and Applied Analysis 3 the scalar product of two vectors x and y. ‖x‖ denotes the Euclidean vector norm of x. λ A denotes the set of all eigenvalues ofA. λmax A is defined by λmax A max{Re λ : λ ∈ λ A }. λmin A is defined by λmin A min{Re λ : λ ∈ λ A }. For h ≥ 0, C −h, 0 , R denotes the set of allR-valued continuous functions mapping −h, 0 into R. A matrixA is semipositive definite A ≥ 0 if 〈A...

The universal thermodynamic driving force for coherent plane interface reorientation (IR) during first-order phase transformations (PT) in solids is derived. The relation between the rates of IR and interface propagation (IP) and the corresponding driving forces are derived for combined athermal and drag interface friction. The coupled evolution of IR and IP during cubic-tetragonal and tetragon...

Event count data are distinguished by being positive and integer valued with often small numbers of unique values. Examples include just about anything measured by counts or summary frequency data. Standard ordinary least squares (OLS) regression modeling requires the assumption that the model errors, εi, are independently and identically distributed, (i.i.d.)~ N(0,σ), normal random variables. ...

We show that for a typical coordinate projection of a subgaussian class of functions, the infimum over signs inf(εi) supf∈F | ∑k i=1 εif(Xi)| is asymptotically smaller than the expectation over signs as a function of the dimension k, if the canonical Gaussian process indexed by F is continuous. To that end, we establish a bound on the discrepancy of an arbitrary subset of R using properties of ...

Let bm(n) denote the number of partitions of n into powers of m. Define σr = ε2m 2 + ε3m 3 + · · · + εrm, where εi = 0 or 1 for each i. Moreover, let cr = 1 if m is odd, and cr = 2 r−1 if m is even. The main goal of this paper is to prove the congruence bm(m n− σr −m) ≡ 0 (mod m/cr). For σr = 0, the existence of such a congruence was conjectured by R. F. Churchhouse some thirty years ago, and i...

Consider measuring a vector x ∈ R through the inner product with several measurement vectors, a1, a2, . . . , am. It is common in both signal processing and statistics to assume the linear response model yi = 〈ai, x〉+ εi, where εi is a noise term. However, in practice the precise relationship between the signal x and the observations yi may not follow the linear model, and in some cases it may ...

Given εi ∈ [0,1) for each 1 < i < n, a particle performs the following random walk on {1,2, . . . ,n}: If the particle is at n, it chooses a point uniformly at random (u.a.r.) from {1, . . . ,n− 1}. If the current position of the particle is m (1 < m < n), with probability εm it decides to go back, in which case it chooses a point u.a.r. from {m + 1, . . . ,n}. With probability 1− εm it decides...

In the present work, two SIR patterns with demography will be considered: classical pattern and a modified linear coefficient of infection transmission. By reformulating each first-order differential systems as system second-order equations, we examine nonlinear dynamics from Jacobi stability perspective through Kosambi–Cartan–Chern (KCC) geometric theory. The intrinsic properties studied by de...

The polyhedral realizations for crystal bases of the integrable highest weight modules Uq(g) have been introduced in Nakashima (1999) [13], which describe as sets lattice points infinite Z-lattice Z∞ given by some system linear inequalities, where g is a symmetrizable Kac-Moody Lie algebra. To construct realization, we need to fix an sequence ι from indices simple roots. If pair (ι,λ) (λ: domin...

Consider measuring a vector x ∈ Rn through the inner product with several measurement vectors, a1, a2, . . . , am. It is common in both signal processing and statistics to assume the linear response model yi = 〈ai, x〉+ εi, where εi is a noise term. However, in practice the precise relationship between the signal x and the observations yi may not follow the linear model, and in some cases it may...