We study the existence and nonexistence of positive singular solutions to second–order non-divergence type elliptic inequalities in the form N ∑ i,j=1 aij(x) ∂u ∂xi∂xj + N ∑ i=1 bi(x) ∂u ∂xi ≥ K(x)u, −∞ < p <∞, with measurable coefficients in a punctured ball BR \{0} of R , N ≥ 1. We prove the existence of a critical value p∗ that separates the existence region from non-existence. In the critic...

where z = (x, t) ∈ RN×R and 1 ≤ p0 ≤ N. By convenience, hereafter the term “Kolmogorov equation” will be shortened to KE. We assume the following hypotheses: (H.1) the matrix A0 = (aij)i,j=1,...,p0 is symmetric and uniformly positive definite in R p0 : there exists a positive constant μ such that |η| μ ≤ p0 ∑ i,j=1 aij(z)ηiηj ≤ μ|η|, ∀η ∈ R0 , z ∈ R; (1.2) (H.2) the matrix B ≡ (bij) has constan...

This condensed summary highlights the results of a 2016 AIJ paper reporting on a successful generalpurpose conjecturing program.

Assume F = {f1, · · · , fn} is a family of nonnegative functions of n− 1 nonnegative variables such that, for every matrix A of order n, |aii| > fi (moduli of off-diagonal entries in row i of A) for all i implies A nonsingular. We show that there is a positive vector x, depending only on F , such that for all A = (aij), and all i, fi = ∑ j |aij | xj xi . This improves a theorem of Ky Fan [F], a...

Exercise 7. Let A be an n × n matrix such that the sum of every row is 0 and the sum of every column is 0. Let Aij be the (n − 1) × (n − 1) matrix obtained by removing row i and column j from A. Prove: det(Aij) = (−1) det(A11). (Note that this result applies in particular to the Laplacian L: the determinant of the reduced Laplacian, obtained by removing the i-th row and the i-th column from L, ...

Author: Jenny Zhang First, let’s preview what mutually orthogonal Latin squares are. Two Latin squares L1 = [aij ] and L2 = [bij ] on symbols {1, 2, ...n}, are said to be orthogonal if every ordered pair of symbols occurs exactly once among the n2 pairs (aij , bij), 1 ≤ i ≤ n, 1 ≤ j ≤ n. Now, let me introduce a related concept which is called transversal. A transversal of a Latin square is a se...

Let A be a connected integer symmetric matrix, i.e., A = (aij) ∈ Mn(Z) for some n, A = AT , and the underlying graph (vertices corresponding to rows, with vertex i joined to vertex j if aij 6= 0) is connected. We show that if all the eigenvalues of A are strictly positive, then tr(A) ≥ 2n− 1. There are two striking corollaries. First, the analogue of the Schur-SiegelSmyth trace problem is solve...

We study uniqueness of parabolic equations for measures μ(dtdx) = μt(dx)dt of the type L∗μ = 0, satisfying μt → ν as t→ 0, where each μt is a probability measure on Rd, L = ∂t + aij(t, x)∂xi∂xj + bi(t, x)∂xj is a differential operator on (0, T ) × Rd and ν is a given initial measure. One main result is that uniqueness holds under uniform ellipticity and Lipschitz conditions on aij but for bi me...

The lower bound is the same as before, due to the fact that Tr(n) is Kr+1-free. Define the adjacency matrix A = A(G) = (aij) for a graph G of order n. Let V = {v1, · · · , vn}. Then A is a n-by-n 0 − 1 matrix such that aij = 1 if and only if vivj ∈ E. Thus A is symmetric. We will be interested in a quadratic form ⟨Ax,x⟩ where x denotes a vector of length n. This is often called the Lagrangian o...

Matrices of dimensions m × 1 and 1 × n are called column and row vectors, respectively. We will typically denote column and row vectors by lower case Latin letters, e.g. a, b, x, y and other matrices by upper case Latin letters, e.g. A, B, X, Y . The scalars (or 1 × 1 matrices) will be frequently denoted by Greek letters α, β, λ, μ, etc. Unless stated otherwise, all scalars will be real numbers...