We prove a separation between monotone and general arithmetic formulas for polynomials of constant degree. We give an example of a polynomial C in n variables and degree k which is computable by a homogeneous arithmetic formula of size O(k2n2), but every monotone formula computing C requires size (n/kc)Ω(log k), with c ∈ (0, 1). Since the upper bound is achieved by a homogeneous arithmetic form...

(Def. 2) lenpr2(h1) = lenh1 and for every n such that n ∈ dompr2(h1) holds pr2(h1)(n) = h1(n)2. Let us consider A, B, let f1 be a homogeneous quasi total non empty partial function from A∗ to A, and let f2 be a homogeneous quasi total non empty partial function from B∗ to B. Let us assume that arity f1 = arity f2. The functor ee f1, f2dd yields a homogeneous quasi total non empty partial functi...

We study the “flat” directions of non-BPS extremal black hole attractors for N = 2, d = 4 supergravities whose vector multiplets’ scalar manifold is endowed with homogeneous symmetric special Kähler geometry. The non-BPS attractors with non-vanishing central charge have a moduli space described by real special geometry (and thus related to the d = 5 parent theory), whereas the moduli spaces of ...

Unless otherwise noted, we consider only n > 0. The parameters in (1.1) are all integers satisfying k, pi and aij > 0. Assume c initial conditions R(1) = ξ1, R(2) = ξ2, . . . , R(c) = ξc, with all ξi > 0. Golomb [6] first solved the simplest example of such a non-homogeneous nested recursion, namely, G(n) = G(n− G(n− 1)) + 1, G(1) = 1; see also [7]. In fact, all of the recursions we find with c...

A homogeneous set is a non-trivial module of a graph, i.e. a non-unitary, proper subset H of a graph’s vertices such that all vertices in H have the same neighbors outside H . Given two graphs G1(V,E1), G2(V,E2), the Homogeneous Set Sandwich Problem asks whether there exists a sandwich graph GS(V,ES), with E1 ⊆ ES ⊆ E2, which contains a homogeneous set. Two are the algorithms which stand for th...

Let X be a homogeneous compact complex manifold, and let Aut(X) be the complex Lie group of holomorphic automorphisms of X. Examples show that dim Aut(X) can grow exponentially in n = dimX. In this note it is shown that dim Aut(X) ≤ n − 1 + (2n− 1 n− 1 ) when n ≥ 3. Thus, dim Aut(X) is at most exponential in n. The proof relies on an upper bound for the dimension of the space of sections of the...

in this paper we study a two-phase free boundary problem for a semilinear elliptic equation on a bounded domain $dsubset mathbb{r}^{n}$ with smooth boundary‎. ‎we give some results on the growth of solutions and characterize the free boundary points in terms of homogeneous harmonic polynomials using a fundamental result of caffarelli and friedman regarding the representation of functions whose ...

Let R be a polynomial ring over a field of dimension n. Let m be the maximal homogeneous ideal of R. Let I be a complete intersection homogeneous ideal of R minimally generated by polynomials of degrees d1, d2, . . . , dn. F. S. Macaulay showed that for all integers i = 0, 1, . . . , δ + 1, I : m = I + m where δ = d1 + d2 + . . .+ dn − n. We provide a modern proof of a generalization of this re...