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The shortage of scientific understanding on physics has hindered chemistry students from understanding quantum mechanics in Japan. On the other hand, many of them studied full course of the high school level math. The author has made use of the math understanding to let the students understand quantum mechanics. As one of examples for usage of the math understanding, I suggest that a teaching method of “bra vector” in quantum mechanics by using axiomatic construction of linear algebra.

In this paper, the physical phenomena in fractional calculus of the space–time fractional Bi-Hamiltonian Boussinesq system involving the Riemann–Liouville derivatives is studied. The wave dispersion relationship in the linear form of the governing system is analyzed and then that relationship is used to obtain the velocity of the phase and group. Soliton solutions are developed using the fractional sub-equation method. The soliton solutions obtained in the form of trigonometry, hyperbolic and rational function illustrate that the approach of fractional sub-equation is very effective in evaluating the physical phenomena.

Let F be a local field of characteristic zero. Let μ be a good character of GL p ( F ) × GL p + 1 ( F ) . We study the generalized linear period problem for the pair ( G , H p , p + 1 ) = ( GL 2 p + 1 ( F ) , GL p ( F ) × GL p + 1 ( F ) ) and we prove that any bi- ( H p , p + 1 , μ ) -equivariant tempered generalized function on G is invariant under the matrix transpose. We also show that any P ∩ H p , p + 1 -invariant linear functional on an H p , p + 1 -distinguished irreducible smooth representation of G is also H p , p + 1 -invariant if F is nonarchimedean, where P is the standard mirabolic subgroup of G consisting of matrices with last row vector ( 0 , ⋯ , 0 , 1 ) .

This paper concentrates on a simple population model incorporating fear. Firstly, positivity and steady state analysis are performed, where the theoretical investigations show that change in the level of fear in prey population does not effect the local stability of the system around each equilibria (either stable or unstable). For the deterministic model, the numerical simulations are plotted for the density of prey species as a function of various system parameters. The stability analysis of the coexisting state shows that only transcritical bifurcation, where the steady states intersect, is observed. Secondly, the model is analysed with Gaussian noise term incorporated in the prey’s death rate. The model comprising noise term turns the system into stochastic differential equations and irregular noise related oscillations are observed in the densities of both species.

The notion of approximate left $$\phi$$ -biflatness for Banach algebras is introduced, where $$\phi :A\longrightarrow {\mathbb {C}}$$ is a non-zero multiplicative linear functional. Under this new concept, the approximate left $$\phi$$ -biflatness for some algebras like group algebras and measure algebras are studied. Moreover, some hereditary properties of this notion are given. Furthermore, some examples to show the differences of our notion and the classical ones are presented.

Assume that $$\Omega \subset \mathbb {R}^k$$ Ω ⊂ R k is an open set, V is a real separable Banach space and $$f_1,\ldots ,f_N :\Omega \rightarrow \Omega $$ f 1 , … , f N : Ω → Ω , $$g_1,\ldots , g_N:\Omega \rightarrow \mathbb {R}$$ g 1 , … , g N : Ω → R , $$h_0:\Omega \rightarrow V$$ h 0 : Ω → V are given functions. We are interested in the existence and uniqueness of solutions $$\varphi :\Omega \rightarrow V$$ φ : Ω → V of the linear equation $$\varphi =\sum _{k=1}^{N}g_k\cdot (\varphi \circ f_k)+h_0$$ φ = ∑ k = 1 N g k · ( φ ∘ f k ) + h 0 in generalized Orlicz spaces.

A novel High-order extended Strong tracking filter (H-STF) is proposed for a class of nonlinear systems. All high-order polynomial terms in the state model are regarded as implicit variables; the original state model is equivalently formulated into a pseudo-linear form; the dynamic relationship between each implicit variable and all variables is modeled; original state model is rewritten into an augmented linear model; the nonlinear measurement model can be rewritten into linear form; taking into account the problems of modeling errors and state mutations that may be caused by the introduction of implicit variables, a high-order extended strong tracking filter is designed. Examples are presented to demonstrate the effectiveness of the new algorithm.

Semidefinite programming is based on optimization of linear functionals over convex sets defined by linear matrix inequalities, namely, inequalities of the form $$L_A(X)=I-A_1X_1-\dots-A_g X_g\succeq0.$$ Here the $X_j$ are real numbers and the set of solutions is called a spectrahedron. These inequalities make sense when the $X_i$ are symmetric matrices of any size, $n\times n$, and enter the formula though tensor product $A_i\otimes X_i$: The solution set of $L_A(X)\succeq0$ is called a free spectrahedron since it contains matrices of all sizes and the defining ``linear pencil" is ``free" of the sizes of the matrices. In this article, we report on empirically observed properties of optimizers obtained from optimizing linear functionals over free spectrahedra restricted to matrices $X_i$ of fixed size $n\times n$. The optimizers we find are always classical extreme points. Surprisingly, in many reasonable parameter ranges, over 99.9\% are also free extreme points. Moreover, the dimension of the active constraint, $\ker(L_A(X^\ell))$, is about twice what we expected. Another distinctive pattern regards reducibility of optimizing tuples $(X_1^\ell,\dots,X_g^\ell)$. We give an algorithm for representing elements of a free spectrahedron as matrix convex combinations of free extreme points; these representations satisfy a very low bound on the number of free extreme points neede, 46 pages body. Includes table of contents and index

In this paper, we investigate the homological notion of left $$\phi $$ -biprojectivity for certain Banach algebras, where $$\phi $$ is a nonzero multiplicative linear functional. Our initial result states that this notion is equivalent to left $$\phi $$ -contractibility, provided that $$\mathcal {A}$$ has a left approximate identity. As an application, we study left $$\phi $$ -biprojectivity of Banach algebras related to locally compact groups. For instance, we show that for a locally compact group $$\mathcal {G}$$ , the Segal algebra $$S(\mathcal {G})$$ is left $$\phi $$ -biprojective if and only if $$\mathcal {G}$$ is compact and the Fourier algebra $$A(\mathcal {G})$$ is left $$\phi $$ -biprojective if and only if $$\mathcal {G}$$ is discrete. Finally, we give some examples which show the differences between our new notion and the classical ones.

In this article, we deal with the consensus problem of multiple partial-state coupled linear systems, which are neutrally stable. These systems communicate over dynamic undirected networks, which change continuously and can be disconnected at any time. We develop an analysis framework from uniform complete observability theory to work out the necessary and sufficient conditions for exponential consensus. It turns out that, with a suitably designed feedback matrix, exponential consensus can be realized globally and uniformly if and only if a jointly $(\delta,T)$ -connected condition and an observability condition relying only on system and input matrices are satisfied. A lower bound of the convergence rate is also provided. We figure out the proof by applying matrix analysis and linear functional analysis. A simulation example is presented to illustrate our theoretical findings.

In this study, a matrix-collocation method is developed numerically to solve the linear Fredholm-Volterra-type functional integral and integro-differential equations. The linear functional integro-differential equations are considered under initial conditions. The mentioned type problems often appear in various branches of science and engineering such as physics, biology, mechanics, electronics. The method essentially is a collocation method based on the Lagrange polynomials and matrix operations. By using presented method, the problem is reduced to a system of linear algebraic equations. The solution of this system gives the coefficients of assumed solution. An error analysis based on the residual function is studied. Some examples are solved to demonstrate the accuracy and efficiency of the method.

This paper is concerned with systems of linear functional differential equations of mixed type, that is, differential equations with delays and advances. In one of the sections, we obtain sufficient conditions under which the system has at least a nonoscillatory solution.These criteria, based on the form of the system matrices, are obtained via the analysis of the characteristic equation. In the other section we obtain criteria for the oscillatory behavior of the solutions. Some of these criteria are also obtained by studying the characteristic equation, while others use the definition of matrix measures. A set of examples illustrates the application of the theoretical results.

We consider questions concerning the solvability and the number of solutions of systems of Boolean equations. Many mathematical models arising in both operations research and cryptography are described in the language of such systems. This is due, in particular, to the fact that, in general, the problem of checking the compatibility of such systems of equations is NP-complete; therefore, the study of the qualitative properties of a system of Boolean equations provides additional information that permits one either to single out polynomially solvable particular cases or to increase the efficiency of enumeration schemes. The focus is on two aspects. The first one concerns the study of the existence and number of solutions of a Boolean equation and a system of equations when parametrizing the problem by the right-hand sides. Formulas and estimates are given for calculating this number both in general and in particular cases. Its maximum is also investigated depending on the specified parameter. The second aspect is devoted to a special case of the problem when the equation is given by the so-called continuous linear form. The properties of such forms and various criteria of continuity are studied.

Although Cattaneo et al. (2019) provided a data-driven framework for power computations for Regression Discontinuity Designs in line with rdrobust Stata and R commands, which allows higher-order fu...

The triply bonded heavier main-group compounds have a textbook trans-bent geometry, in contrast to a familiar linear form found for the lightest analogues. Strikingly, the unexpected linear group 13 E≡E triple bonds were herein found in the D4h -symmetry E2 Li62+ clusters, and they possess a large barrier (>18.0 kcal/mol) towards the dissociation of Li+ . The perfectly surrounded Li4 motifs and two linear coordinated Li atoms strongly suppress the increasing nonbonded electron density of heavier E atoms, making two degenerate π bonds and one multi-center σ bond in linear heavier main-group triple bonds. The surrounding Li6 motifs not only creates an effective electronic structure to form a linear E≡E triple bond, but the resulting electrostatic interactions account for the highly stable global E2 Li62+ clusters.

We study the nonparametric estimation and specification testing for partially linear functional-coefficient dynamic panel data models, where the effects of some covariates on the dependent variable...

As a mathematical model of mechanical and electronic oscillation, the study and analysis of the oscillation characteristics of the solution of the non-zero continuous linear functional equation are of great significance in theory and practice. In view of the oscillation characteristics of the solutions of the second and third order non-zero continuous functional equations, this paper puts forward a hypothesis, studies the oscillation and asymptotics of the non-zero continuous linear functional differential equations by using the generalized Riccati transformation and the integral average technique, and establishes some new sufficient conditions for the oscillation or convergence to zero of all solutions of the equations, so as to obtain a new theorem for the solutions of the non-zero continuous linear functional equations.

This paper is concerned with the event-triggered control of positive semi-Markovian jump systems without/with input saturation. The considered systems are subject to a stochastic semi-Markovian process whose sojourn time is dependent on a non-exponential distribution. First, an event-triggering condition is introduced in a linear form for the systems. A class of event-triggered feedback controllers is proposed using matrix decomposition technique. By using a stochastic co-positive Lyapunov function, the systems’ positivity and stability are guaranteed. Then, the obtained results are developed for the systems with input saturation. A cone set is chosen as the attraction domain and the corresponding attraction domain gain matrix is designed in terms of standard linear programming approach. Finally, two numerical examples are provided to verify the validity and effectiveness of the presented theoretical findings.

In this brief, the prescribed finite-time $H_{\infty }$ problems of nonlinear descriptor systems (NDSs) with and without actuator saturation are respectively investigated. When addressing this problem of NDSs subject to actuator saturation, it is pointed out that the existing treatments for the saturation nonlinearities are not so effective. The reason is that some of them cannot ensure that the resulting closed-loop systems are impulse controllable, and others cannot make sure that the closed-loop system can achieve prescribed finite-time stable. To overcome this issue, an available and interesting treatment is proposed. The main characteristic of this treatment is that the saturation nonlinearity is decomposed into a linear form consisting of the control signal and its constant constraints. Based on this treatment, the state feedback scheme is adopted for the considered problem, and some sufficient conditions are obtained. To illustrate the efficiency of the proposed results, an example about circuit systems is provided.

We consider a model of intertemporal choice where time is a continuum, the set of instantaneous outcomes (e.g., consumption bundles) is a topological space, and intertemporal plans (e.g., consumption streams) must be continuous functions of time. We assume that the agent can form preferences over plans defined on open time intervals. We axiomatically characterize the intertemporal preferences that admit a representation via discounted utility integrals. In this representation, the utility function is continuous and unique up to positive affine transformations, and the discount structure is represented by a unique Riemann–Stieltjes integral plus a unique linear functional measuring the long-run asymptotic utility.

Many studies take sampling to apply fuzzy C-means (FCM) to very large data sets. However, comparing with the plenty algorithms for sampling-based FCM, few research works theoretically study the generalization error. In this article, we take the concept probably approximately correct learning to study it, and define the generalization error as the maximum difference between the empirical risks and the real risks of solutions in solution space. First, we analyze this generalization error under finite solution space, and prove two theorems by Hoeffding's inequality, where one of them relies on a reasonable hypothesis. Then, we discuss the situation in which the solution space is infinite, and propose a theorem and a corollary by another hypothesis, where the results under finite solution space are used in the proofs. Finally, we bound the generalization error from the perspective of the FCM algorithm's convergence, where we take Taylor expansion to transform the risk function to the linear form and estimate the upper bound of its Vapnik-Chervonenkis (VC) dimension. The hypotheses proposed in this article are all intuitive and common phenomena in practice. Our results show the upper bound of the generalization error under the given minimum probability, which can offer insight into the stability of sampling-based FCM and can guide in its application.

In this paper, we study two properties viz., property-U and property-SU of a subspace Y of a Banach space X, which correspond to the uniqueness of the Hahn–Banach extension of each linear functional in (Formula presented.) and when this association forms a linear operator of norm-1 from (Formula presented.) to (Formula presented.). It is proved that, under certain geometric assumptions on (Formula presented.) these properties are stable with respect to the injective tensor product; Y has property-U (SU) in Z if and only if (Formula presented.) has property-U (SU) in (Formula presented.). We prove that when (Formula presented.) has the Radon–Nikod (Formula presented.) m Property for (Formula presented.), (Formula presented.) has property-U (property-SU) in (Formula presented.) if and only if Y is so in X. We show that if (Formula presented.) and Y has property-U (SU) in X then Y/Z has property-U (SU) in X/Z. On the other hand, Y has property-SU in X if Y/Z has property-SU in X/Z and (Formula presented.) is an M-ideal in X. This partly solves the 3-space problem for property-SU. We characterize all hyperplanes in (Formula presented.) which have property-SU. We derive necessary and sufficient conditions for all finite codimensional proximinal subspaces of (Formula presented.) which have property-U (SU).