Semigroup identities and proofs

Tamura proved that for any semigroup word U(x, y), if every group satisfying an identity of the form yx ~ xU(x, y)y is abelian, then so is every semigroup that satisfies that identity. Because a group has an identity element and the cancellation property, it is easier to show that a group is abelian than that a semigroup is. If we know that it is, then there must be a sequence of substitutions using xU(x, y)y ~ yx that transforms xy to yx. We examine such sequences and propose finding them as a challenge to proof by computer. Also, every model of y ~ xU(x, y)x is a group. This raises a similar challenge, which we explore in the special case y ~ x ^m y ^p x ⁿ . In addition, we determine the free model with two generators of some of these identities. In particular, we find that the free model for y ~ x ² yx ² has order 32 and is the product of D ₄ (the symmetries of a square), C ₂, and C ₄, and point out relations between such identities and Burnside’s Problem concerning models of x ⁿ ~ y ⁿ . We also examine several identities not related to groups.     

A Dini type test on the pointwise convergence of double Fourier integrals

We give sufficient conditions for the convergence of the double Fourier integral of a complex-valued function f ∈ L ¹(?²) with bounded support at a given point (x ₀,y ₀) ∈ ?². It turns out that this convergence essentially depends on the convergence of the single Fourier integrals of the marginal functions f(x,y ₀), x ∈ ?, and f(x ₀,y), y ∈ ?, at the points x:= x ₀ and y:= y ₀, respectively. Our theorem applies to functions in the multiplicative Zygmund classes of functions in two variables.     

The diophantine equation x3 + 3y3 = 2n

It is proved that the equation of the title has a finite number of integral solutions (x, y, n) and necessary conditions are given for (x, y, n) in order that it can be a solution (Theorem 2). It is also proved that for a given odd x₀ there is at most one integral solution (y, n), n ≥ 3, to x₀³ + 3y³ = 2ⁿ and for a given odd y₀ there is at most one integral solution (x, n), n ≥ 3, to x³ + 3y₀³ = 2ⁿ.     

Monochromatic Vs multicolored paths

Letl andk be positive integers, and letX={0,1,...,l ^k?1}. Is it true that for every coloring δ:X×X→{0,1,...} there either exist elementsx ₀<x ₁<...<x _l ofX with δ(x ₀,x ₁)=δ(x ₁,x ₂)=...=δ(x _l?1,x _l), or else there exist elementsy ₀<y ₁<...<y _k ofX with δ(y _i?1,y _i) ∈ δ(y _j?1,y _j) for all 1<-i<j≤k? We prove here that this is the case if eitherl≤2, ork≤4, orl≥(3k)^2k . The general question remains open.     

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

     

Positive integer solutions of the diophantine equation x 2 − L n xy + (−1) n y 2 = ±5 r

In this paper, we consider the equation x ²?L _n x y+(?1) ⁿ y ² = ±5 ^r and determine the values of n for which the equation has positive integer solutions x and y. Moreover, we give all positive integer solutions of the equation x ²?L _n x y+(?1) ⁿ y ² = ±5 ^r when the equation has positive integer solutions.     

Pringsheim type tests on the pointwise convergence of integrals conjugate to double fourier integrals

We prove sufficient conditions for the convergence of the integrals conjugate to the double Fourier integral of a complex-valued function f ∈ L ¹ (?²) with bounded support at a given point (x ₀, g ₀) ∈ ?². It turns out that this convergence essentially depends on the convergence of the integral conjugate to the single Fourier integral of the marginal functions f(x, y ₀), x ∈ ?, and f(x ₀, y), y ∈ ?, at x:= x ₀ and y:= y ₀, respectively. Our theorems apply to functions in the multiplicative Lipschitz and Zygmund classes introduced in this paper.     

Theorems on asymptotics for singular Sturm-Liouville operators with various boundary conditions

We consider the Sturm-Liouville operator L(y) = ?d ² y/dx ² + q(x)y in the space L ₂[0, π], where the potential q(x) is a complex-valued distribution of the first order of singularity; i.e., q(x) = u′(x), u ∈ L ²[0, π]. (Here the derivative is understood in the sense of distributions.) We obtain asymptotic formulas for the eigenvalues and eigenfunctions of the operator in the case of the Neumann-Dirichlet conditions [y ^[1](0) = 0, y(π) = 0] and Neumann conditions [y ^[1](0) = 0, y ^[1](π) = 0] and refine similar formulas for all types of boundary conditions. The leading and second terms of asymptotics are found in closed form.     

Fundamental solutions of some degenerate operators

In this paper we modify the usual series computation for a parametrix of an elliptic operator and find a method that is applicable to some simple degenerate operators. In fact, we are able to construct fundamental solutions for operators such as D_x + ixD_y and D_x² + x²D_y² + cD_y which explain the lack of local solvability in a natural way.     

Optimal interval lengths for nonlocal boundary value problems associated with third order Lipschitz equations

For the third order differential equation, y^?=f(x,y,y^′,y^″), where f(x,y₁,y₂,y₃) is Lipschitz continuous in terms of yi, i=1,2,3, we obtain optimal bounds on the length of intervals on which there exist unique solutions of certain nonlocal three and four point boundary value problems. These bounds are obtained through an application of the Pontryagin Maximum Principle from the theory of optimal control.     

On the algebraic structure of Abelian integrals for a kind of perturbed cubic Hamiltonian systems

The finite generators of Abelian integral are obtained, where Γh is a family of closed ovals defined by H(x,y)=x²+y²+ax⁴+bx²y²+cy⁴=h, h∈Σ, ac(4ac−b²)≠0, Σ=(0,h₁) is the open interval on which Γh is defined, f(x,y), g(x,y) are real polynomials in x and y with degree 2n+1 (n?2). And an upper bound of the number of zeros of Abelian integral I(h) is given by its algebraic structure for a special case a>0, b=0, c=1.     

Sixth-order superstable two-step methods for second-order initial-value problems

For the numerical integration of general second-order initial-value problems y″ = f(x, y, y′), y(x₀) = y₀, y′(x₀) = y′₀, we report a family of two-step sixth-order methods which are superstable for the test equation y″ + 2αy′ + β²y = 0, α, β ⩾ 0, α + β\s>0, in the sense of Chawla [1].     

An approximation property of exponential functions

We solve the inhomogeneous linear first order differential equations of the form y′(x) ? λy(x) = Σ _m=0 ^∞ a _m (x ? c) ^m , and prove an approximation property of exponential functions. More precisely, we prove the local Hyers-Ulam stability of linear first order differential equations of the form y′(x) = λy(x) in a special class of analytic functions.     

Boundary data smoothness for solutions of nonlocal boundary value problems for second order differential equations

Under certain conditions, solutions of the boundary value problem, y^″=f(x,y,y^′), y(x₁)=y₁, and , are differentiated with respect to boundary conditions, where a<x₁<η₁<?<ηm<x₂<b, r₁,…,rm∈R, and y₁,y₂∈R.     

On a conjecture of R.F. Scott (1881)

A formula is given for the permanent of a general Cauchy matrix ((x_i?y_j)^?1). In a special case, where the x_i and the y_j are the distinct nth roots of 1 and ?1 respectively, a formula for the permanent, conjectured by R. F. Scott, is proved by computing the eigenvalues of related circulants.     

Three-variable Mahler measures and special values of modular and Dirichlet L-series

In this paper we prove that the Mahler measures of the Laurent polynomials (x+x ^?1)(y+y ^?1)(z+z ^?1)+k ^1/2, (x+x ^?1)²(y+y ^?1)²(1+z)³ z ^?2?k, and x ⁴+y ⁴+z ⁴+1+k ^1/4 xyz, for various values of k, are of the form r ₁ L′(f,0)+r ₂ L′(χ,?1), where $r_{1},r_{2}\in \mathbb{Q}$ , f is a CM newform of weight 3, and χ is a quadratic character. Since it has been proved that these Mahler measures can also be expressed in terms of logarithms and ₅ F ₄-hypergeometric series, we obtain several new hypergeometric evaluations and transformations from these results.     

Ramanujan’s identities and representation of integers by certain binary and quaternary quadratic forms

We revisit old conjectures of Fermat and Euler regarding the representation of integers by binary quadratic form x ²+5y ². Making use of Ramanujan’s ₁ ψ ₁ summation formula, we establish a new Lambert series identity for $\sum_{n,m=-\infty }^{\infty}q^{n^{2}+5m^{2}}We revisit old conjectures of Fermat and Euler regarding the representation of integers by binary quadratic form x ²+5y ². Making use of Ramanujan’s ₁ ψ ₁ summation formula, we establish a new Lambert series identity for ?_n,m=-￥^￥q^n2+5m2\sum_{n,m=-\infty }^{\infty}q^{n^{2}+5m^{2}} . Conjectures of Fermat and Euler are shown to follow easily from this new formula. But we do not stop there. Employing various formulas found in Ramanujan’s notebooks and using a bit of ingenuity, we obtain a collection of new Lambert series for certain infinite products associated with quadratic forms such as x ²+6y ², 2x ²+3y ², x ²+15y ², 3x ²+5y ², x ²+27y ², x ²+5(y ²+z ²+w ²), 5x ²+y ²+z ²+w ². In the process, we find many new multiplicative eta-quotients and determine their coefficients.     

Unit groups of group algebras of some small groups

Let FG be a group algebra of a group G over a field F and U (FG) the unit group of FG. It is a classical question to determine the structure of the unit group of the group algebra of a finite group over a finite field. In this article, the structure of the unit group of the group algebra of the non-abelian group G with order 21 over any finite field of characteristic 3 is established. We also characterize the structure of the unit group of FA ₄ over any finite field of characteristic 3 and the structure of the unit group of FQ ₁₂ over any finite field of characteristic 2, where Q ₁₂ = 〈x, y; x ⁶ = 1, y ² = x ³, x ^y = x ^?1〉.     

Asymptotic behaviour of three-dimensional singularly perturbed convection-diffusion problems with discontinuous data

We consider three singularly perturbed convection-diffusion problems defined in three-dimensional domains: (i) a parabolic problem −?(uxx+uyy)+ut+v₁ux+v₂uy=0 in an octant, (ii) an elliptic problem −?(uxx+uyy+uzz)+v₁ux+v₂uy+v₃uz=0 in an octant and (iii) the same elliptic problem in a half-space. We consider for all of these problems discontinuous boundary conditions at certain regions of the boundaries of the domains. For each problem, an asymptotic approximation of the solution is obtained from an integral representation when the singular parameter ?→⁰⁺. The solution is approximated by a product of two error functions, and this approximation characterizes the effect of the discontinuities on the small ?− behaviour of the solution and its derivatives in the boundary layers or the internal layers.