Related papers: Generalized Miller Formulae
We study multiplicative dependence between terms of the $k$-generalized Pell sequence $(P_n^{(k)})_{n\ge 2-k}$, defined by the linear recurrence \[ P_n^{(k)} = 2P_{n-1}^{(k)} + P_{n-2}^{(k)} + \dots + P_{n-k}^{(k)}, \] with initial…
We establish a new generalized Taylor's formula for power fractional derivatives with nonsingular and nonlocal kernels, which includes many known Taylor's formulas in the literature. Moreover, as a consequence, we obtain a general version…
In this paper we generalize the notion of the Taylor spectrum to modules over an arbitrary Lie algebra and study it for finite-dimensional modules. We show that the spectrum can be described as the set of simple submodules in case of…
We present a general theory of the magnetic susceptibility of insulators that can be extended to treat spatially varying and finite frequency fields. While there are existing results in the literature for the zero frequency response that…
We study the perturbation of bound states embedded in the continuous spectrum which are unstable by the Fermi Golden Rule. The approach to resonance theory based on spectral deformation is extended to a more general class of quantum systems…
The uncertainty principle lies at the heart of quantum physics, and is widely thought of as a fundamental limit on the measurement precisions of incompatible observables. Here we show that the traditional uncertainty relation in fact…
In this paper, we introduce a concept of non-dependence of variables in formulas. A formula in first-order logic is non-dependent of a variable if the truth value of this formula does not depend on the value of that variable. This variable…
The formula for the frequency shift of a plane-polarized light wave reflected from a uniformly moving mirror is derived directly from the constant light speed postulate and the basic principles of wave optics. Unlike the original derivation…
We define spherical diffraction measures for a wide class of weighted point sets in commutative spaces, i.e. proper homogeneous spaces associated with Gelfand pairs. In the case of the hyperbolic plane we can interpret the spherical…
The notion of f-oscillators generalizing q-oscillators is introduced. For classical and quantum cases, an interpretation of the f-oscillator is provided as corresponding to a special nonlinearity of vibration for which the frequency of…
We present a systematic formulation of scattering theory for nonlinear interactions in one dimension and develop a nonlinear generalization of the transfer matrix that has a composition property similar to its linear analog's. We offer…
In this paper, we show that a partitioned formula \phi is dependent if and only if \phi has uniform definability of types over finite partial order indiscernibles. This generalizes our result from a previous paper [1]. We show this by…
We derive conditions for $L_2$ differentiability of generalized linear models with error distributions not necessarily belonging to exponential families, covering both cases of stochastic and deterministic regressors. These conditions…
A scalar Wigner distribution function for describing polarized light is proposed in analogy with the treatment of spin variables in quantum kinetic theory. The formalism is applied to the propagation of circularly polarized light in…
A general formalism to solve nonlinear differential equations is given. Solutions are found and reduced to those of second order nonlinear differential equations in one variable. The approach is uniformized in the geometry and solves…
In classical field theory, radiation does not reflect off of reflectionless kinks. In quantum field theory, radiation quanta, called mesons, can be reflected. We provide a general analytical formula for the leading order amplitude and…
A quantum theory of dispersion for an inhomogeneous solid is obtained, from a starting point of multipolar coupled atoms interacting with an electromagnetic field. The dispersion relations obtained are equivalent to the standard classical…
We consider an ordinary nonlinear differential equation with generalized coefficients as an equation in differentials in algebra of new generalized functions. Then the solution of such equation will be a new generalized function. In the…
The survey is devoted to the combinatorial and metric theory of filtrations, i.\,e., decreasing sequences of $\sigma$-algebras in measure spaces or decreasing sequences of subalgebras of certain algebras. One of the key notions, that of…
The Westervelt equation describes the propagation of pressure waves in continuous nonlinear and, eventually, diffusive media. The classical framework of this equation corresponds to fluid dynamics theory. This work seeks to connect this…