The discrete analogue of a differential operator and its applications

We construct a discrete analogue D _m (hβ) of the differential operator d^2m /dx ^2m + 2ω ²d^2m?2 /dx ^2m?2 + ω ⁴d^2m?4 /dx ^2m?4 for any m ≥ 2. In the case m = 2, we apply in the Hilbert space K ₂(P ₂) the discrete analogue D ₂(hβ) for construction of optimal quadrature formulas and interpolation splines minimizing the seminorm, which are exact for trigonometric functions sin ωx and cos ωx.     

Exact solution to a nonlinear Klein-Gordon equation

The nonlinear Klein-Gordon equation ?^μ?_μΦ + M²Φ + λ₁Φ^1?m + λ₂Φ^1?2m = 0 has the exact formal solution Φ = [u^2m ?λ₁u^m/(m ? 2)M²+λ₁²/(m?2)²M⁴?λ₂/4(m ? 1)M²]^1/mu^?1, m ≠ 0, 1, 2, where u and v^?1 are solutions of the linear Klein-Gordon equation. This equation is a simple generalization of the ordinary second order differential equation satisfied by the homogeneous function y = [au^m + b(uv)^m/2 + cv^m]^k/m, where u and v are linearly independent solutions of y″ + r(x) y′ + q(x) y = 0.     

On integers which are the sum of a power of 2 and a polynomial value

Here, we show that if f (x) ∈ ?[x] has degree at least 2 then the set of integers which are of the form 2 ^k + f (m) for some integers k ≥ 0 and m is of asymptotic density 0. We also make some conjectures and prove some results about integers not of the form |2 ^k ± m ^a (m ? 1)|.     

On the enumeration of certain weighted graphs

We enumerate weighted simple graphs with a natural upper bound condition on the sum of the weight of adjacent vertices. We also compute the generating function of the numbers of these graphs, and prove that it is a rational function. In particular, we show that the generating function for connected bipartite simple graphs is of the form p₁(x)/(1-x)^m+1. For nonbipartite simple graphs, we get a generating function of the form p₂(x)/(1-x)^m+1(1+x)^l. Here m is the number of vertices of the graph, p₁(x) is a symmetric polynomial of degree at most m, p₂(x) is a polynomial of degree at most m+l, and l is a nonnegative integer. In addition, we give computational results for various graphs.     

Mixed sums of squares and triangular numbers (III)

In this paper we confirm a conjecture of Sun which states that each positive integer is a sum of a square, an odd square and a triangular number. Given any positive integer m, we show that p=2m+1 is a prime congruent to 3 modulo 4 if and only if Tm=m(m+1)/2 cannot be expressed as a sum of two odd squares and a triangular number, i.e., p²=x²+8(y²+z²) for no odd integers x,y,z. We also show that a positive integer cannot be written as a sum of an odd square and two triangular numbers if and only if it is of the form 2Tm(m>0) with 2m+1 having no prime divisor congruent to 3 modulo 4.     

Arithmetic progressions that consist only of primes

Let N_m(x) be the number of arithmetic progressions that consist of m terms, all primes and not larger than x, and set $\text{F}{}_{\mathrm{m}}\text{(x) =}\text{C}{}_{\mathrm{m}}\text{x}{}^2\text{log}{}^{\mathrm{m}}\text{x}$ (C_m explicitly given). It is shown that Hardy and Littlewood's prime k-tuple conjecture implies that N_m(x) = F_m(x){1 + Σ_j=1^Na_jlog^?jx + O((log x)^?N?1)}, (here the bracket represents an asymptotic series with explicitly computable coefficients). This formula holds rather trivially for m = 1 and m = 2. It is proved here for m = 3, by the Vinogradov version of the Hardy-Ramanujan-Littlewood circle method.     

On a functional equation related to derivations in prime rings

In this paper we prove the following result. Let m ≥ 1, n ≥ 1 be fixed integers and let R be a prime ring with m + n + 1 ≤ char(R) or char(R) = 0. Suppose there exists an additive nonzero mapping D : R → R satisfying the relation 2D(x ^n+m+1) = (m + n + 1)(x ^m D(x)x ⁿ + x ⁿ D(x)x ^m ) for all \({x\in R}\). In this case R is commutative and D is a derivation.     

The algebra of hyperboloids of revolution

If we change the sign of p ? m columns (or rows) of an m × m positive definite symmetric matrix A, the resultant matrix B has p negative eigenvalues. We give systems of inequalities for the eigenvalues of B and of the matrix obtained from B by deleting one row and column. To obtain these, we first develop characterizations of the eigenvalues of B which are analogous to the minimum-maximum properties of the eigenvalues of a symmetric A, i.e. the Courant-Fischer theorem. These results arose from studying probability distributions on the hyperboloid of revolution

$\text{x}{}^2{}_1\text{+ ? + x}{}^2{}_{\mathrm{m?p}}\text{? x}{}^2{}_{\mathrm{m\; ?\; p\; +\; 1}}\text{? ? ? x}{}^2{}_{\mathrm{m}}\text{= 1}$

. By contrast, the familiar results are associated with the sphere x²₁ + ? + x²_m = 1.     

On the finiteness of certain Rabinowitsch polynomials II

Let m be a positive integer and f_m(x) be a polynomial of the form f_m(x)=x²+x−m. We call a polynomial f_m(x) a Rabinowitsch polynomial if for and consecutive integers is either 1 or prime. In Byeon (J. Number Theory 94 (2002) 177), we showed that there are only finitely many Rabinowitsch polynomials f_m(x) such that 1+4m is square free. In this note, we shall remove the condition that 1+4m is square free.     

Schrödinger operators with singular potentials

Recently B. Simon proved a remarkable theorem to the effect that the Schrödinger operatorT=?Δ+q(x) is essentially selfadjoint onC ₀ ^∞ (R ^m if 0≦q ∈L ²(R ^m). Here we extend the theorem to a more general case,T=?Σ _j ^=1/m (?/?x _j ?ib _j(x))² +q ₁(x) +q ₂(x), whereb _j, q₁,q ₂ are real-valued,b _j ∈C(R ^m),q ₁ ∈L _loc ² (R ^m),q ₁(x)≧?q*(|x|) withq*(r) monotone nondecreasing inr ando(r ²) asr → ∞, andq ₂ satisfies a mild Stummel-type condition. The point is that the assumption on the local behavior ofq ₁ is the weakest possible. The proof, unlike Simon’s original one, is of local nature and depends on a distributional inequality and elliptic estimates.     

A necessary condition for the local solvability of the operator Pm2(x, D) + P2m − 1(x, D)

It is shown that a necessary condition for the local solvability of the operator P(x, D) = P_m²(x, D) + P_{2m ? 1}(x, D), where P_m(x, D) is an mth-order homogeneous differential operator of principal type with real coefficients, is that along any null-bicharacteristic strip of P_m(x, ξ) the imaginary part of the sub-principal symbol cannot have an odd-order zero where its real part does not vanish.     

Note on primitive permutation groups and a diophantine equation

It is shown that there are no transitive rank 3 extensions of the projective linear groups H, PSL(m,q) ? H ? PFL(m,q), for any prime power q and integer m ? 3. In the course of the proof the diophantine equation 5^m + 11 = x^p2, where m, x are positive integers, arose. As such equations can now be solved completely we had the choice of using number theory or geometry to complete the proof.     

The Jacobian problem as a system of ordinary differential equations

LetP=x ⁿ +P _n?1(y)x ^n?1+…+P ₀(y),Q=x ^m +Q _m?2(y)x ^m?2+…+Q ₀(y) belong toK[x, y], whereK is a field of characteristic zero. The main result of this paper is the following: Assume thatP _x Q _y ?P _y Q _x =1. Then:*

1. K[Q _m?2(y), …,Q ₀(y)]=K[y],
2. K[P, Q]=K[x, y] ifQ=x ^m +Q _k (y)x ^k +Q _r (y)x ^r

     

Tight bounds for rational sums of squares over totally real fields

Let K be a totally real Galois number field. Hillar proved that if f ∈ ?[x ₁, ..., x _n ] is a sum of m squares in K[x ₁, ..., x _n ], then f is a sum of N(m) squares in ?[x ₁, ..., x _n ], where N(m) ≤ 2^[K:?]+1 · $ \left( {_2^{[K:\mathbb{Q}] + 1} } \right) $ · 4m. We show in fact that N(m) ≤ m + 4 $ \left\lceil {\tfrac{m} {4}} \right\rceil $ ([K: ?] ? 1), our proof being constructive too. Moreover, we give some examples where this bound is sharp, for instance in the case of quadratic extensions. We also extend our results to the setting of non-commutative polynomials over ?.     

A simplex variant solving an m × d linear program in O(min(m2, d2) expected number of pivot steps

We present a variant of the Simplex method which requires on the average at most 2 (min(m, d) + 1)² pivots to solve the linear program min c^T, Ax ≥ b, x ≥ 0 with A ε R^m×d. The underlying probabilistic distribution is assumed to be invariant under inverting the sense of any subset of the inequalities. In particular, this implies that under Smale's spherically symmetric model this variant requires an average of no more than 2(d + 1)² pivots, independent of m, where d ≤ m.     

Estimates for nonlinear harmonic measures on trees

In this paper we give some estimates for nonlinear harmonic measures on trees. In particular, we estimate in terms of the size of a set D the value at the origin of the solution to u(x) = F((x, 0),...,(x,m ? 1)) for every x ∈ \(\mathbb{T}_m \) , a directed tree with m branches with initial datum f + χD. Here F is an averaging operator on ? ^m , x is a vertex of a directed tree \(\mathbb{T}_m \) with regular m-branching and (x, i) denotes a successor of that vertex for 0 ≤ i ≤ m ? 1.     

A Gerschgorin theorem for difference equations—II

It is shown that the bounded solutions of the difference equation x(m + 1) = A(m + 1)x(m) arise as fixed points of a contraction mapping when A(m) satisfies a diagonal dominance condition on {0, 1, 2,…}.     

On Maillet determinant

For a positive integer m, let $\text{A = {1 ≤ a <}\text{m}\text{2}\text{| (a, m) = 1}}$ and let n = |A|. For an integer x, let R(x) be the least positive residue of x modulo m and if (x, m) = 1, let x′ be the inverse of x modulo m. If m is odd, then |R(ab′)|_a,b∈A = ?2^1?n(∏_χ(Σ_{a = 1}^{m ? 1}aχ(a))), where χ runs over all the odd Dirichlet characters modulo m.     

Analogues to Fermat primes related to Pell��s equation

The coefficients of some weight 3 modular forms give reason to study primes of the form p = 2x ² ? 1 = 2dy ² + 1. If x _a , y _a are the positive solutions of Pell??s equation x ² ? dy ² = 1, given by ${x_a + y_a \sqrt{d} = (x_1 + y_1 \sqrt{d})^a}The coefficients of some weight 3 modular forms give reason to study primes of the form p = 2x ² − 1 = 2dy ² + 1. If x _a , y _a are the positive solutions of Pell’s equation x ² − dy ² = 1, given by x_a + y_a ?d = (x₁ + y₁ ?d)^a{x_a + y_a \sqrt{d} = (x_1 + y_1 \sqrt{d})^a}, and if p_a = 2 x_a² - 1{p_a = 2 x_a^2 - 1} is prime, then a = 2 ^m is a power of 2. So there are analogues to the Fermat numbers 2 ^a + 1.