The second order linear differential equation (p(x)y′)′ + q(x)y = 0 , x ∈ (0, x0] is considered, where p, q ∈ C1(0, x0], p(x) > 0, q(x) > 0 for x ∈ (0, x0]. Sufficient conditions are established for every nontrivial solutions to be nonrectifiable oscillatory near x = 0 without the Hartman–Wintner condition.

Let q be a power of a prime p, and let k = Fq be an algebraic closure of Fq. For objects over Fq we use a subscript 0, and unadorned letters denote the base change to k. For example, X0 denotes a scheme (or stack) over Fq and X denotes the fiber product X0 ×Spec(Fq) Spec(k). Let X0 be a separated finite type Fq-scheme. A correspondence on X0 is a diagram of separated finite type Fq-schemes C0 c...

where g : R × R × R → R is a C function. Once we specify a parameter path u(t), t ≥ t1, in the parameter space we can solve (1.1) uniquely for x(t), t ≥ t1, with the given initial condition x(t1) = x0. Alternatively if we specify a parameter path u(t), t ≤ t1, in the parameter space we can solve (1.1) uniquely for x(t), t ≤ t1, with the given terminal condition x(t1) = x0. Suppose now that for ...

X∗ : X0 ⊆ X1 ⊆ X2 ⊆ · · · ⊆ Xn ⊆ · · · ⊆ X. We replace the fundamental group π1(X, ∗) by the homotopy crossed complex πX∗ which consists of the family of groups Cn(p) = πn(Xn, Xn−1, p) for n > 2 and all p ∈ X0, together with the fundamental groupoid C1 = π1(X1, X0) over C0 = X0, all with the standard boundary maps Cn(p) → Cn−1(p) and action of C1. ∗This research was supported by the Science Res...

Let f : [−1, 1] → IR, x0 ∈ (−1, 1), r ≥ 0 be an integer. The point x0 is called a singularity of f of order r if the derivative f (r) has a jump discontinuity at x0, but is continuous at every other point of some neighborhood of x0. In this paper, we propose a sequence of polynomial operators {τj} with the following properties. Each τj is computed using the values f(cos(kπ/2 j)), k = 1, · · · ,...

Various kinds of closed centred sets in normed spaces are considered. Necessary and sufficient conditions are obtained for every decreasing sequence of such sets to have nonempty intersection. Let X be a real or complex normed space. Recall that a set S ⊆ X is symmetric if S = −S and balanced if tS ⊆ S for all t ∈ [−1, 1]. It is convenient to introduce the following definitions. Definitions. 1....

We say that a simple graph G is Seidel integral if its spectrum consists entirely of integers. If ?Ka,a ? ?Kb,b integral, we show it belongs the class graphs [kt/? x0 + mt/?z]Ka,a y0 a/?z]nKb,b, where (i) = (t+?n)k+?m and b ?m; (ii) t, k, ?,m, n N such (m, n) 1, (n, t) 1 (?, 1; (iii) (a,mt) | kt; (iv) (x0, y0) particular solution linear Diophantine equation ax (mt)y (v) z z0 least integer (kt/?...

We find a new Tribonacci-like sequence of positive integers 〈x0, x1, x2, . . .〉 given by xn = xn−1 + xn−2 + xn−3 , n ≥ 3, and gcd(x0, x1, x2) = 1 that contains no prime numbers. We show that the sequence with initial values x0 = 151646890045, x1 = 836564809606, x2 = 942785024683 is the current record in terms of the number of digits.

By a result due to Furstenberg, a homeomorphism T of a compact space is distal if and only if it possesses the property of IP-recurrence, meaning that for any x0 ∈ X, for any open neighborhood U of x0, and for any sequence (ni) in Z, the set RU (x0) = {n ∈ Z : T x0 ∈ U} has a non-trivial intersection with the set of finite sums {ni1 +ni2 + · · ·+nis : i1 < i2 < . . . < is, s ∈ N}. We show that ...

For simplicity, we adopt the following rules: x, x0, y, y0, z, z0, r denote real numbers, u, u0 denote elements of R3, f , f1, f2 denote partial functions from R3 to R, R denotes a rest, and L denotes a linear function. Let f be a partial function from R3 to R and let u be an element of R3. We say that f is partial differentiable on 1st-1st coordinate in u if and only if the condition (Def. 1) ...