Recently Swatee Naik and Theodore Stanford proved that two Sequivalent knots are related by a finite sequence of doubled-delta moves on their knot diagrams. We show that classical S-equivalence is not sufficient to extend their result to ordered links. We define a new algebraic relation on Seifert matrices, called Strong S-equivalence, and prove that two oriented, ordered links L and L are rela...

Models of bipedal robots in motion are fundamentally hybrid due to the presence of continuous phases, discrete transitions, and unilateral constraints arising from the contact forces between the robot and the ground. A major challenge in the control of bipedal robots has been to create a feedback theory that provides systematic synthesis methods, provable correctness and computational tools for...

1. Let f, g ∈ C[T ] be nonconstant and relatively prime. a) If f−g 6= 0, show deg(f−g) ≥ (1/2) deg f+1, or equivalently deg f ≤ 2(deg(f−g)−1). b) Find infinitely many examples where equality occurs in the conclusion of part a. Start with an example where deg f = 3 and deg g = 2. c) When the hypothesis of relative primality is dropped, is part a still true? d) Find a lower bound on deg(f − g) in...

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Given a prime p, the Fermat quotient qp(u) of u with gcd(u, p)= 1 is defined by the conditions qp(u)≡ u p−1 − 1 p mod p, 0≤ qp(u)≤ p − 1. We derive a new bound on multiplicative character sums with Fermat quotients qp(`) at prime arguments `. 2010 Mathematics subject classification: primary 11A07; secondary 11L40, 11N25.

Let n be a positive odd integer and let p > n + 1 be a prime. We mainly derive the following congruence: ∑ 0<i1<···<in<p ( i1 3 ) (−1)i1 i1 · · · in ≡ 0 (mod p).

We prove that, for any > 0, there exist a constant C > 0 such that the interval [x, x + C √ x] contains numbers whose all prime factors are smaller than x √ e)+ .

Let p > 3 be a prime. We derive the following new congru-ences: p−1 n=0 (2n + 1)A n ≡ p (mod p 4) and p−1 n=0