We show that the known bounded-depth proofs of the Weak Pigeonhole Principle PHP 2n n in size n O(log(n)) are not optimal in terms of size. More precisely, we give a size-depth trade-oo upper bound: there are proofs of size n O(d(log(n)) 2=d) and depth O(d). This solves an open problem of Maciel, Pitassi and Woods (2000). Our technique requires formalizing the ideas underlying Nepomnja s cij's ...

where A is the infinitesimal generator of a compact analytic semigroup of linear operators, (T (t))t≥0, on a Banach space X; the history xt, xt(θ) = x(t+θ), belongs to an appropriate phase space D and G,F : R×D → X are continuous functions. The Massera Theorem [10] is a pioneer work in the study of the relations between the boundedness of solutions and the existence of periodic solutions. In Ma...

Lemma 3. Theorem 1 is true if and only if Theorem 2 is true. Proof. Assume that Theorem 2 is true and let θ′ 1, . . . , θ ′ k, 1 be real numbers that are linearly independent over Z, let α1, . . . , αk be real numbers, let N > 0 and let 0 < < 1. Let θm = θ ′ m − qm with 0 < θm ≤ 1. Because θ′ 1, . . . , θ′ k, 1 are linearly independent over Z, so are θ1, . . . , θk, 1. Using Theorem 2 with k + ...

In this paper we provide sufficient conditions that ensure the convexity of the inverse images of an operator, monotone in some sense. Further, conditions that ensure the monotonicity, respectively the local injectivity of an operator are also obtained. Combining the conditions that provide the local injectivity, respectively the convexity of the inverse images of an operator, we are able to ob...

In this paper we study linear fractional relations defined in the following way. Let Bi, B′ i, i = 1, 2, be Banach spaces. We denote the space of bounded linear operators by L. Let T ∈ L(B1 ⊕ B2,B 1 ⊕ B′ 2). To each such operator there corresponds a 2 × 2 operator matrix of the form T = ( T11 T12 T21 T22 ) , (∗) where Tij ∈ L(Bj,B i), i, j = 1, 2. For each such T we define a set-valued map GT f...

Using the properties of theta-series and Schwarz reflection principle, a proof for Riemann hypothesis (RH) is directly presented and the first ten nontrivial zeros are easily obtained. From now on RH becomes Riemann Theorem (RT) and all its equivalent results and the consequences assuming RH are true.

Improper priors are used frequently, but often formally and without reference to a sound theoretical basis. The present paper demonstrates that Kolmogorov’s (1933) formulation of probability theory admits a minimal generalization which includes improper priors and a general Bayes theorem. The resulting theory is closely related to the theory of conditional probability spaces formulated by Renyi...

We provide a different approach to and prove a (partial) generalisation of a recent theorem on the structure of low energy solutions of the compatible two well problem in two dimensions [Lor05], [CoSc06]. More specifically we will show that a “quantitative” two well Liouville theorem holds for the set of matrices K = SO (2) ∪ SO (2) H where H = ( σ 0 0 σ−1 ) under a constraint on the L norm of ...

We first examine points and lines within projective spaces. Then we classify affine conics based on the classification of projective conics. Based on the parametrization of conics, we also prove two easy cases of Bézout’s Theorem. In the end we turn to the discussion of the space of conics and the notion of pencils of conics.