotechnology, and energy harvesting. Conventional cell separation methods are fluorescent activated cell sorting and magnetic-activated cell sorting based on fluorescent probes and magnetic particles on cell surfaces. Label-free cell separation methods such as Raman-activated cell sorting, electro-physiologically activated cell sorting, dielectric-activated cell sorting, or inertial microfluidic cell sorting are, however, limited when separating cells of the same kind or cells with similar sizes and dielectric properties, as well as similar electrophysiological phenotypes. Here we report a label-free density difference amplification-based cell sorting (dDACS) without using any external optical, magnetic, electrical forces, or fluidic activations. The conceptual microfluidic design consists of an inlet, hydraulic jump cavity, and multiple outlets. Incoming particles experience gravity, buoyancy, and drag forces in the separation chamber. The height and distance that each particle can reach in the chamber are different and depend on its density, thus allowing for the separation of particles into multiple outlets. The separation behavior of the particles, based on the ratio of the channel heights of the inlet and chamber and Reynolds number has been systematically studied. Numerical simulation reveals that the difference between the heights of only lighter particles with densities close to that of water increases with increasing the ratio of the channel heights, while decreasing Reynolds number can amplify the difference in the heights between the particles considered irrespective of their densities.Separating specific cells from heterogeneous or homogeneous mixtures has been considered as a key step in a wide variety of applications ranging from biomedicine to energy harvesting. For example, the separation and sorting of rare circulating tumor cells (CTCs) from whole blood has gained significant importance in the potential diagnosis and treatment of metastatic cancers.1,2 Similarly, malaria detection relies on the collection of infected red blood cells (RBCs) from whole blood.3,4 In addition, the selective separation of lipid-rich microalgae from homogeneous mixtures of microalgae is a promising technique in biomass conversion.5To date, conventional cell separation can be done by labelling cells with biomolecules to induce differences in physical properties. For instance, in a fluorescence-activated cell sorter (FACS), cells to be separated are labelled with antibodies or aptamers with fluorescent molecules, and then sorted by applying an electrical potential.6,7 Similarly, magnetic-activated cell sorter (MACS) uses magnetic.8,9 Alternatively, label-free cell separation methods have exploited inherent differences in the physical properties (e.g., size and dielectric properties) of different kinds of cells. For example, acoustophoresis forces particles larger than a desired size to move into the center of a fluidic channel by using ultrasonic standing waves.10–12 Inertial microfluidics takes advantage of curved fluidic channels in order to amplify the size differences between particles.13,14 Mass-dependent separation of particles based on gravity and hydrodynamic flow was also reported.15 Particles with different dielectric properties can also be sorted by dielectrophoresis which induces the movement of polarizable particles.16–18The disadvantage of these methods, however, is that they require external forces and labels that may cause unexpected damage to biological cells.19–21 More importantly, most methods are limited in separating cells of the same kind or cells with similar sizes and dielectric properties.Here we designed a novel, label-free density difference amplification-based cell sorting (dDACS) that allows the separation of particles with the same size and charge by exploiting subtle differences in density without the use of external forces. Figure 1(a) illustrates the proposed microfluidic model and its underlying mechanism. The conceptual microfluidic system consists of an inlet, a separation chamber (hydraulic jump cavity), and multiple outlets. Particles entering through the inlet experience gravity (FG), buoyancy (FB), and drag (FD) forces in the separation chamber. The net force acting on the particles can be described as
F = FG + FB + FD.(1)As particles enter the separation chamber (i.e., hydraulic jump cavity), FD acting on the particles changes its direction along the streamline. The particles experience additional forces in the y direction due to large tangential angle (Fig. 1(b)). For lighter particles, whose densities are close to that of the surrounding water, FD becomes comparable to FG (i.e., in the y direction), while the net force for heavier particles is less affected by this additional contribution of FD due to a large FG. As a result, the height (H) and distance (D) that each particle can travel are different depending on its density. The difference in the maximum height (ΔHmax) between two particles with different density (ρp1 and ρp2) can be further approximated as
(2)where vyp0 and vyf represent the velocity of particle and fluid along the y direction at the entrance of hydraulic jump cavity, respectively.Open in a separate windowFIG. 1.Schematic illustration of label-free density difference amplification-based cell sorting (dDACS), which exploits differences in the densities (ρ1 > ρ2) of particles with similar diameters (d) and charge. (a) The conceptual microfluidic design consists of an inlet, a separation chamber (hydraulic jump cavity), and multiple outlets. Incoming particles experience gravity (FG), buoyancy (FB), and drag (FD) forces in the separation chamber, and depending on their densities, the height (H) and distance (D) that each particle is able to reach will be different, allowing the particles to be separated into multiple outlets. (b) Possible microfluidic channel configurations for density-based separation: Uniform channel height (left), gradual channel expansion (middle), and hydraulic jump cavity with sudden channel expansion (right). The height difference between particles with different densities can be amplified by the sudden channel expansion compared to the other two cases due to the relatively large tangential angle, θ of FD. (|θ1|≪ |θ2|) (see Fig. S1 in the supplementary material22).In comparison with the other two cases (Fig. 1(b) uniform channel height and gradual channel expansion), the height difference between the particles with different densities can be amplified by the sudden channel expansion in the hydraulic jump cavity due to relatively large tangential angle (see supplementary material22). Therefore, the particles can be separated through the multiple outlets, depending on their height and distance.In order to analyze the separation behavior of particles in the chamber according to differences in their densities, H and D are systematically investigated. The numerical simulations are performed using a commercial CFD software (CFX 14.0; ANSYS 14.0; ANSYS, Inc.). Particles with the same density may have different trajectories in the separation chamber depending on their inlet positions (Fig. 2(a)). Prior to this investigation, the maximum height (Hmax) and distance (Dmax) for each particle are compared by examining H and D of 100 identical particles at different inlet positions since the inlet position of particles could be controlled.20 Fig. 2(b) shows Hmax and Dmax of particles with respect to density at a fixed Reynolds number (Re = 0.1). Note that Reynolds number is defined as Re = ρfvfDh/μ, where ρf, vf, Dh, μ are density of fluid, velocity of the fluid, hydraulic diameter of a channel, and dynamic viscosity of the fluid, respectively. The hydraulic diameter in the Reynolds number is determined with the inlet channel. Particle densities in the range of 1.1 to 2.0 g/cm3 are chosen with the increase of 0.1 g/cm3. These values are quite reasonable in that the densities of many microorganisms such as microalgae are typically within this range and their densities can be varied by 0.2 g/m3 depending on their cellular context.23 The lighter particles travel with a higher Hmax, and longer Dmax. With the separation chamber, the height difference between particles with densities of 1.1 and 1.2 g/cm3 can be amplified by about 10 times as compared to that in a channel without the chamber, judging from the position where the 1.1 g/cm3 particle reaches its Hmax.Open in a separate windowFIG. 2.Microfluidic particle separation with respect to Reynolds number (Re). (a) Trajectories in the separation chamber of a hundred particles with the same density starting from inlet positions chosen arbitrarily in order to investigate the effect of the inlet positions on the maxima of the height (Hmax) and distance (Dmax) prior to further simulation. (b) Representative trajectories of particles having different densities from 1.1 to 2.0 g/cm3. (c) The maximum height (Hmax) of each particle with respect to Re. (d) Representative maximum distance (Dmax) of each particle at Re = 0.1. (Left) Streamline of fluid and representative trajectories of particles with densities of 1.1 and 2.0 g/cm3 in the separation chamber at Re = 0.1 (right).In Fig. 2(c), the values for Hmax of particles with respect to Reynolds number (Re) are presented. Since in our study, the maximum height (Hmax) and distance (Dmax) for each particle were compared by examining H and D of 100 identical particles that are randomly distributed in the channel (throughout all figures), there is little variation in Hmax and Dmax between each simulation. However, the standard deviation between each simulation is quite small and can be negligible. The Hmax values particles at Re = 0.5 with densities of 1.1 g/cm3 and 1.2 g/cm3 are 2.21 × 103 μm and 2.17 × 103 μm, respectively. The difference between Hmax of different particles, ΔHmax, increases with decreasing Re. For example, ΔHmax between particles with densities of 1.1 and 2.0 g/cm3 becomes 0.26 × 103 μm at Re = 1.0, but increases to 1.38 × 103 μm as Re decreases to 0.1. As Re increases (velocity of fluid increases), the relative velocity in the y direction between the fluid and the particle increases resulting in increasing of FD in the y direction since the velocity of particle in the y direction is very small at the entrance of the separation chamber. Thus, contribution of FD becomes comparable to the net force in the y direction. As a result, most of the particles even in the case of heavier ones travel quite similarly with the streamline, and ΔHmax subsequently decreases. On the other hand, as Re decreases, the contribution of FG becomes dominant due to the decrease of FD in the y direction. Consequently, the particles start to cross downwards streamlines as the density of the particles increases and Hmax gradually decreases. In addition, irrespective of their densities, ΔHmax of the particles increases with decreasing Re.Fig. 2(d) shows Dmax with respect to the density of the particles (left). Different densities of particles show different trajectories due to the relative contribution of FD to the net force in the y direction depending on the particle density (right). At Re = 0.1, Dmax of particles with densities of 1.1 cm3 and 1.2 g/cm3 are 2.91 × 104 μm and 1.43 × 104 μm, respectively. As the density of a particle increases, its Dmax dramatically decreases. The difference in Dmax between particles with densities of 1.1 and 1.2 g/cm3 is 1.48 × 104 μm, and 0.0037 × 104 μm for particles with densities of 1.9 and 2.0 g/cm3. The effect of FD is stronger compared to that of FG on lighter particles. Thus, lighter particles travel quite similarly with the streamline and finally have a large Dmax. On the other hand, heavier particles where effect of FG is stronger compared to that of FD cross downwards streamlines and finally have a small Dmax.Next, in order to investigate the separation behavior of particles with respect to the geometry of the microfluidic device, the effect of the ratio of the height of the separation chamber (hc) to the inlet (hi) on Hmax is investigated as shown in Fig. Fig.3.3. Interestingly, Hmax of particles with density of 1.1 g/cm3 increases from 1.93 × 103 μm to 6.48 × 103 μm while that of particles with density of 1.9 g/cm3 slightly changes from 0.70 × 103 μm to 0.73 × 103 μm as hc/hi increases from 5 to 20.Open in a separate windowFIG. 3.Microfluidic particle separation with respect to the ratio of the height of the inlet (hi) to the separation chamber (hc).This result can be attributed to two effects: (1) the change in the streamline and (2) the relative contribution of drag force to the net force depending on the density. With increasing hc/hi, dramatic increase in Hmax for lighter particles is because the streamline for the lighter ones experiences more vertical displacement in the separation chamber and the contribution of FD to the net force acting on the lighter one is more significant (see Fig. S2 in the supplementary material22).Based on this approach, we propose a microfluidic device for the selective separation of the lightest particle. Fig. 4(a) shows one unit (with three outlets) of the proposed microfluidic device that can be connected in series. The ratio of channel heights (hc/hi) is set to 20, and the particle densities are in the range of 1.1 ∼ 1.5 g/m3. Fig. 4(b) shows the representative separation behavior of the particles. A portion of the lightest particles (1.1 g/cm3) is selectively separated into the upper and middle outlets, while remaining light particles together with four other heavier particles with densities in the range of 1.2 to 1.5 g/cm3 leave through the lowest outlet. With a single operation of this unit, 40% of the lightest particles are recovered. In addition, the yield increases with increasing number of cycles (Fig. 4(c)).Open in a separate windowFIG. 4.(a) One unit of the proposed microfluidic device for the selective separation of the lightest particle based on the simulation results. Particles are separated into two outlets based on differences in both the height and distance travelled stemming from differences in density. (b) Representative separation behavior of particles observed in the device. (c) The yield of the lightest particle (1.1 g/cm3) with the proposed microfluidic device according to the number of cycles (i.e., this unit is assumed to be connected in series).In summary, we have demonstrated a label-free microfluidic system for the separation of particles according to subtle differences in their densities without external forces. Our microfluidic design consists simply of an inlet, a separation chamber, and multiple outlets. When entering the separation chamber, the particles experience an additional drag force in the y direction, amplifying the difference in both the height and the distance that the particles with different densities can travel within the chamber. At a fixed Reynolds number, with increasing particle density, Hmax decreases monotonously, and Dmax decreases dramatically. On the other hand, as Reynolds number increases, the difference between the heights of particles with different densities is attenuated. In addition, the simulation reveals that increasing the ratio of the channel heights increases the difference between the heights of particles only when their densities are close to that of the surrounding water. Based on this approach, a microfluidic device for the separation of the lightest particles has been proposed. We expect that our density-based separation design can be beneficial to the selective separation of specific microorganisms such as lipid-rich microalgae for energy harvesting application. 相似文献
The main goal of the present paper is twofold: (i) to establish the well-posedness of a class of nonlinear degenerate parabolic equations and (ii) to investigate the related null controllability and decay rate properties. In a previous step, we consider an appropriate regularized system, where a small parameter α is involved. More precisely, the usual nonlinear term b(x)uux is replaced by b(x)zux, where and A is a Poisson–Dirichlet operator. We investigate the behavior of the null controls and their associated states as α → 0. 相似文献
By means of Mawhin's continuation theorem, we study a kind of fourth-order p-Laplacian neutral functional differential equation with a deviating argument in the form:
Logarithmic finite-size scaling of the O(n) universality class at the upper critical dimensionality (dc = 4) has a fundamental role in statistical and condensed-matter physics and important applications in various experimental systems. Here, we address this long-standing problem in the context of the n-vector model (n = 1, 2, 3) on periodic four-dimensional hypercubic lattices. We establish an explicit scaling form for the free-energy density, which simultaneously consists of a scaling term for the Gaussian fixed point and another term with multiplicative logarithmic corrections. In particular, we conjecture that the critical two-point correlation g(r, L), with L the linear size, exhibits a two-length behavior: follows governed by the Gaussian fixed point at shorter distances and enters a plateau at larger distances whose height decays as with a logarithmic correction exponent. Using extensive Monte Carlo simulations, we provide complementary evidence for the predictions through the finite-size scaling of observables, including the two-point correlation, the magnetic fluctuations at zero and nonzero Fourier modes and the Binder cumulant. Our work sheds light on the formulation of logarithmic finite-size scaling and has practical applications in experimental systems. 相似文献
Let {Πτ(m, n): m?≥?n?≥?0} be the family of periodic discrete transition matrices generated by bounded valued square matrices Λτ(n), where is an arbitrary switching signal. We prove that the family {Πτ(m, n): m?≥?n?≥?0} of bounded linear operator is uniformly exponentially stable if and only if the sequence is bounded. 相似文献
