On the classification of conditionally integrable evolution systems in (1 + 1) dimensions

We generalize earlier results of Fokas and Liu and find all locally analytic (1 + 1)-dimensional evolution equations of order n that admit an N-shock-type solution with N ≤ n + 1. For this, we develop a refinement of the technique from our earlier work, where we completely characterized all (1+1)-dimensional evolution systems u _t = F (x, t, u, ?u/?x,..., ?ⁿ u/? x ⁿ) that are conditionally invariant under a given generalized (Lie-Bäcklund) vector field Q(x, t, u, ?u/?x,..., ?^k u/?x ^k)?/?u under the assumption that the system of ODEs Q = 0 is totally nondegenerate. Every such conditionally invariant evolution system admits a reduction to a system of ODEs in t, thus being a nonlinear counterpart to quasi-exactly solvable models in quantum mechanics.     

Classification theory for non-elementary classes I: The number of uncountable models ofψ ∈L ω_1, ω. Part A

Assuming that 2^Nn < 2^Nn+1 forn < ω, we prove that everyψ ∈L _{ω_1, ω} has many non-isomorphic models of powerN _n for somen>0or has models in all cardinalities. We can conclude that every such Ψ has at least 2 _N 1 non-isomorphic uncountable models. As for the more vague problem of classification, restricting ourselves to the atomic models of some countableT (we can reduce general cases to this) we find a cutting line named “excellent”. Excellent classes are well understood and are parallel to totally transcendental theories, have models in all cardinals, have the amalgamation property, and satisfy the Los conjecture. For non-excellent classes we have a non-structure theorem, e.g., if they have an uncountable model then they have many non-isomorphic ones in someN _n (provided {ie212-7}).     

On the Diophantine equation x 2 − kxy + y 2 − 2 n = 0

In this study, we determine when the Diophantine equation x ²?kxy+y ²?2 ⁿ = 0 has an infinite number of positive integer solutions x and y for 0 ? n ? 10. Moreover, we give all positive integer solutions of the same equation for 0 ? n ? 10 in terms of generalized Fibonacci sequence. Lastly, we formulate a conjecture related to the Diophantine equation x ² ? kxy + y ² ? 2 ⁿ = 0.     

Radial growth,Lipschitz and Dirichlet spaces on solutions to the non-homogenous Yukawa equation

In this paper, we investigate some properties of solutions f to the nonhomogenous Yukawa equation Δf(z) = λ(z)f(z) in the unit ball \(\mathbb{B}^n\) of ? ⁿ , where λ is a real function from \(\mathbb{B}^n\) into ?. First, we prove that a main result of Girela, Pavlovi? and Peláez (J. Analyse Math. 100 (2006), 53–81) on analytic functions can be extended to this more general setting. Then we study relationships on such solutions between the bounded mean oscillation and Lipschitz-type spaces. The obtained result generalized the corresponding result of Dyakonov (Acta Math. 178 (1997), 143–167). Finally, we discuss Dirichlet-type energy integrals on such solutions in the unit ball of ? ⁿ and give an application.     

Hirzebruch Functional Equation: Classification of Solutions

The Hirzebruch functional equation is \(\sum\nolimits_{i = 1}^n {\prod\nolimits_{j \ne i} {(1/f({z_j} - {z_i})) = c} } \) with constant c and initial conditions f(0) = 0 and f'(0) = 1. In this paper we find all solutions of the Hirzebruch functional equation for n ≤ 6 in the class of meromorphic functions and in the class of series. Previously, such results have been known only for n ≤ 4. The Todd function is the function determining the two-parameter Todd genus (i.e., the χ_a,b-genus). It gives a solution to the Hirzebruch functional equation for any n. The elliptic function of level N is the function determining the elliptic genus of level N. It gives a solution to the Hirzebruch functional equation for n divisible by N. A series corresponding to a meromorphic function f with parameters in U ? ?^k is a series with parameters in the Zariski closure of U in ?^k, such that for the parameters in U it coincides with the series expansion at zero of f. The main results are as follows: (1) Any series solution of the Hirzebruch functional equation for n = 5 corresponds either to the Todd function or to the elliptic function of level 5. (2) Any series solution of the Hirzebruch functional equation for n = 6 corresponds either to the Todd function or to the elliptic function of level 2, 3, or 6. This gives a complete classification of complex genera that are fiber multiplicative with respect to ?P^n?1 for n ≤ 6. A topological application of this study is an effective calculation of the coefficients of elliptic genera of level N for N = 2,..., 6 in terms of solutions of a differential equation with parameters in an irreducible algebraic variety in ?⁴.     

Variational Inequalities: an Elementary Approach

We will deal with the following problem: Let M be an n×n matrix with real entries. Under which conditions the family of inequalities: x∈? ⁿ ;x?0;M·x?0has non–trivial solutions? We will prove that a sufficient condition is given by m_i,j+m_j,i?0 (1?i,j?n); from this result we will derive an elementary proof of the existence theorem for Variational Inequalities in the framework of Monotone Operators.     

The number of large prime divisors of consecutive integers

Let ?(N) > 0 be a function of positive integers N and such that ?(N) → 0 and N^?(N) → ∞ as N → + ∞. Let Nν_N(n:…) be the number of positive integers n ≤ N for which the property stated in the dotted space holds. Finally, let g(n; N, ?, z) be the number of those prime divisors p of n which satisfy N^Z?(N) ? p ? N^?(N), 0 < z < 1 In the present note we show that for each k = 0, ±1, ±2,…, as N → ∞, limv_N(n : g(n; N, ?, z) ? g(n + 1; N, ?z) = k) exists and we determine its actual value. The case k = 0 induced the present investigation. Our solution for this value shows that the natural density of those integers n for which n and n + 1 have the same number of prime divisors in the range (1) exists and it is positive.     

Number-theoretic degeneracy of the energy levels of a perturbed N-dimensional isotropic harmonic oscillator

The problem of constructing all integer solutions n₁ ≥ n₂ ≥ … ≥ n_N to the pair of Diophantine equations n = n₁ + … + n_N, m = n₁² + … + n_N² arises in the determination of the degeneracy of a given energy level of an N-dimensional isotropic quantum oscillator that is perturbed by an isotropic quartic potential energy term. This problem is solved recursively (in N) using the concept of a multiplet, which is a finite set of points in a lattice space L^N whose points are N-tuples of integers that sum to zero. The basic definition and properties of multiplets are given and then used to obtain the solutions to the Diophantine equations described above. The classification of multiplets into two types, fundamental and nonfundamental, is shown to have an important role in elucidating the structure of multiplets. The concept of a fundamental multiplet is demonstrated to be an important characterization of the solutions to a pair of Diophantine equations that are closely related to those of the original problem.     

How Close is the Sample Covariance Matrix to?the?Actual Covariance Matrix?

Given a probability distribution in ? ⁿ with general (nonwhite) covariance, a classical estimator of the covariance matrix is the sample covariance matrix obtained from a sample of N independent points. What is the optimal sample size N=N(n) that guarantees estimation with a fixed accuracy in the operator norm? Suppose that the distribution is supported in a centered Euclidean ball of radius $O(\sqrt{n})$ . We conjecture that the optimal sample size is N=O(n) for all distributions with finite fourth moment, and we prove this up to an iterated logarithmic factor. This problem is motivated by the optimal theorem of Rudelson (J. Funct. Anal. 164:60?C72, 1999), which states that N=O(nlog?n) for distributions with finite second moment, and a recent result of Adamczak et al. (J. Am. Math. Soc. 234:535?C561, 2010), which guarantees that N=O(n) for subexponential distributions.     

An abstract approach to Loewner chains

We present a new geometric construction of Loewner chains in one and several complex variables which holds on complete hyperbolic complex manifolds and prove that there is essentially a one-to-one correspondence between evolution families of order d and Loewner chains of the same order. As a consequence, we obtain a univalent solution (f _t : M → N) of any Loewner-Kufarev PDE. The problem of finding solutions given by univalent mappings (f _t : M → ? ⁿ ) is reduced to investigating whether the complex manifold ∪ _t≥0 f _t (M) is biholomorphic to a domain in ? ⁿ . We apply such results to the study of univalent mappings f: B ⁿ → ? ⁿ .     

Linear equations with unknowns from a multiplicative group whose solutions lie in a small number of subspaces

Let K be a field of characteristic 0 and let (K^*)ⁿ denote the n-fold Cartesian product of K^*, endowed with coordinatewise multiplication. Let Γ be a subgroup of (K^*)ⁿ of finite rank. We consider equations (*) a₁x₁ + … + a_nx_n = 1 in x = (x₁x_n)Γ, where a = (a₁,a_n)(K^*)ⁿ. Two tuples a, b(K^*)ⁿ are called Γ-equivalent if there is a uΓ such that b = u · a. Gy?ry and the author [Compositio Math. 66 (1988) 329-354] showed that for all but finitely many Γ-equivalence classes of tuples a(K^*)ⁿ, the set of solutions of (*) is contained in the union of not more than 2⁽ⁿ⁺¹! proper linear subspaces of Kⁿ. Later, this was improved by the author [J. reine angew. Math. 432 (1992) 177-217] to (n!)²ⁿ⁺². In the present paper we will show that for all but finitely many Γ-equivalence classes of tuples of coefficients, the set of non-degenerate solutions of (*) (i.e., with non-vanishing subsums) is contained in the union of not more than 2ⁿ proper linear subspaces of Kⁿ. Further we give an example showing that 2ⁿ cannot be replaced by a quantity smaller than n.     

A note on the number of solutions of the generalized Ramanujan-Nagell equation x 2 ? D = p n

Let D be a positive integer, and let p be an odd prime with p ? D. In this paper we use a result on the rational approximation of quadratic irrationals due to M. Bauer, M.A. Bennett: Applications of the hypergeometric method to the generalized Ramanujan-Nagell equation. Ramanujan J. 6 (2002), 209–270, give a better upper bound for N(D, p), and also prove that if the equation U ² ? DV ² = ?1 has integer solutions (U, V), the least solution (u ₁, v ₁) of the equation u ² ? pv ² = 1 satisfies p ? v ₁, and D > C(p), where C(p) is an effectively computable constant only depending on p, then the equation x ² ? D = p ⁿ has at most two positive integer solutions (x, n). In particular, we have C(3) = 10⁷.     

On well-posedness of the semilinear heat equation on the sphere

We are concerned with existence, uniqueness and nonuniqueness of nonnegative solutions to the semilinear heat equation in open subsets of the n-dimensional sphere. Existence and uniqueness results are obtained using L ^p ?? L ^q estimates for the semigroup generated by the Laplace?CBeltrami operator. Moreover, under proper assumptions on the nonlinear function, we establish nonuniqueness of weak solutions, when n??? 3; to do this, we shall prove uniqueness of positive classical solutions and nonuniqueness of singular solutions of the corresponding semilinear elliptic problem.     

On exact recovery of sparse vectors from linear measurements

Let 1 ≤ k ≤ n < N. We say that a vector x ∈ ? ^N is k-sparse if it has at most k nonzero coordinates. Let Φ be an n × N matrix. We consider the problem of recovery of a k-sparse vector x ∈ ? ^N from the vector y = Φx ∈ ? ⁿ . We obtain almost-sharp necessary conditions for k, n, N for this problem to be reduced to that of minimization of the ?₁-norm of vectors z satisfying the condition y = Φz.     

On the Diophantine equation ax + by = ck

In this paper we investigate the solvability and the representation of the solutions of the equation ax² +by² = ckⁿ. We extend and improve many known results. In particular, we completely solve the equation (a ± 1)x² + (3a ? 1) = 4aⁿ, 2 ? n.     

Uniqueness implies existence and uniqueness conditions for nonlocal boundary value problems for nth order differential equations

For the nth order differential equation, y(n)=f(x,y,y^′,…,y(n−1)), we consider uniqueness implies existence results for solutions satisfying certain nonlocal (k+2)-point boundary conditions, 1?k?n−1. Uniqueness of solutions when k=n−1 is intimately related to uniqueness of solutions when 1?k?n−2. These relationships are investigated as well.     

Isolated singularities of some semilinear elliptic equations

Let Ω be an open subset of $\text{R}{}^{\mathrm{<mi\; mathvariant="italic"\; is="true">N</mi>}}$, N ? 3, containing 0. We consider the solutions of ?Δu(x) + g(u(x)) = f(x) in Ω-{0}, where g is nondecreasing and f is bounded and we study the possible singularities at 0: when u(x) = o(|x|^{1 ? N}) we prove that u is isotropic near 0 and show that either it is a C¹ function in Ω (removable singularity) or |x|^{N ? 2}u(x) → c, c ≠ 0 (weak singularity) or |x|^{N ? 2} |u(x) |→ + ∞ (strong singularity). We also characterize the g's for which solutions with a weak singularity exist and improve a previous removability result of H. Brézis and L. Véron (Arch. Rational Mech. Anal.23 (1979), 153–166).     

About some cyclic properties in digraphs

In this article we are concerned with digraphs in which any two vertices are on a common cycle. For example, we prove that, in a strong digraph of order n and half degrees at least 2 with at least n² ? 5n + 15 arcs, any two vertices are on a common cycle. We also consider related properties and give sufficient conditions on half degrees and the number of arcs to insure these properties. In particular, we show that every digraph of order n with half degrees at least r and with at least n² ? rn + r² arcs is 2-linked.     

A new technique of integral representations in ℂ n

A new technique of integral representations in ? ⁿ , which is different from the well-known Henkin technique, is given. By means of this new technique, a new integral formula for smooth functions and a new integral representation of solutions of the ?-equations on strictly pseudoconvex domains in ? ⁿ are obtained. These new formulas are simpler than the classical ones, especially the solutions of the ?-equations admit simple uniform estimates. Moreover, this new technique can be further applied to arbitrary bounded domains in ? ⁿ so that all corresponding formulas are simplified.     

Strong approximation by Fourier--Laplace series on the unit sphere S n-1

We study the strong approximation properties of the Cesáro means of order δ of the Fourier--Laplace expansion of functions integrable on the unit sphere S ^n-1, where δ ≥λ? (n-2)/2, the latter being the critical index for Cesáro summability of Fourier--Laplace series on S ^n-1. The main purpose of this paper is to extend known results from the unit circle S ¹to the general sphere S ^n-1 with n≥3. We prove six theorems. To prove Theorems 1-3, our machinery is based on the equiconvergent operator E ^δ _N (f) of the Cesáro means σ^δ _N (f) on S ^n-1 introduced by Wang Kunyang for δ>-1. We prove in Theorem 6 that E ^δ _N (f) is also equiconvergent with σ^δ _N (f) for δ>0 in the case of strong approximation. To prove Theorems 4 and 5, we rely on known equivalence relations between K-functionals and moduli of continuity.