Characterization and Transformation of Invariants

We consider difference equations of order k _n+k ≥ 2 of the form: y_n+k = f(yn,…,yn+k-1), n= 0,1,2,… where f: D ^k → D is a continuous function, and D?R. We develop a necessary and sufficient condition for the existence of a symmetric invariant I(x ₁,…,x_k ) ∈C^∞[D^k,D]. This condition will be used to construct invariants for linear and rational difference equations. Also, we investigate the transformation of invariants under invertible maps. We generalize and extend several results that have been obtained recently.     

The asymptotic formula in Waring’s problem for one square and seventeen fifth powers

Let R _k,s (n) denote the number of solutions of the equation n = x² + y₁^k + y₂^k + ?+ y_s^k{n= x^2 + y_1^k + y_2^k + \cdots + y_s^k} in natural numbers x, y ₁, . . . , y _s . By a straightforward application of the circle method, an asymptotic formula for R _k,s (n) is obtained when k ≥ 3 and s ≥ 2 ^k–1 + 2. When k ≥ 6, work of Heath-Brown and Boklan is applied to establish the asymptotic formula under the milder constraint s ≥ 7 · 2 ^k–4 + 3. Although the principal conclusions provided by Heath-Brown and Boklan are not available for smaller values of k, some of the underlying ideas are still applicable for k = 5, and the main objective of this article is to establish an asymptotic formula for R _5,17(n) by this strategy.     

Quasi‐symmetric designs with fixed difference of block intersection numbers

The following results for proper quasi‐symmetric designs with non‐zero intersection numbers x,y and λ > 1 are proved.

• (1) Let D be a quasi‐symmetric design with z = y ? x and v ≥ 2k. If x ≥ 1 + z + z³ then λ < x + 1 + z + z³.
• (2) Let D be a quasi‐symmetric design with intersection numbers x, y and y ? x = 1. Then D is a design with parameters v = (1 + m) (2 + m)/2, b = (2 + m) (3 + m)/2, r = m + 3, k = m + 1, λ = 2, x = 1, y = 2 and m = 2,3,… or complement of one of these design or D is a design with parameters v = 5, b = 10, r = 6, k = 3, λ = 3, and x = 1, y = 2.
• (3) Let D be a triangle free quasi‐symmetric design with z = y ? x and v ≥ 2k, then x ≤ z + z².
• (4) For fixed z ≥ 1 there exist finitely many triangle free quasi‐symmetric designs non‐zero intersection numbers x, y = x + z.
• (5) There do not exist triangle free quasi‐symmetric designs with non‐zero intersection numbers x, y = x + 2.

© 2006 Wiley Periodicals, Inc. J Combin Designs 15: 49–60, 2007     

Investigation of solvable (120, 35, 10) difference sets

Abelian difference sets with parameters (120, 35, 10) were ruled out by Turyn in 1965. Turyn's techniques do not apply to nonabelian groups. We attempt to determine the existence of (120, 35, 10) difference sets in the 44 nonabelian groups of order 120. We prove that if a solvable group admits a (120, 35, 10) difference set, then it admits a quotient group isomorphic to the cyclic group of order 24 or to U₂₄ ? 〈x,y : x⁸ = y³ = 1, xyx^?1 = y^?1〉. We describe a computer search, which rules out solutions with a ?₂₄ quotient. The existence question remains undecided in the three solvable groups admitting a U₂₄ quotient. The question also remains undecided for the three nonsolvable groups of order 120. © 2004 Wiley Periodicals, Inc.     

Thompson’s metric and global stability of difference equations

We apply the Thompson’s metric to study the global stability of the equilibium of the following difference equation for fixed integers k with k≠0,±1 in the quasi-Banach spaces.     

18.

Nonvanishing derivatives and normal families     

Walter Bergweiler  J. K. Langley 《Journal d'Analyse Mathématique》2003,91(1):353-367

We consider the differential operators Ψ _k , defined by Ψ₁(y) =y and Ψ _k+1(y)=yΨ _k y+d/dz(Ψ _k (y)) fork ∈ ℕ fork∈ ℕ. We show that ifF is meromorphic in ℂ and Ψ _k F has no zeros for somek≥3, and if the residues at the simple poles ofF are not positive integers, thenF has the formF(z)=((k-1)z+a)/(z ²+β z+γ) orF(z)=1/(az+β) where α, β, γ ∈ ℂ. If the residues at the simple poles ofF are bounded away from zero, then this also holds fork=2. We further show that, under suitable additional conditions, a family of meromorphic functionsF is normal if each Ψ _k (F) has no zeros. These conditions are satisfied, in particular, if there exists δ>0 such that Re (Res(F, a)) <−δ for all polea of eachF in the family. Using the fact that Ψ _k (f ^′/f) =f ^(k)/f, we deduce in particular that iff andf ^(k) have no zeros for allf in some familyF of meromorphic functions, wherek≥2, then {f ^′/f :f ∈F} is normal. The first author is supported by the German-Israeli Foundation for Scientific Research and Development G.I.F., G-643-117.6/1999, and INTAS-99-00089. The second author thanks the DAAD for supporting a visit to Kiel in June–July 2002. Both authors thank Günter Frank for helpful discussions.     

19.

ON PSEUDO-COMMUTATMTY AND COMMUTATIVITY IN RINGS     

《Quaestiones Mathematicae》2013,36(2):173-182

Abstract

A ring R is called pseudo-commutative if for each x,y ε R there exists an integer n = n(x, y) for which xy = nyx. We first show that a generalization of a commutativity condition of Chacron and Thierrin implies pseudo-commutativity in rings; we then study pseudo-commutativity and commutativity in rings with constraints of the form xy = σk_iyⁱxⁱ, where the k_i are integers.     

20.

Spectral parameter power series for arbitrary order linear differential equations     

Vladislav V. Kravchenko  R. Michael Porter  Sergii M. Torba 《Mathematical Methods in the Applied Sciences》2019,42(15):4902-4908

Let L be the n‐th order linear differential operator Ly=?₀y⁽ⁿ⁾+?₁y^(n?1)+?+?_ny with variable coefficients. A representation is given for n linearly independent solutions of Ly=λry as power series in λ, generalizing the SPPS (spectral parameter power series) solution that has been previously developed for n=2. The coefficient functions in these series are obtained by recursively iterating a simple integration process, beginning with a solution system for λ=0. It is shown how to obtain such an initializing system working upwards from equations of lower order. The values of the successive derivatives of the power series solutions at the basepoint of integration are given, which provides a technique for numerical solution of n‐th order initial value problems and spectral problems.     

Nonvanishing derivatives and normal families

We consider the differential operators Ψ _k , defined by Ψ₁(y) =y and Ψ _k+1(y)=yΨ _k y+d/dz(Ψ _k (y)) fork ∈ ℕ fork∈ ℕ. We show that ifF is meromorphic in ℂ and Ψ _k F has no zeros for somek≥3, and if the residues at the simple poles ofF are not positive integers, thenF has the formF(z)=((k-1)z+a)/(z ²+β z+γ) orF(z)=1/(az+β) where α, β, γ ∈ ℂ. If the residues at the simple poles ofF are bounded away from zero, then this also holds fork=2. We further show that, under suitable additional conditions, a family of meromorphic functionsF is normal if each Ψ _k (F) has no zeros. These conditions are satisfied, in particular, if there exists δ>0 such that Re (Res(F, a)) <−δ for all polea of eachF in the family. Using the fact that Ψ _k (f ^′/f) =f ^(k)/f, we deduce in particular that iff andf ^(k) have no zeros for allf in some familyF of meromorphic functions, wherek≥2, then {f ^′/f :f ∈F} is normal. The first author is supported by the German-Israeli Foundation for Scientific Research and Development G.I.F., G-643-117.6/1999, and INTAS-99-00089. The second author thanks the DAAD for supporting a visit to Kiel in June–July 2002. Both authors thank Günter Frank for helpful discussions.     

ON PSEUDO-COMMUTATMTY AND COMMUTATIVITY IN RINGS

Abstract

A ring R is called pseudo-commutative if for each x,y ε R there exists an integer n = n(x, y) for which xy = nyx. We first show that a generalization of a commutativity condition of Chacron and Thierrin implies pseudo-commutativity in rings; we then study pseudo-commutativity and commutativity in rings with constraints of the form xy = σk_iyⁱxⁱ, where the k_i are integers.     

Spectral parameter power series for arbitrary order linear differential equations

Let L be the n‐th order linear differential operator Ly=?₀y⁽ⁿ⁾+?₁y^(n?1)+?+?_ny with variable coefficients. A representation is given for n linearly independent solutions of Ly=λry as power series in λ, generalizing the SPPS (spectral parameter power series) solution that has been previously developed for n=2. The coefficient functions in these series are obtained by recursively iterating a simple integration process, beginning with a solution system for λ=0. It is shown how to obtain such an initializing system working upwards from equations of lower order. The values of the successive derivatives of the power series solutions at the basepoint of integration are given, which provides a technique for numerical solution of n‐th order initial value problems and spectral problems.     

Matrices dont deux lignes quelconque coïncident dans un nombre donne de positions communes

Le nombre maximal de lignes de matrices seront désignées par:

1. (a) R(k, λ) si chaque ligne est une permutation de nombres 1, 2,…, k et si chaque deux lignes différentes coïncide selon λ positions;

2. (b) S⁰(k, λ) si le nombre de colonnes est k et si chaque deux lignes différentes coïncide selon λ positions et si, en plus, il existe une colonne avec les éléments y₁, y₂, y₃, ou y₁ = y₂ ≠ y₃;

3. (c) T⁰(k, λ) si c'est une (0, 1)-matrice et si chaque ligne contient k unités et si chaque deux lignes différentes contient les unités selon λ positions et si, en plus, il existe une colonne avec les éléments 1, 1, 0.

La fonction T⁰(k, λ) était introduite par Chvátal et dans les articles de Deza, Mullin, van Lint, Vanstone, on montrait que T⁰(k, λ) max(λ + 2, (k − λ)² + k − λ + 1). La fonction S⁰(k, λ) est introduite ici et dans le Théorème 1 elle est étudiée analogiquement; dans les remarques 4, 5, 6, 7 on donne les généralisations de problèmes concernant T⁰(k, λ), S⁰(k, λ), dans la remarque 9 on généralise le problème concernant R(k, λ). La fonction R(k, λ) était introduite et étudiée par Bolton. Ci-après, on montre que R(k, λ) S⁰(k, λ) T⁰(k, λ) d'où découle en particulier: R(k, λ) λ + 2 pour λ k + 1 − (k + 2)^1/2; R(k, λ) = 0(k²) pour k − λ = 0(k); R(k, λ) (k − 1)² − (k + 2) pour k 1191.     

Approximations of Two-Place Functions by Finite Sums of Terms with Separable Variables

It is known that if a smooth function h in two real variables x and y belongs to the class Σ_n of all sums of the form Σⁿ_k=1ƒ_k(x) g_k(y), then its (n + 1)th order "Wronskian" det[h_xⁱy^j]ⁿ_i,j=0 is identically equal to zero. The present paper deals with the approximation problem h(x, y) Σⁿ_k=1ƒ_k(x) g_k(y) with a prescribed n, for general smooth functions h not lying in Σ_n. Two natural approximation sums T=T(h) Σ_n, S=S(h) Σ_n are introduced and the errors |h-T|, |h-S| are estimated by means of the above mentioned Wronskian of the function h. The proofs utilize the technique of ordinary linear differential equations.     

Existence and Uniqueness of Strong Solutions for a Class of Quasi‐Linear Hyperbolic Equations with Order Degeneration

Let k(y) > 0, 𝓁(y) > 0 for y > 0, k(0) = 𝓁(0) = 0 and lim_{y → 0}k(y)/𝓁(y) exists; then the equation L(u) ≔ k(y)u_xx – ∂_y(𝓁(y)u_y) + a(x, y)u_x = f(x, y, u) is strictly hyperbolic for y > 0 and its order degenerates on the line y = 0. Consider the boundary value problem Lu = f(x, y, u) in G, u|_AC = 0, where G is a simply connected domain in ℝ² with piecewise smooth boundary ∂G = AB∪AC∪BC; AB = {(x, 0) : 0 ≤ x ≤ 1}, AC : x = F(y) = ∫^y₀(k(t)/𝓁(t))^1/2dt and BC : x = 1 – F(y) are characteristic curves. Existence of generalized solution is obtained by a finite element method, provided f(x, y, u) satisfies Carathéodory condition and |f(x, y, u)| ≤ Q(x, y) + b|u| with Q ∈ L²(G), b = const > 0. It is shown also that each generalized solution is a strong solution, and that fact is used to prove uniqueness under the additional assumption |f(x, y, u₁) – f(x, y, u₂| ≤ C|u₁ – u₂|, where C = const > 0.     

High‐order local discontinuous Galerkin method for a fractal mobile/immobile transport equation with the Caputo–Fabrizio fractional derivative

In this article, a local discontinuous Galerkin (LDG) method is studied for numerically solving the fractal mobile/immobile transport equation with a new time Caputo–Fabrizio fractional derivative. The stability of the LDG scheme is proven, and a priori error estimates with the second‐order temporal convergence rate and the (k + 1) th order spatial convergence rate are derived in detail. Finally, numerical experiments based on P^k, k = 0, 1, 2, 3, elements are provided to verify our theoretical results.     

On the cost of controlling unstable systems: The case of boundary controls

We discuss the cost of controlling parabolic equations of the formy _t + δ² y +kδy = 0 in a bounded smooth domain Ώ ofℝ ^d by means of a boundary control. More precisely, we are interested in the cost of controlling from zero initial state to a given final state (in a suitable approximate sense) at timeT > 0 and in the behavior of this cost ask → ∞. We introduce finite-dimensional Galerkin approximations and prove that they are exactly controllable. Moreover, we also prove that the cost of controlling converges exponentially to zero ask → ∞. This proves, roughly speaking, that when the system becomes more unstable it is easier to control. The system under consideration does not admit a variational formulation. Thus, in order to introduce its Galerkin approximation, we first approximate it by means of a singular perturbation. We also develop a method for the construction of special Galerkin bases well adapted to the control problem. Dedicated to John E. Lagnese on his 60th Birthday. Supported by project PB93-1203 of the DGICYT (Spain) and grant CHRX-CT94-0471 of the European Union.     

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.     

Inequalities and bounds for quasi-symmetric 3-designs

Quasi-symmetric 3-designs with block intersection numbers x and y(0x<y<k) are studied, several inequalities satisfied by the parameters of a quasi-symmetric 3-designs are obtained. Let D be a quasi-symmetric 3-design with the block size k and intersection numbers x, y; y>x1 and suppose D′ denote the complement of D with the block size k′ and intersection numbers x′ and y′. If k −1 x + y then it is proved that x′ + y′ k′. Using this it is shown that the quasi-symmetric 3-designs corresponding to y = x + 1, x + 2 are either extensions of symmetric designs or designs corresponding to the Witt-design (or trivial design, i.e., v = k + 2) or the complement of above designs.     

Coin flipping from a cosmic source: On error correction of truly random bits

We study a problem related to coin flipping, coding theory, and noise sensitivity. Consider a source of truly random bits x ∈ {0, 1}ⁿ, and k parties, who have noisy version of the source bits yⁱ ∈ {0, 1}ⁿ, when for all i and j, it holds that P [y = x_j] = 1 ? ?, independently for all i and j. That is, each party sees each bit correctly with probability 1 ? ?, and incorrectly (flipped) with probability ?, independently for all bits and all parties. The parties, who cannot communicate, wish to agree beforehand on balanced functions f_i: {0, 1}ⁿ → {0, 1} such that P [f₁(y¹) = … = f_k(y^k)] is maximized. In other words, each party wants to toss a fair coin so that the probability that all parties have the same coin is maximized. The function f_i may be thought of as an error correcting procedure for the source x. When k = 2,3, no error correction is possible, as the optimal protocol is given by f_i(yⁱ) = y. On the other hand, for large values of k, better protocols exist. We study general properties of the optimal protocols and the asymptotic behavior of the problem with respect to k, n, and ?. Our analysis uses tools from probability, discrete Fourier analysis, convexity, and discrete symmetrization. © 2005 Wiley Periodicals, Inc. Random Struct. Alg., 2005     

Upper bounds on the sum of powers of the degrees of a simple planar graph

For a simple planar graph G and a positive integer k, we prove the upper bound 2(n ? 1)^k + 4^k(n ? 4) + 2·3^k ? 2((δ + 1)^k ? δ^k)(3n ? 6 ? m) on the sum of the kth powers of the degrees of G, where n, m, and δ are the order, the size, and the minimum degree of G, respectively. The bound is tight for all m with 0?3n ? 6 ? m≤?n/2? ? 2 and δ = 3. We also present upper bounds in terms of order, minimum degree, and maximum degree of G. © 2010 Wiley Periodicals, Inc. J Graph Theory 67:112‐123, 2011     

Remarks on rings of constants of derivations II

Let k be a field of characteristic p>0 and D≠0 a family of k-derivations of k[x,y]. It is proved in [1] that k[x,y]^D, the ring of constants with respect to D, can be generated, as a k[x ^p,y ^p]-algebra, by p - 1 elements. In this note we prove that p - 1 is the sharp upper bound of numbers of generators.     

Periodic Solutions of Non-linear Anisotropic Partial Differential Equations

Periodic solutions of arbitrary period to semilinear partial differential equations of Zabusky or Boussinesq type are obtained. More generally, for a linear differential operator A(y,∂), the equation A(y,∂)u = (−1)^∣γ∣∂^γf(y,∂^γu), y = (t,x)∈ℝ^k×G is studied, where homogeneous boundary conditions on ∂G and periodicity conditions on t are imposed. The solutions are obtained by variational methods in anisotropic Sobolev spaces.     

Solution and stability of generalized mixed type cubic, quadratic and additive functional equation in quasi-Banach spaces

In this paper, we achieve the general solution and the generalized Hyers–Ulam–Rassias stability of the following functional equation

f(x+ky)+f(x−ky)=k²f(x+y)+k²f(x−y)+2(1−k²)f(x)