On the Obstruction to Linearizability of Second-Order Ordinary Differential Equations

In this paper, we investigate the action of the pseudogroup of all point transformations on the bundle of equations y″=u ⁰(x,y)+u ¹(x,y)y′+u ²(x,y)(y′)²+u ³(x,y)(y′)³. We calculate the 1st nontrivial differential invariant of this action. It is a horizontal differential 2-form with values in some algebra, it is defined on the bundle of 2-jets of sections of the bundle under consideration. We prove that this form is a unique obstruction to linearizability of these equations by point transformations.     

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.     

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.     

An integral geometry lemma and its applications: The nonlocality of the Pavlov equation and a tomographic problem with opaque parabolic objects

Written in the evolutionary form, the multidimensional integrable dispersionless equations, exactly like the soliton equations in 2+1 dimensions, become nonlocal. In particular, the Pavlov equation is brought to the form v_t = v_xv_y - ?_x^-1?_y[v_y + v_x²], where the formal integral ?_x^?1 becomes the asymmetric integral \( - \int_x^\infty {dx'} \). We show that this result could be guessed using an apparently new integral geometry lemma. It states that the integral of a sufficiently general smooth function f(X, Y) over a parabola in the plane (X, Y) can be expressed in terms of the integrals of f(X, Y) over straight lines not intersecting the parabola. We expect that this result can have applications in two-dimensional linear tomography problems with an opaque parabolic obstacle.     

Nonexistence of Isochronous Centers in Planar Polynomial Hamiltonian Systems of Degree Four

In this paper we study centers of planar polynomial Hamiltonian systems and we are interested in the isochronous ones. We prove that every center of a polynomial Hamiltonian system of degree four (that is, with its homogeneous part of degree four not identically zero) is nonisochronous. The proof uses the geometric properties of the period annulus and it requires the study of the Hamiltonian systems associated to a Hamiltonian function of the form H(x, y)=A(x)+B(x) y+C(x) y²+D(x) y³.     

Heisenberg Uniqueness Pairs for the Parabola

Let Γ denote the parabola y=x ² in the plane. For some simple sets Λ in the plane we study the question whether (Γ,Λ) is a Heisenberg uniqueness pair. For example we shall consider the cases where Λ is a straight line or a union of two straight lines.     

Finite embedding theorems for partial Latin squares,quasi-groups,and loops

In this paper we prove that a finite partial commutative (idempotent commutative) Latin square can be embedded in a finite commutative (idempotent commutative) Latin square. These results are then used to show that the loop varieties defined by any non-empty subset of the identities {x(xy) = y, (yx)x = y} and the quasi-group varieties defined by any non-empty subset of {x² = x, x(xy) = y, (yx)x = y}, except possibly {x(xy) = y, (yx)x = y}, have the strong finite embeddability property. It is then shown that the finitely presented algebras in these varities are residually finite, Hopfian, and have a solvable word problem.     

Two-point Taylor expansions in the asymptotic approximation of double integrals. Application to the second and fourth Appell functions

The main difficulty in Laplace's method of asymptotic expansions of double integrals is originated by a change of variables. We consider a double integral representation of the second Appell function F₂(a,b,b^′,c,c^′;x,y) and illustrate, over this example, a variant of Laplace's method which avoids that change of variables and simplifies the computations. Essentially, the method only requires a Taylor expansion of the integrand at the critical point of the phase function. We obtain in this way an asymptotic expansion of F₂(a,b,b^′,c,c^′;x,y) for large b, b^′, c and c^′. We also consider a double integral representation of the fourth Appell function F₄(a,b,c,d;x,y). We show, in this example, that this variant of Laplace's method is uniform when two or more critical points coalesce or a critical point approaches the boundary of the integration domain. We obtain in this way an asymptotic approximation of F₄(a,b,c,d;x,y) for large values of a,b,c and d. In this second example, the method requires a Taylor expansion of the integrand at two points simultaneously. For this purpose, we also investigate in this paper Taylor expansions of two-variable analytic functions with respect to two points, giving Cauchy-type formulas for the coefficients of the expansion and details about the regions of convergence.     

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.     

A three-point Taylor algorithm for three-point boundary value problems

We consider second-order linear differential equations φ(x)y^″+f(x)y^′+g(x)y=h(x) in the interval (−1,1) with Dirichlet, Neumann or mixed Dirichlet-Neumann boundary conditions given at three points of the interval: the two extreme points x=±1 and an interior point x=s∈(−1,1). We consider φ(x), f(x), g(x) and h(x) analytic in a Cassini disk with foci at x=±1 and x=s containing the interval [−1,1]. The three-point Taylor expansion of the solution y(x) at the extreme points ±1 and at x=s is used to give a criterion for the existence and uniqueness of the solution of the boundary value problem. This method is constructive and provides the three-point Taylor approximation of the solution when it exists. We give several examples to illustrate the application of this technique.     

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.     

Applications of utility functions defined on quasi-metric spaces

A quasi-metric space (X,d) is called sup-separable if (X,d^s) is a separable metric space, where d^s(x,y)=max{d(x,y),d(y,x)} for all x,y∈X. We characterize those preferences, defined on a sup-separable quasi-metric space, for which there is a semi-Lipschitz utility function. We deduce from our results that several interesting examples of quasi-metric spaces which appear in different fields of theoretical computer science admit semi-Lipschitz utility functions. We also apply our methods to the study of certain kinds of dynamical systems defined on quasi-metric spaces.     

Pillai's conjecture revisited

We prove a generalization of an old conjecture of Pillai (now a theorem of Stroeker and Tijdeman) to the effect that the Diophantine equation 3^x−2^y=c has, for |c|>13, at most one solution in positive integers x and y. In fact, we show that if N and c are positive integers with N?2, then the equation |(N+1)^x−N^y|=c has at most one solution in positive integers x and y, unless (N,c)∈{(2,1),(2,5),(2,7),(2,13),(2,23),(3,13)}. Our proof uses the hypergeometric method of Thue and Siegel and avoids application of lower bounds for linear forms in logarithms of algebraic numbers.     

On an internal property of absolute retracts,II

Let X be a (metrizable) space. A mixer for X is, roughly speaking, a map μ:X³→X such that μ(x, x, y) = μ(x, y, x) = μ(y, x, x) = x for all x, y ∈ X. We show that each AR has a mixer and that a finite dimensional path connected space with a mixer is an AR. Our main result is that each separable space with a mixer and having an open cover by sets contractible within the whole space, is LEC.     

Some properties for the intersection of Moran sets with their translates

Motivated by Mandelbrot’s idea of referring to lacunarity of Cantor sets in terms of departure from translation invariance, Nekka and Li studied the properties of these translation sets and showed how they can be used for the classification purpose. In this paper, we pursue this study on a class of Moran sets with their rational translates. We also get the fractal structure of intersection I(x, y) of a class of Moran sets with their rational translates, and the formula of the box-counting dimension. We find that the Hausdorff measures of these sets form a discrete spectrum whose non-zero values come only from shifting vector with the expansion in fraction of (x, y). Concretely, when (x, y) has a finite expansion in fraction, a very brief calculation formula of the measure is given.     

On the measurability of functions with quasi-continuous and upper semi-continuous vertical sections

Let f: ?² → ? be a function with upper semicontinuous and quasi-continuous vertical sections f _x (t) = f(x, t), t, x ∈ ?. It is proved that if the horizontal sections f ^y (t) = f(t, y), y, t ∈ ?, are of Baire class α (resp. Lebesgue measurable) [resp. with the Baire property] then f is of Baire class α + 2 (resp. Lebesgue measurable and sup-measurable) [resp. has Baire property].     

Pseudo-linear superposition principle for the Monge-Ampère equation based on generated pseudo-operations

Through this paper, we consider generated pseudo-operations of the following form: x⊕y=g⁻¹(g(x)+g(y)), x⊙y=g⁻¹(g(x)g(y)), where g is a continuous generating function. Pseudo-linear superposition principle, i.e., the superposition principle with this type of pseudo-operations in the core, for the Monge-Ampère equation is investigated.     

On p−q=c and related three term exponential Diophantine equations with prime bases

Using a theorem on linear forms in logarithms, we show that the equation p^x−2^y=p^u−2^v has no solutions (p,x,y,u,v) with x≠u, where p is a positive prime and x,y,u, and v are positive integers, except for four specific cases, or unless p is a Wieferich prime greater than 10¹⁵. More generally, we obtain a similar result for p^x−q^y=p^u−q^v>0 where q is a positive prime, . We solve a question of Edgar showing there is at most one solution (x,y) to p^x−q^y=2^h for positive primes p and q and positive integer h. Finally, we use elementary methods to show that, with a few explicitly listed exceptions, there are at most two solutions (x,y) to |p^x±q^y|=c and at most two solutions (x,y,z) to p^x±q^y±2^z=0, for given positive primes p and q and integer c.     

Extrema of functions of eigenvalues

We consider the question of finding an extreme value for some function of the eigenvalues of the differential equation y″ + λφ(x) y = 0,y(0) = y(1) = 0, as φ(x) varies over a region in a function space. A characterization of the φ(x) at which the function of the eigenvalues achieves its extremum is derived.