A note on a conjecture of Borwein and Choi

A polynomial P(X) with coefficients {ǃ} of odd degree N - 1 is cyclotomic if and only if¶¶P(X) = ±F_p1 (±X)F_p2(±X^p1) ?F_pr(±X^{p1 p2 ?p_r-1}) P(X) = \pm \Phi_{p1} (\pm X)\Phi_{p2}(\pm X^{p1}) \cdots \Phi_{p_r}(\pm X^{p1 p2 \cdots p_r-1}) ¶where N = p₁ p₂ · · · p_r and the p_i are primes, not necessarily distinct, and where F_p(X) : = (X^p - 1) / (X - 1) \Phi_{p}(X) := (X^{p} - 1) / (X - 1) is the p-th cyclotomic polynomial. This is a conjecture of Borwein and Choi [1]. We prove this conjecture for a class of polynomials of degree N - 1 = 2^r p^l - 1 N - 1 = 2^{r} p^{\ell} - 1 for any odd prime p and for integers r, l\geqq 1 r, \ell \geqq 1 .     

On minimal p-degrees in 2-transitive permutation groups

Suppose G is a transitive permutation group on a finite set W\mit\Omega of n points and let p be a prime divisor of |G||G|. The smallest number of points moved by a non-identity p-element is called the minimal p-degree of G and is denoted m_p (G). ¶ In the article the minimal p-degrees of various 2-transitive permutation groups are calculated. Using the classification of finite 2-transitive permutation groups these results yield the main theorem, that m_p(G) 3 [(p-1)/(p+1)] ·|W|m_{p}(G) \geq {{p-1} \over {p+1}} \cdot |\mit\Omega | holds, if Alt(W) \nleqq G {\rm Alt}(\mit\Omega ) \nleqq G .¶Also all groups G (and prime divisors p of |G||G|) for which m_p(G) ￡ [(p-1)/(p)] ·|W|m_{p}(G)\le {{p-1}\over{p}} \cdot |\mit\Omega | are identified.     

On the class number of p-th cyclotomic field

Let h^[-(p)h^-(p) be the relative class number of the p-th cyclotomic field. We show that logh^-(p) = [(p+3)/4] logp - [(p)/2] log2p+ log(1-b) + O(log₂² p)\log h^-(p) = {{p+3} \over {4}} \log p - {{p} \over {2}} \log 2\pi + \log (1-\beta ) + O(\log _2^2 p), where b\beta denotes a Siegel zero, if such a zero exists and p o -1 mod 4p\equiv -1\pmod {4}. Otherwise this term does not appear.     

Prime Chains and Pratt Trees

Prime chains are sequences $p_{1}, \ldots , p_{k}Prime chains are sequences p₁, ?, p_kp_{1}, \ldots , p_{k} of primes for which p_j+1 o 1{p_{j+1} \equiv 1} (mod p _j ) for each j. We introduce three new methods for counting long prime chains. The first is used to show that N(x; p) = O_e(x^1+e){N(x; p) = O_{\varepsilon}(x^{1+\varepsilon})}, where N(x; p) is the number of chains with p ₁ = p and p_k ￡ p_x{p_k \leq p_x}. The second method is used to show that the number of prime chains ending at p is ≍ log p for most p. The third method produces the first nontrivial upper bounds on H(p), the length of the longest chain with p _k = p, valid for almost all p. As a consequence, we also settle a conjecture of Erdős, Granville, Pomerance and Spiro from 1990. A probabilistic model of H(p), based on the theory of branching random walks, is introduced and analyzed. The model suggests that for most p ￡ x{p \leq x}, H(p) stays very close to e log log x.     

Large time asymptotics of the doubly nonlinear equation in the non-displacement convexity regime

We study the long-time asymptotics of the doubly nonlinear diffusion equation ${\rho_t={\rm div}(|\nabla\rho^m |^{p-2} \nabla\left(\rho^m\right))}We study the long-time asymptotics of the doubly nonlinear diffusion equation r_t=div(|?r^m |^p-2 ?(r^m)){\rho_t={\rm div}(|\nabla\rho^m |^{p-2} \nabla\left(\rho^m\right))} in \mathbbRⁿ{\mathbb{R}^n}, in the range \fracn-pn(p-1) < m < \fracn-p+1n(p-1){\frac{n-p}{n(p-1)} < m < \frac{n-p+1}{n(p-1)}} and 1 < p < ∞ where the mass of the solution is conserved, but the associated energy functional is not displacement convex. Using a linearisation of the equation, we prove an L ¹-algebraic decay of the non-negative solution to a Barenblatt-type solution, and we estimate its rate of convergence. We then derive the nonlinear stability of the solution by means of some comparison method between the nonlinear equation and its linearisation. Our results cover the exponent interval \frac2nn+1 < p < \frac2n+1n+1{\frac{2n}{n+1} < p < \frac{2n+1}{n+1}} where a rate of convergence towards self-similarity was still unknown for the p-Laplacian equation.     

Optimal Extension of Fourier Multiplier Operators in L p (G)

Given 1 ≤ p < ∞, a compact abelian group G and a p-multiplier ${\psi : \Gamma \to {\mathbb C}}Given 1 ≤ p < ∞, a compact abelian group G and a p-multiplier y: G? \mathbb C{\psi : \Gamma \to {\mathbb C}} (with Γ the dual group), we study the optimal domain of the multiplier operator T^(p)_y : L^p (G) ? L^p (G){T^{(p)}_\psi : L^p (G) \to L^p (G)}. This is the largest Banach function space, denoted by L¹(m^(p)_y){L^1(m^{(p)}_\psi)}, with order continuous norm into which L ^p (G) is embedded and to which T^(p)_y{ T^{(p)}_\psi} has a continuous L ^p (G)-valued extension. Compactness conditions for the optimal extension are given, as well as criteria for those ψ for which L¹(m^(p)_y) = L^p (G){L^1(m^{(p)}_\psi) = L^p (G)} is as small as possible and also for those ψ for which L¹(m^(p)_y) = L¹ (G){L^1(m^{(p)}_\psi) = L^1 (G)} is as large as possible. Several results and examples are presented for cases when L^p (G) \subsetneqq L¹(m^(p)_y) \subsetneqq L¹ (G){L^p (G) \subsetneqq L^1(m^{(p)}_\psi) \subsetneqq L^1 (G)}.     

Diophantine approximation with one prime and three squares of primes

We show that if λ ₁,λ ₂,λ ₃,λ ₄ are nonzero real numbers, not all of the same sign, η is real, and at least one of the ratios λ ₁/λ _j (j=2,3,4) is irrational, then given any real number ω>0, there are infinitely many ordered quadruples of primes (p ₁,p ₂,p ₃,p ₄) for which

p-Frames and Shift Invariant Subspaces of Lp

In this article we investigate the frame properties and closedness for the shift invariant space V_p(F) = { ?_i=1^r ?_{j ? \Zd} d_i(j) f_i (·-j):  ( d_i(j) )_{j ? \Zd} ? l^p }, \q 1 ￡ p ￡ ￥ . \displaystyle V_p(\Phi) = \left\{ \sum_{i=1}^r \sum_{j\in \Zd} d_i(j) \phi_i (\cdot-j): \ \left( d_i(j) \right)_{j\in \Zd}\in \ell^p \right\}, \q 1\le p \le \infty~. We derive necessary and sufficient conditions for an indexed family {f_i(·-j): 1 ￡ i ￡ r, j ? \Zd}\{\phi_i(\cdot-j):\ 1\le i\le r, j\in \Zd\} to constitute a pp-frame for V_p(F)V_p(\Phi), and to generate a closed shift invariant subspace of L^pL^p. A function in the L^pL^p-closure of V_p(F)V_p(\Phi) is not necessarily generated by l^p\ell^p coefficients. Hence we often hope that V_p(F)V_p(\Phi) itself is closed, i.e., a Banach space. For p 1 2p\ne 2, this issue is complicated, but we show that under the appropriate conditions on the frame vectors, there is an equivalence between the concept of pp-frames, Banach frames, and the closedness of the space they generate. The relation between a function f ? V_p(F)f \in V_p(\Phi) and the coefficients of its representations is neither obvious, nor unique, in general. For the case of pp-frames, we are in the context of normed linear spaces, but we are still able to give a characterization of pp-frames in terms of the equivalence between the norm of ff and an l^p\ell^p-norm related to its representations. A Banach frame does not have a dual Banach frame in general, however, for the shift invariant spaces V_p(F)V_p(\Phi), dual Banach frames exist and can be constructed.     

On the size of the set A(A + 1)

Let F _p be the field of a prime order p. For a subset A ì F_p{A \subset F_p} we consider the product set A(A + 1). This set is an image of A ×  A under the polynomial mapping f(x, y) = xy +  x : F _p ×  F _p → F _p . In the present note we show that if |A| <  p ^1/2, then

Global asymptotic stability for half-linear differential systems with generalized almost periodic coefficients

The following system considered in this paper:

Blocks with cyclic defect of Hecke orders of Coxeter groups

Let B\cal B be a p-block of cyclic defect of a Hecke order over the complete ring \Bbb Z[q] _{áq-1,p ?}\Bbb {Z}[q] _{\langle q-1,p \rangle}; i.e. modulo áq-1 ?\langle q-1 \rangle it is a p-block B of cyclic defect of the underlying Coxeter group G. Then B\cal B is a tree order over \Bbb Z[q]_{áq-1, p ?}\Bbb {Z}[q]_{\langle q-1, p \rangle } to the Brauer tree of B. Moreover, in case B\cal B is the principal block of the Hecke order of the symmetric group S(p) on p elements, then B\cal B can be described explicitly. In this case a complete set of non-isomorphic indecomposable Cohen-Macaulay B\cal B-modules is given.     

A partial ordering of some of the elements of Algol

Them _Algol productions of Algol are of the formX _p0 ::=X _p1 X _p2 ...X _{pn p} , where 1pm _Algol, 1n _p,X _p0 is a defined type andX _pj, 1jn _p is either a defined type or a basic symbol or possibly, representing the empty string ifn _p=1 [1]. A partial ordering of that subset of Algol's basic symbols and defined types which for somep are eitherX _p0 orX _p1 is exhibited. This ordering is of interest in implementing the syntax-oriented translator described by Ingerman.     

Local Hardy–Littlewood maximal operator

In this article we define and investigate a local Hardy–Littlewood maximal operator in Euclidean spaces. It is proved that this operator satisfies weighted L ^p , p > 1, and weighted weak type (1,1) estimates with weight function ${w \in A^p_{\rm{loc}}}In this article we define and investigate a local Hardy–Littlewood maximal operator in Euclidean spaces. It is proved that this operator satisfies weighted L ^p , p > 1, and weighted weak type (1,1) estimates with weight function w ? A^p_loc{w \in A^p_{\rm{loc}}}, the class of local A _p weights which is larger than the Muckenhoupt A _p class. Also, the condition w ? A^p_loc{w \in A^p_{\rm{loc}}} turns out to be necessary for the weighted weak type (p,p), p ≥ 1, inequality to hold.     

Ultrafilters, monotone functions and pseudocompactness

In this article we, given a free ultrafilter p on , consider the following classes of ultrafilters:(1) T(p) - the set of ultrafilters Rudin-Keisler equivalent to p,(2) S(p)={q *: f , strictly increasing, such that q=f(p)},(3) I(p) - the set of strong Rudin-Blass predecessors of p,(4) R(p) - the set of ultrafilters equivalent to p in the strong Rudin-Blass order,(5) P_RB(p) - the set of Rudin-Blass predecessors of p, and(6) P_RK(p) - the set of Rudin-Keisler predecessors of p,and analyze relationships between them. We introduce the semi-P-points as those ultrafilters p * for which P_RB(p)=P_RK(p), and investigate their relations with P-points, weak-P-points and Q-points. In particular, we prove that for every semi-P-point p its -th left power p is a semi-P-point, and we prove that non-semi-P-points exist in ZFC. Further, we define an order in T(p) by rq if and only if r S(q). We prove that (S(p),) is always downwards directed, (R(p),) is always downwards and upwards directed, and (T(p),) is linear if and only if p is selective.We also characterize rapid ultrafilters as those ultrafilters p * for which R(p)S(p) is a dense subset of *.A space X is M-pseudocompact (for ) if for every sequence (U_n)_n< of disjoint open subsets of X, there are q M and x X such that x=q-lim (U_n); that is, for every neighborhood V of x. The P_RK(p)-pseudocompact spaces were studied in [ST].In this article we analyze M-pseudocompactness when M is one of the classes S(p), R(p), T(p), I(p), P_RB(p) and P_RK(p). We prove that every Frolik space is S(p)-pseudocompact for every p *, and determine when a subspace with is M-pseudocompact.The first authors research was partially supported by a grant GAR 201/00/1466Mathematics Subject Classification (2000): 54D80, 03E05, 54A20, 54D20     

Computations on the growth of the first factor for prime cyclotomic fields

Denote byh(p) the first factor of the class number of the prime cyclotomic fieldk(exp (2i/p)). The theorem:h(p ₂)>h(p ₁) if 641 p ₂>p ₁ 19 is proved by straightforward computation.     

On a Diophantine Inequality Over Primes (II)

In this short note we prove that if 1 < c < 81/40, c ≠ 2, N is a large real number, then the Diophantine inequality |p₁^c+p₂^c+p₃^c+p₄^c+p₅^c-N| < log^-1 N \vert p_1^c+p_2^c+p_3^c+p_4^c+p_5^c-N\vert < \log^{-1} N is solvable, where p ₁,···,p ₅ are primes.     

Nonexistence of backward self-similar blowup solutions to a supercritical semilinear heat equation

We consider a Cauchy problem for a semilinear heat equation

with p>p_S where p_S is the Sobolev exponent. If u(x,t)=(T−t)^−1/(p−1)φ((T−t)^−1/2x) for xR^N and t[0,T), where φ is a regular positive solution of

(P)

then u is called a backward self-similar blowup solution. It is immediate that (P) has a trivial positive solution κ≡(p−1)^−1/(p−1) for all p>1. Let p_L be the Lepin exponent. Lepin obtained a radial regular positive solution of (P) except κ for p_S<p<p_L. We show that there exist no radial regular positive solutions of (P) which are spatially inhomogeneous for p>p_L.     

Oscillations of first-order neutral delay differential equations

Consider the neutral delay differential equation (*) (d/dt)[y(t) + py(t − τ)] + qy(t − σ) = 0, t t₀, where τ, q, and σ are positive constants, while p ε (−∞, −1) (0, + ∞). (For the case p ε [−1, 0] see Ladas and Sficas, Oscillations of neutral delay differential equations (to appear)). The following results are then proved. Theorem 1. Assume p < − 1. Then every nonoscillatory solution y(t) of Eq. (*) tends to ± ∞ as t → ∞. Theorem 2. Assume p < − 1, τ > σ, and q(σ − τ)/(1 + p) > (1/e). Then every solution of Eq. (*) oscillates. Theorems 3. Assume p > 0. Then every nonoscillatory solution y(t) of Eq. (*) tends to zero as t → ∞. Theorem 4. Assume p > 0. Then a necessary condition for all solutions of Eq. (*) to oscillate is that σ > τ. Theorem 5. Assume p > 0, σ > τ, andq(σ − τ)/(1 + p) > (1/e). Then every solution of Eq. (*) oscillates. Extensions of these results to equations with variable coefficients are also obtained.     

Approximation by the Bernstein–Durrmeyer Operator on a Simplex

For the Jacobi-type Bernstein–Durrmeyer operator M _n,κ on the simplex T ^d of ℝ ^d , we proved that for f∈L ^p (W _κ ;T ^d ) with 1<p<∞,

A survey of orthogonal arrays of strength two

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