A simple model for dendritic growth is given by S2d'" + 9' — cos(9). For S ss 1 we prove that there is no bounded, monotonic solution which satisfies d(-oo) = -7t/2 and Q(oo) = n/2. We also investigate the existence of bounded, monotonic solutions of an equation derived from the Kuramoto-Sivashinsky equation, namely y" + y = 1 y1 /2. We prove that there is no monotonic solution which satisfies ...

A collection C of finite L-structures is a 1-dimensional asymptotic class if for every m ∈ N and every formula φ(x, ȳ), where ȳ = (y1, . . . , ym): (i) There is a positive constant C and a finite set E ⊂ R>0 such that for every M ∈ C and ā ∈ Mm, either |φ(M, ā)| ≤ C, or for some μ ∈ E, ∣∣|φ(M, ā)| − μ|M |∣∣ ≤ C|M | 2 . (ii) For every μ ∈ E, there is an L-formula φμ(ȳ), such that φμ(M) is precis...

Using the Bethe ansatz, we obtain the exact solution of the master equation for the totally asymmetric exclusion process on an infinite one-dimensional lattice. We derive explicit expressions for the conditional probabilities P (x1, . . . , xN ; t|y1, . . . , yN ; 0) of finding N particles on lattice sites x1, . . . , xN at time t with initial occupation y1, . . . , yN at time t = 0.

We study Minkowski’s inequality Da b(x1 + x2, y1 + y2) ≤ Da b(x1, y1) +Da b(x2, y2) (x1, x2, y1, y2 ∈ R+) and its reverse where Da b is the difference mean introduced by Stolarsky. We give necessary and sufficient conditions (concerning the parameters a, b) for the inequality above (and for its reverse) to hold.

Suppose y1, . . . , yn are independent random variables. The density pθi(·) of yi is supposed to be known up to a parameter θi. Let θ̂ = ( θ̂1(y), . . . , θ̂n(y) ) be an estimate of θ constructed from the sample y = (y1, . . . , yn). Note that the estimate of θi can depend on yj , j 6= i. Let `i(·, ·) be a loss function so that `i(θi, θ̂i) represents the loss of using θ̂i as the estimate of θi. The ...

F (t, x, y) = det(tIn + x(T + T )/2 + y(T − T )/(2i)). Let ΓF be the algebraic curve of F (t, x, y), i.e., ΓF = {[(t, x, y)] ∈ CP : F (t, x, y) = 0}, where [(t, x, y)] is the equivalence class containing (t, x, y) ∈ C − (0, 0, 0) under the relation (t1, x1, y1) ∼ (t2, x2, y2) if (t2, x2, y2) = k(t1, x1, y1) for some nonzero complex number k. The dual curve Γ∧F of ΓF is defined by Γ∧F = {[(T,X, ...

We recall from [1] that a line segment in Rmax, and consiquently Rmax has a similar case, has one of the three following forms: When X ≤ Y and x1 − y1 ≤ x2 − y2, then [X,Y ] = [X, (y1 + x2 − y2, x2)] ∪ [(y1 + x2 − y2, x2), Y ] (2) When X ≤ Y and x2 − y2 ≤ x1 − y1, then [X,Y ] = [X, (x1, y2 + x1 − y1)] ∪ [(x1, y2 + x1 − y1), Y ] (3) When X Y and Y X, then [X,Y ] = [X,max(X,Y )] ∪ [max(X,Y ), Y ]...

where Y = (y1, · · · , yp) and ỹi = √ σyi,w̃i = wi/σ . These properties are used for the proof of the main results. Note: throughout the supplementary material, when evaluation is taken place at σ = σ̄, sometimes we omit the argument σ in the notation for simplicity. Also we use Y = (y1, · · · , yp) to denote a generic sample and use Y to denote the p× n data matrix consisting of n i.i.d. such sa...