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A simple necessary and sufficient condition ensuring that a real matrix of dimension 3 is similar to a Metzler matrix is exhibited. When this condition is satisfied, a construction of the transfer matrix is given. This construction is used to design an interval observer for a family of continuous-time systems. An example is provided with interval observer design for the so-called love dynamics in the case of limit cycles.

The synthesis of information and energy schemes is posed as an extreme problem, the purpose of which is a weighted directed graph of the minimum length from the input value to the output value of the device. The nodes of the graph are the physical effects included in the given database, and the branches are the input and output values of the effects. Nodes and branches are mathematically defined by diagonal multidimensional matrices, whose elements are determined by the dimensions of the quantities in the selected system of physical coordinates with a given number of basic units of measurement. The weight or resource intensity of the graph elements is determined by the norm of the corresponding matrices. The resulting circuit is suitable for use in technical documentation to explain the operating principle of the device, as well as for patent protection.In the enhanced formulation of the extreme problem, restrictions are introduced on the numerical values of the input and output values of the effect and its dynamic properties in the form of the transfer function of the effect. In this case, the size of the transfer matrices of nodes and branches is expanded by one. As a result, the transfer matrix of the effect contains information not only about the dynamic properties of the effect, but also about the dimensions of the physical quantities at its input and output.In a detailed example, the case of searching for the operating principle of a measuring-converting device of a pressure sensor with an electric current output is considered. To simplify the geometric representation of graph vectors on a plane, the problem is considered for a two-dimensional system of physical quantities with basic units of length and time. The calculation of the resource capacity is carried out according to the scheme of dimensional simulation, in which the phase variables of the differential equation enter with their physical dimensions. According to the numerical value of the resource capacity, you can compare different versions of the implementation of the operating physical principle of the device.

We present a new optimal power flow (OPF) design that not only optimizes fuel cost but also enhances dynamic performance of a power system. Performance is quantified by the $H_2$ -norm of the transfer matrix from any disturbance input to a set of performance outputs, which in this case are chosen as the frequencies of the generators. The $H_2$ -norm models the attenuation of the frequency amplitudes following the disturbance, and thereby quantifies the amount of damping torque induced on the tie-line flows. The method, referred to as $H_2$ -power flow modification ( $H_2$ -PFM) is carried out in two steps. First, the regular OPF is solved to obtain the optimal generator setpoints for active and reactive power dispatch. Second, the load setpoints are re-tuned to minimize the aforesaid $H_2$ -norm while keeping the generator setpoints fixed. In particular, manipulating the load reactive power in this way is found to reduce the norm remarkably, improving disturbance attenuation and damping. A gradient descent algorithm is proposed for this minimization. Results are validated using the 68-bus test system with a solar farm.

Subject of Research. The research of deterministic systems is a topical problem of natural science. The paper presents an approach for behavior study of the deterministic systems. The work is aimed at creation of the evolution equation for deterministic systems, which shows that part of the macroscopic processes complies with quantum logic. Method. A new algebraic approach is proposed based on non-commutative algebra. It is shown that for any deterministic system in case of setting its change for a short time period, it is possible to create a system of differential equations describing the evolution of a given system in time. Implemented apparatus of non-commutative multiplication is an alternative to the operator calculus for quantum mechanics. Main Results. An associative non-commutative ring is built describing the evolution of arbitrary deterministic system. The proposed algebra is an isomorphic Heisenberg algebra. It is shown that all elements of algebraic rings are functions of numerical variables unlike the mathematical apparatus of quantum mechanics and, therefore, it is possible to give them different physical meaning. Practical Relevance. An example of differential equation creation is considered describing the motion of a classical particle in the presence of random forces. The obtained equation describes the probability density of a classical particle location at an arbitrary point of time in phase space.

For an arbitrary rational matrix (RM), a new descriptor realization of its all-pass factorization is presented. The coefficients of the factors are expressed explicitly in the solution of a linear matrix inequality, which is well known in control theory. Using the all-pass factorization, necessary and sufficient conditions for existence of a solution of an equation with RMs where the unknown RM has its poles in the left closed complex half plane and it can be improper are presented, as well as all solutions. As an application of the latter result, the notion of almost disturbance decoupling, i.e., when the infimum of the ${\mathscr {H}}_{\infty}$ norm of the transfer matrix from the disturbance input to the controlled output over the stabilizing controllers is zero, is generalized. The generalization is such that instead of the whole frequency axis, the frequency axis with excluded intervals of small lengths, is considered. The controller consists of output feedback plus disturbance feedforward designed independently. A necessary and sufficient solvability condition is presented, which is expressed in terms of the given data. The three main results are illustrated by an example.

Calculating the genus distributions of ladder graphs is a concerned topic in topological graph theory. In this paper, we formulate several ladder-class graphs by using a starting graph iterative amalgamation with copies of a path to construct a base graph and then adding some edges to the appointed root-vertices of the base graph. By means of transfer matrix and a finer partition of the embeddings, the explicit formulas for the genus distribution polynomials of four types of ladder-class graphs are derived.

The problem of synthesizing an asymptotic observer for a linear system with $$l$$ measurable outputs $$y=Cx$$ and $$m$$ unknown inputs $$f(t)$$ acting on the system through the matrix $$D$$ is considered. A case of hyperoutput systems is examined; such systems have output dimensions larger than that of an unknown input ($$l>m$$). Ways of constructing observers are well-known for such systems when matrix $$CD$$ (the transfer matrix from input $$f(t)$$ to output $$y(t)$$) is of full rank. The case $$\textrm{rank}(CD)

The dynamic stiffness method and Transfer method is applied to study the vibration characteristics of the Euler-Bernoulli pipe conveying fluid in this paper. According to the dynamics equation of the pipe conveying fluid, the element dynamic stiffness is established. The vibration characteristic of the single-span pipe is analyzed under two kinds of boundary conditions. The results compared with the literature, which has a good consistency. Based on this method, natural frequency and the critical speed of the two types of multi-span pipe are deserved. This paper shows that the dynamic stiffness method and transfer matrix is an effective method to deal with the vibration problem of pipe conveying fluid.

The transfer matrix is a powerful technique that can be applied to statistical mechanics systems as, for example, in the calculus of the entropy of the ice model. One interesting way to study such systems is to map it onto a three-color problem. In this paper, we explicitly build the transfer matrix for the three-color problem in order to calculate the number of possible configurations for finite systems with free, periodic in one direction and toroidal boundary conditions (periodic in both directions)

Three-dimensional Ising model in zero external field is exactly solved by operator algebras, similar to the Onsager's approach in two dimensions. The partition function of the simple cubic crystal imposed by the periodic boundary condition along two directions and the screw boundary condition along the third direction is calculated rigorously. In the thermodynamic limit an integral replaces a sum in the formula of the partition function. The critical temperatures, at which order-disorder transitions in the infinite crystal occur along three axis directions, are determined. The analytical expressions for the internal energy and the specific heat are also presented., 10 pages, 2 figures

In this paper, a general solution was developed by the Laplace transform applying to the one-dimensional (1D) consolidation equation in a multi-layered unsaturated soil layer, and the analytical solutions of excess pore pressures as well as settlements under a semi-permeable boundary were derived in the Laplace domain employing the transfer matrix technique. Then to acquire the final solution in the time domain, a numerical computation was implemented by Crump’s method. Moreover, the proposed semi-analytical result is in good agreement with that obtained from the finite difference method. It validated that the present mathematical methodology is accessible, efficient, and reliable for deriving a semi-analytical solution of the 1D consolidation equation. A parametric study was conducted to describe the consolidation characteristic of a two-layered unsaturated soil. It is shown that both the boundary permeability and the soil-layer properties play a significant role in the consolidation of multi-layered unsaturated soils.

In this paper, the use of the Smith-McMillan form in decoupling multiple-input multiple-output system dynamics is analyzed. In short, from a transfer matrix plant model one can obtain a decoupling compensator which leads to a decoupled plant that contains the same transmission poles and zeros of the transfer matrix of the original system. As a result, full decoupling of the plant transfer matrix is obtained for all frequencies, which can be individually addressed by single-input single-output control. The aim of this paper is to present conditions for the decoupled system that guarantee internal stability and performance requirements for the overall control system. Performance specifications are defined in terms of magnitude limits for the maximum singular value of the closed-loop transfer matrices. The potential of the decoupling procedure is shown in a simulation study of a mechanical system., 6 pages, 9 figures, American control conference 2021

Purpose Partial-volume correction (PVC) using the Geometric Transfer Matrix (GTM) method is used in positron emission tomography (PET) to compensate for the effects of spatial resolution on quantitation. We evaluate the effect of misspecification of scanner point-spread function (PSF) on GTM results in amyloid imaging, including the effect on amyloid status classification (positive or negative). Methods Twenty-nine subjects with Pittsburgh Compound B ([11C]PiB) PET and structural T1 MR imaging were analyzed. FreeSurfer 5.3 (FS) was used to parcellate MR images into regions-of-interest (ROIs) that were used to extract radioactivity concentration values from the PET images. GTM PVC was performed using our “standard” PSF parameterization [3D Gaussian, full-width at half-maximum (w) of approximately 5 mm]. Additional GTM PVC was performed with “incorrect” parameterizations, taken around the correct value. The result is a set of regional activity values for each of the GTM applications. For each case, activity values from various ROIs were combined and normalized to produce standardized uptake value ratios (SUVRs) for nine standard [11C]PiB quantitation ROIs and a global region. GTM operating-point characteristics were determined from the slope of apparent SUVR versus w curves. Results Errors in specification of w on the order of 1 mm (3D) mainly produce only modest errors of up to a few percent. An exception was the anterior ventral striatum in which fractional errors of up to 0.29 per millimeter (3D) of error in w were observed. Conclusion While this study does not address all the issues regarding the quantitative strengths and weakness of GTM PVC, we find that with reasonable caution, the unavoidable inaccuracies associated with PSF specification do not preclude its use in amyloid quantitation.

In this study, the dynamic response of beams resting on two-parameter elastic foundation subjected to moving load is investigated by using the transfer matrix method (TMM). The Timoshenko beam theory (TBT) which considers shear deformation and rotational inertia is used to model the beam. The two-parameter elastic foundation model is selected as Pasternak foundation that takes into account a shear layer at the end of linear springs of Winkler foundation. The TMM which uses the relation between analytically obtained state vectors of each end of the beam is applied to solve the free vibration problem. After performing the free vibration analysis, the mathematical model is simplified into an equivalent single degree of freedom (SDOF) system by using the exact mode shapes to obtain dynamic responses. The generalized displacement is calculated for each mode by using the Runge-Kutta algorithm. A numerical case study is presented for a simply-supported Timoshenko beam on the Pasternak foundation subjected to a concentrated load. The natural frequencies obtained from finite element method (FEM) results of SAP2000 are presented with the results of TMM for comparison purposes using the Winkler foundation. The effects of shear layer on the natural frequencies of the model are revealed. The mode shapes are plotted. The proposed approach for calculating dynamic responses is validated by using the results of FEM for Winkler foundation model. Then, the effects of Winkler springs and shear layer of the foundation model on the dynamic responses are presented in figures. The effects of modal damping are discussed. Finally, the critical velocities for the model are calculated for various elastic foundation scenarios and the effects of elastic foundation parameters on the dynamic response of beam model subjected to moving load with high velocity are observed.

We introduce a dense and a dilute loop model on causal dynamical triangulations. Both models are characterised by a geometric coupling constant $g$ and a loop parameter $\alpha$ in such a way that the purely geometric causal triangulation model is recovered for $\alpha=1$. We show that the dense loop model can be mapped to a solvable planar tree model, whose partition function we compute explicitly and use to determine the critical behaviour of the loop model. The dilute loop model can likewise be mapped to a planar tree model; however, a closed-form expression for the corresponding partition function is not obtainable using the standard methods employed in the dense case. Instead, we derive bounds on the critical coupling $g_c$ and apply transfer matrix techniques to examine the critical behaviour for $\alpha$ small., Comment: 30 pages, v2: minor changes

he inverse spectral problem for the Sturm-Liouville equation with a piecewise-entire potential function and the discontinuity conditions for solutions on a rectifiable curve \(\gamma \subset \textbf{C}\) by the transfer matrix along this curve is studied. By the method of a unit transfer matrix the uniqueness of the solution to this problem is proved with the help of studying of the asymptotic behavior of the solutions to the Sturm-Liouville equation for large values of the spectral parameter module.

The 16th-order transfer matrix of the three-dimensional Ising model in the particular case n = m = 2 (n × m is number of spins in a layer) is specified by the interaction parameters of three basis vectors. The matrix eigenvectors are divided into two classes, even and odd. Using the symmetry of the eigenvectors, we find their corresponding eigenvalues in general form. Eight of the sixteen eigenvalues related to odd eigenvectors are found from quadratic equations. Four eigenvalues related to even eigenvectors are found from a fourth-degree equation with symmetric coefficients. Each of the remaining four eigenvalues is equal to unity.

In noise source identification based on the inverse boundary element method (IBEM), the boundary vibration velocity is predicted based on the field pressure through a transfer matrix of the vibration velocity and field pressure established on the Helmholtz integral equation. Because the matrix is often ill-posed, it needs to be regularized before reconstructing the vibration velocity. Two regularization methods and two methods of selecting the regularization parameter are investigated through the simulation analysis of a pulsating sphere. The result of transfer matrix regularization is further verified through the reconstruction of the vibration of an aluminum plate. Additionally, to reduce the large errors at some frequencies in the reconstruction result, increasing the number of measuring points is more effective than reducing the distance between the measurement plane and the sound source.

A 2-dimensional ( 1 , ∞ ) -runlength limited constraint S h s is the set of ( 0 , 1 ) -matrices/arrays that do not have adjacent horizontal or vertical 1’s. Let f ( m , n ) be the number of ( m + 1 ) × ( n + 1 ) ( m , n ≥ 0 ) matrices that satisfy the constraint S h s . It is known (Calkin and Wilf, 1998) that f ( m , n ) is equal to the number of independent sets in an ( m + 1 ) × ( n + 1 ) grid graph and f ( m , n ) = 1 t T m n 1 , where T m is an associated transfer matrix and 1 is the all 1’s vector. The capacity of the constraint S h s , c a p ( S h s ) = lim m , n → + ∞ log 2 f ( m , n ) m n , has been bounded actively in the last decades (e.g., Calkin and Wilf, 1998; Engel, 1990; Nagy and Zeger, 2000; Weber, 1988), with the most recent bounds being 0 . 5878911617 . . . ≤ c a p ( S h s ) ≤ 0 . 5878911618 . . . (Nagy and Zeger, 2000). In this paper we derive various algebraic and combinatorial properties of the transfer matrix T m and by which we show that lim m , n → + ∞ f ( m , n ) 1 m n = lim m → + ∞ λ m λ m − 1 , and c a p ( S h s ) ≈ 0 . 587891161775 . . . , where λ m is the largest eigenvalue of T m . Interestingly, our approach can be applied to study other related 2-dimensional or multi-dimensional constraints (for example, when applied to the 3-dimensional dimer problem, it generates better bounds than the ones given in Lundow, 2001).

The asymptotics of the transfer matrix of Sturm-Liouville equation with piecewise-entire potential function on a curve in the complex plane is obtained and studied for large absolute values of the spectral parameter.

In this paper, the stabilization problems for singular Markovian systems with Brownian motion are studied by use of sliding mode control (SMC) with the jumping transfer matrix known, unknown, or partly unknown. At first, a new stochastic sliding mode surface (SSS) is introduced. The problem of state trajectories moving among several sliding mode surfaces can be settled perfectly. By use of the proposed SSS, the sliding mode dynamics of singular Markovian systems with Brownian motion is asymptotically mean square admissible and sufficient design conditions are derived in linear matrix inequalities. Second, by use of a new approach different from the normal Lyapunov method, an SMC law is provided to avoid the state trajectories escaping from the sliding mode surface. Finally, three numerical simulations and one practical simulation are provided to illustrate the validity of the proposed method.

An effective method is proposed in this paper for the symplectic eigenvalue problem of a periodic structure with uniform stochastic properties. Since structural parameters are uncertain, the symplectic transfer matrix of a typical substructure, which is described in the state space, also has stochastic properties. An effective spectral stochastic finite element method is adopted to ensure the symplectic orthogonality of the random symplectic matrix on the premise of certain precision. By means of the Rayleigh quotient method, the symplectic eigenvalue problem of random symplectic matrix is investigated. On the basis of this, the mean value and the standard deviation of the random eigenvalues are fully discussed. The comparison between the numerical results derived from the proposed method and the Monte-Carlo simulation indicates that the proposed method has high precision. This research provides a useful guidance for the dynamic analysis of periodic structures with stochastic properties.

This paper proposes a new approach that can generate a low-order Fornasini–Marchesini (F–M) model realization for a given n -D system by taking into account the column structural properties of its transfer matrix. Specifically, a new necessary and sufficient realization condition is developed for the F–M model realization based on a resolvent invariant space associated with the Gleason’s problem specified by the given n -D transfer matrix. Then, a new constructive procedure with respect to the columns of a given transfer matrix is proposed for constructing a low-order F–M model realization. In order to apply this procedure to the multiple-input multiple-output case more effectively, an improved realization procedure based on a polynomial description is also proposed. It will be shown, by both algorithmic analysis and illustrative examples, that for a transfer matrix having more rows than columns the proposed realization approach may generate a much lower realization order than the existing one with respect to the rows of a given n -D transfer matrix recently developed by Cheng et al. Particularly, for a given scalar transfer function, our new approach always generates an F–M model realization with lower order than the existing methods.

Two forms of dynamic transfer matrix are derived for a one-dimensional (beam-like) repetitive pin-jointed structure with point masses located at nodal cross-sections, the displacement-force transfer matrix G, and the displacement-displacement transfer matrix, H. Similarity matrices are introduced to relate G and H, together with their respective metrics. Symplectic orthogonality relationships for the eigenvectors of both G and H are derived, together with relationships between their respective sets of eigenvectors. New expressions for the group velocity are derived. For repetitive structures of finite length, natural frequency equations are derived employing both G and H, including phase-closure and the direct application of boundary (end) conditions. Besides an exposition of the theory, some familiar but much new, the focus of the present paper is on the relationships between the two forms of transfer matrix, including their respective (dis)advantages. Numerical results, together with further theory necessary for interpretation, are presented in companion papers.

Sparse random linear network coding (SRLNC) is a promising solution for reducing the complexity of random linear network coding (RLNC). RLNC can be modeled as a linear operator channel (LOC). It is well known that the normalized channel capacity of LOC is characterized by the rank distribution of the transfer matrix. In this paper, we study the rank distribution of SRLNC. By exploiting the definition of linear dependence of the vectors, we first derive a novel approximation to the probability of a sparse random matrix being non-full rank. By using the Gauss coefficient, we then provide a closed approximation to the rank distribution of a sparse random matrix over a finite field. The simulation and numerical results show that our proposed approximation to the rank distribution of sparse matrices is very tight and outperforms the state-of-the-art results, except for the finite field size and the number of input packets are small, and the sparsity of the matrices is large.

We introduce a new robust stability measure for retarded type systems, which is called smoothed spectral abscissa. The main characteristics of the smoothed spectral abscissa are that it is smooth with respect to the system parameters, and it provides a trade-off between the spectral abscissa and the H2 norm of a transfer matrix related with the system. It is given by an implicit relation with the H2 norm of an auxiliary system, and its computation is based on the delay Lyapunov matrix. We show that these features make the smoothed spectral abscissa suitable for the design of robust controllers by using standard gradient-based optimization techniques.

We devise an algorithm to calculate upper bounds on the growth rates of the number of independent sets on a variety of regular two-dimensional lattices, using an amalgamation of techniques from linear algebra, combinatorics, and statistical mechanics. Our method uses Calkin and Wilf's transfer matrix eigenvalue upper bound together with the Collatz–Wielandt formula from linear algebra. To obtain a good bound, we need an approximate eigenvector, which we find using Baxter's corner transfer matrix ansatz and Nishino and Okunishi's corner transfer matrix renormalisation group method. This results in an algorithm for computing upper bounds which is far faster in practice than all other known methods. It is also the first algorithm for this problem with a polynomial, rather than exponential, memory requirement, and it is extremely parallelisable. This allows us to make dramatic improvements to the previous best known upper bounds. We apply our algorithm to five models, including independent sets on the square lattice (also known as the hard squares lattice gas from statistical mechanics). In all cases we extend the number of rigorously known digits of the growth rate by 4–6 digits.

Wave-based method for dynamic responses of net structures can provide analytic out-of-plane responses in full frequency range, helping to avoid the problem of control spillover, comparing with traditional modal analysis. However, as the scale of the net structure increases, more computing resources are required. Here, a wave-based method inspired by transfer matrix method is proposed for both orthogonal and non-orthogonal symmetric net structures. According to the symmetric characteristics of the net structure, we first divide the whole structure into several periodic elements and calculate the transfer matrix of the periodic element by using the basic wave-based method. Then by multiplying transfer matrixes of the periodic element and using all the boundary condition of net structure, the out-of-plane displacments of the net structure can be finally obtained. Our proposed wave-based transfer matrix method is proved to be more accurate and efficient than the finite element analysis for large net structures.

We describe the extension, beyond fundamental representations of the Yang-Baxter algebra, of our new construction of separation of variables bases for quantum integrable lattice models. The key idea underlying our approach is to use the commuting conserved charges of the quantum integrable models to generate bases in which their spectral problem is separated, i.e. in which the wave functions are factorized in terms of specific solutions of a functional equation. For the so-called "non-fundamental" models we construct two different types of SoV bases. The first is given from the fundamental quantum Lax operator having isomorphic auxiliary and quantum spaces and that can be obtained by fusion of the original quantum Lax operator. The construction essentially follows the one we used previously for fundamental models and allows us to derive the simplicity and diagonalizability of the transfer matrix spectrum. Then, starting from the original quantum Lax operator and using the full tower of the fused transfer matrices, we introduce a second type of SoV bases for which the proof of the separation of the transfer matrix spectrum is naturally derived. We show that, under some special choice, this second type of SoV bases coincides with the one associated to Sklyanin's approach. Moreover, we derive the finite difference type (quantum spectral curve) functional equation and the set of its solutions defining the complete transfer matrix spectrum. This is explicitly implemented for the integrable quantum models associated to the higher spin representations of the general quasi-periodic Y(gl2) Yang-Baxter algebra. Our SoV approach also leads to the construction of a Q-operator in terms of the fused transfer matrices. Finally, we show that the Q-operator family can be equivalently used as the family of commuting conserved charges enabling to construct our SoV bases., 41 pages, presentation of sections 2 and 4 improved, conclusion section added, some typos corrected, several references added

A tensor network renormalization algorithm with global optimization based on the corner transfer matrix is proposed. Since the environment is updated by the corner transfer matrix renormalization group method, the forward-backward iteration is unnecessary, which is a time-consuming part of other methods with global optimization. In addition, a further approximation reducing the order of the computational cost of contraction for the calculation of the coarse-grained tensor is proposed. The computational time of our algorithm in two dimensions scales as the sixth power of the bond dimension while the higher-order tensor renormalization group and the higher-order second renormalization group methods have the seventh power. We perform benchmark calculations in the Ising model on the square lattice and show that the time-to-solution of the proposed algorithm is faster than that of other methods., 6 pages, 9 figures

In this paper, DCT (discrete cosine transform) with matrix technique for a fractional-order Riesz differentiator is presented. First, review the fractional-order Riesz differentiator definition. Next, transfer matrix of fractional-order Riesz differentiator is obtained by DCT interpolation method. Finally, some computative problems show the importance of DCT interpolation method are discussed.

It is well known that a set of non-defect matrices can be simultaneously diagonalized if and only if the matrices commute. In the case of non-commuting matrices, the best that can be achieved is simultaneous block diagonalization. Here we give an efficient algorithm to explicitly compute a transfer matrix which realizes the simultaneous block diagonalization of unitary matrices whose decomposition in irreducible blocks (common invariant subspaces) is known from elsewhere. Our main motivation lies in particle physics, where the resulting transfer matrix must be known explicitly in order to unequivocally determine the action of outer automorphisms such as parity, charge conjugation, or time reversal on the particle spectrum., 6 pages, accepted by J.Phys.A

This paper is described DST interpolation techniques for Reisz based matrix digital differenrtiator of non-integer order. First, definition of reisz based FOD is discussed in continuous time domain and frequency domain. Next, using the DST interpolation approach the transfer matrix is obtained of matrix reisz fractional order differentiator. Finally, some analytical problems based on matrix reisz FOD is discussed that shows the greatness of DST interpolation approach.

Using the framework of the quantum separation of variables (SoV) for higher rank quantum integrable lattice models [1], we introduce some foundations to go beyond the obtained complete transfer matrix spectrum description, and open the way to the computation of matrix elements of local operators. This first amounts to obtain simple expressions for scalar products of the so-called separate states (transfer matrix eigenstates or some simple generalization of them). In the higher rank case, left and right SoV bases are expected to be pseudo-orthogonal, that is for a given SoV co-vector, there could be more than one non-vanishing overlap with the vectors of the chosen right SoV basis. For simplicity, we describe our method to get these pseudo-orthogonality overlaps in the fundamental representations of the $\mathcal{Y}(gl_3)$ lattice model with $N$ sites, a case of rank 2. The non-zero couplings between the co-vector and vector SoV bases are exactly characterized. While the corresponding SoV-measure stays reasonably simple and of possible practical use, we address the problem of constructing left and right SoV bases which do satisfy standard orthogonality. In our approach, the SoV bases are constructed by using families of conserved charges. This gives us a large freedom in the SoV bases construction, and allows us to look for the choice of a family of conserved charges which leads to orthogonal co-vector/vector SoV bases. We first define such a choice in the case of twist matrices having simple spectrum and zero determinant. Then, we generalize the associated family of conserved charges and orthogonal SoV bases to generic simple spectrum and invertible twist matrices. Under this choice of conserved charges, and of the associated orthogonal SoV bases, the scalar products of separate states simplify considerably and take a form similar to the $\mathcal{Y}(gl_2)$ rank one case., v3, 65 pages, a section "conclusions and perspectives" added, a few typos corrected

By extending the method of the author’s earlier paper, a purely recursive approach is devised for counting the total number of unfolded self-avoiding walks terminating along the line x=n, within the finite lattice strip {0,1,…,n}×{0,1,2,3} of width three. This approach yields a generating function for the sequence of numbers just described, without the need for any exact enumeration, usually entailed with an application of the transfer matrix algorithm. As a result of determining this generating function, the total number of unfolded walks within the lattice strip can then be easily computed.

We consider the XXZ spin-1/2 Heisenberg chain with antiperiodic boundary conditions. The inhomogeneous version of this model can be solved by Separation of Variables (SoV), and the eigenstates can be constructed in terms of Q-functions, solution of a Baxter TQ-equation, which have double periodicity compared to the periodic case. We compute in this framework the scalar products of a particular class of separate states which notably includes the eigenstates of the transfer matrix. We also compute the form factors of local spin operators, i.e. their matrix elements between two eigenstates of the transfer matrix. We show that these quantities admit determinant representations with rows and columns labelled by the roots of the Q-functions of the corresponding separate states, as in the periodic case, although the form of the determinant are here slightly different. We also propose alternative types of determinant representations written directly in terms of the transfer matrix eigenvalues., 40 pages, important corrections in the computations and in the final formulas, new alternative representations added

This paper proposes a method of H 2 -optimal vibration control using regional eigenvalue assignment based only on measured receptances, typically from a modal test. The sum of the squared weighted H 2 -norm of closed-loop receptance matrix and the transfer matrix from the external input to the control input is minimized by exploiting the flexibility from assigning the closed-loop eigenvalues to prescribed stable regions with single- or multiple-input control. It is found that regional eigenvalue assignment and the optimization of a free parameter matrix (in the multiple-input case) provide extra flexibility for improving the closed-loop performance over and above the selection of weighting matrices, as in standard H 2 control. Therefore, the internal stability, satisfactory response, and minimum control energy are readily guaranteed by the proposed method, whereas the standard approach is dependent upon the arbitrarily chosen weighting matrices. The method inherits the advantages of both optimal control and eigenvalue assignment by the receptance method. The working of the method and its advantages over the standard H 2 control are demonstrated numerically.

In this paper, a high-accuracy manifold element modelling scheme for dynamic fracture analysis of two-dimensional elastic cracked bodies is presented. The displacement discontinuity across crack surfaces is naturally simulated attributing to the dual cover systems in the numerical manifold method. The singularity of field gradients at crack tips is appropriately captured through the use of asymptotic basis functions. The discrete equations for dynamic analysis are solved by a new fully explicit direct time integration algorithm based on precise time step integration method, in which Taylor expansion theorem and downscaling time step integration method are used to calculate the exponential transfer matrix with high accuracy. To eliminate the sensitivity of accuracy of results to time-step size, Fourier expansion with trigonometric basis functions is utilized to obtain the approximate expression for different time-dependent loadings, the exponential transfer matrix is also further utilized to calculate the response matrix of inhomogeneous term. A matrix-based spectral analysis is used to prove the stability and convergence of proposed scheme, the results demonstrate that the present scheme is unconditionally stable and convergent. Finally, the applicability and accuracy of the proposed scheme for calculating dynamic stress intensity factors are examined by two numerical examples with different crack configurations and time-dependent loadings. By comparing with the analytical/reference solutions, it is proved that the present scheme has significant advantages in accuracy and efficiency than Newmark scheme for dynamic fracture analysis.