Possible values for 2

From GCH and P^{m(κ)-hypermeasurable} (1 <m<gw), we construct a model satisfying 2_ⁿ = _a(n) and 2_^ω = _ω+m for a monotone a:ω→ω satisfying a(n)>n.     

On neighborhood condition for graphs to have [a,b]-factors

Let G be a graph of order n, and let a and b be integers such that 1a<b. Let δ(G) be the minimum degree of G. Then we prove that if δ(G)(k−1)a, n(a+b)(k(a+b)−2)/b, and |N_G(x₁)N_G(x₂)N_G(x_k)|an/(a+b) for any independent subset {x₁,x₂,…,x_k} of V(G), where k2, then G has an [a,b]-factor. This result is best possible in some sense.     

The W-convexity and normal structure in banach spaces

Let X be a Banach space, S(X) - x ε X : #x02016; = 1 be the unit sphere of X.The parameter, modulus of W^*-convexity, W^*(ε) = inf <(x − y)/2, f_x> : x, y S(X), x − y ≥ ε, f_x Δ_x , where 0 ≤ ε ≤ 2 and Δ_x S(X^*) be the set of norm 1 supporting functionals of S(X) at x, is investigated_ The relationship among uniform nonsquareness, uniform normal structure and the parameter W^*(ε) are studied, and a known result is improved. The main result is that for a Banach space X, if there is ε, where 0 < ε < 1/2, such that W^*(1 + ε) > ε/2 where W^*(1 + ε) = lim_→ε W^* (1 + ), then X has normal structure.     

On cardinal sequences of scattered spaces

It was proved by Dow and Simon that there are 2^ω₁ (as many as possible) pairwise nonhomeomorphic compact, T₂, scattered spaces of height ω₁ and width ω. In this paper, we prove that if is an ordinal withω₁ < ω₂ and θ = κ_ξ: ξ < is a sequence of cardinals such that either κ_ξ = ω or κ_ξ = ω₁ for every ξ < , then there are 2^ω₁ pairwise nonhomeomorphic compact, T₂, scattered spaces whose cardinal sequence is θ.     

L2-approximation of real-valued functions with interpolatory constraints

We construct the polynomial p_m,n^* of degree m which interpolates a given real-valued function f L²[a, b] at pre-assigned n distinct nodes and is the best approximant to f in the L₂-sense over all polynomials of degree m with the same interpolatory character. It is shown that the L₂-error p_m,n^* − f → 0 as m → ∞ if f C[a, b].     

Full quadrature sums for pth powers of polynomials with Freud weights

In this paper, we provide a solution of the quadrature sum problem of R. Askey for a class of Freud weights. Let r> 0, b (− ∞, 2]. We establish a full quadrature sum estimate

1 p < ∞, for every polynomial P of degree at most n + rn^1/3, where W² is a Freud weight such as exp(−¦x¦), > 1, λ_jn are the Christoffel numbers, x_jn are the zeros of the orthonormal polynomials for the weight W², and C is independent of n and P. We also prove a generalisation, and that such an estimate is not possible for polynomials P of degree M = m(n) if m(n) = n + ξ_nn^1/3, where ξ_n → ∞ as n → ∞. Previous estimates could sum only over those x_jn with ¦x_jn¦ σx_1n, some fixed 0 < σ < 1.     

A note on the stationarity of a threshold first-order bilinear process

In the present note we study the threshold first-order bilinear model

X(t)=aX(t−1)+(b₁1_{X(t−1)<c}+b₂1_{X(t−1)c})X(t−1)e(t−1)+e(t), tεN

where {e(t), tεN} is a sequence of i.i.d. absolutely continuous random variables, X(0) is a given random variable and a, b₁, b₂ and c are real numbers. Under suitable conditions on the coefficients and lower semicontinuity of the densities of the noise sequence, we provide sufficient conditions for the existence of a stationary solution process to the present model and of its finite moments of order p.     

On the surface viscosity at the boundary between phases

The equations of motion of the interphase boundary are considered. It is shown that the conditions at the surface separating the phases obtained in /1, 2/ by different methods, are identical. The study of the dynamics of the fluid-fluid interface was initiated by Bussinesq /3/ who postulated a linear relationship between the surface stress tensor T_β and the strain rate tensor S_β, assigning two viscosity coefficients to the surface, the dilatation coefficient k (the analog of volume viscosity) and the two-dimensional shear viscosity . In the three-dimensional coordinate system two of whose axes u¹ and usu2 coincide with the axes of any coordinate system at the surface and whose third axis u³ is perpendicular to the surface, his results have the form Tβ = [γ + (k - )θ]a_β + S_β , θ = a^βS_β, V_{, β} = r_{. β} — v_sb_β,      

On the degree of regularity of some equations

In this paper we investigate the behaviour of the solutions of equations Σ_I=1ⁿ a_ix_i = b, where Σ_i=1ⁿ, a_i = 0 and b ≠ 0, with respect to colorings of the set N of positive integers. It turns out that for any b ≠ 0 there exists an 8-coloring of N, admitting no monochromatic solution of x₃ − x₂ = x₂ − x₁ + b. For this equation, for b odd and 2-colorings, only an odd-even coloring prevents a monochromatic solution. For b even and 2-colorings, always monochromatic solutions can be found, and bounds for the corresponding Rado numbers are given. If one imposes the ordering x₁ < x₂ < x₃, then there exists already a 4-coloring of N, which prevents a monochromatic solution of x₃ − x₂ = x₂ − x₁ + b, where b ε N.     

Extremal properties of strong quadrature weights and maximal mass results for truncated strong moment problems

A bisequence of complex numbers {μ_n}_−∞^∞ determines a strong moment functional satisfying L[xⁿ] = μ_n. If is positive-definite on a bounded interval (a,b) R{0}, then has an integral representation , n=0, ±1, ±2,…, and quadrature rules {w_ni,x_ni} exist such that μ_k = ∑_i=i^n_ns_ni^kw_ni. This paper is concerned with establishing certain extremal properties of the weights w_ni and using these properties to obtain maximal mass results satisfied by distributions ψ(x) representing when only a finite bisequence of moments {μ_k}_k=−nⁿ⁻¹ is given.     

The thermal equilibrium state of semiconductor devices

The thermal equilibrium state of two oppositely charged gases confined to a bounded domain , m = 1,2 or m = 3, is entirely described by the gases' particle densities p, n minimizing the total energy (p, n). it is shown that for given P, N > 0 the energy functional admits a unique minimizer in {(p, n) ε L²(Ω) x L ²(Ω) : p, n ≥ 0, _Ωp = P, _Ωn = N} and that p, n ε C(Ω) ∩ L^∞(Ω).

The analysis is applied to the hydrodynamic semiconductor device equations. These equations in general possess more than one thermal equilibrium solution, but only the unique solution of the corresponding variational problem minimizes the total energy. It is equivalent to prescribe boundary data for electrostatic potential and particle densities satisfying the usual compatibility relations and to prescribe V^e and P, N for the variational problem.     

Weak convergence to the multiple Stratonovich integral

We have considered the problem of the weak convergence, as tends to zero, of the multiple integral processes

in the space , where fL²([0,T]ⁿ) is a given function, and {η(t)}_>0 is a family of stochastic processes with absolutely continuous paths that converges weakly to the Brownian motion. In view of the known results when n2 and f(t₁,…,t_n)=1_{t1<t2<<tn}, we cannot expect that these multiple integrals converge to the multiple Itô–Wiener integral of f, because the quadratic variations of the η are null. We have obtained the existence of the limit for any {η}, when f is given by a multimeasure, and under some conditions on {η} when f is a continuous function and when f(t₁,…,t_n)=f₁(t₁)f_n(t_n)1_{t1<t2<<tn}, with f_iL²([0,T]) for any i=1,…,n. In all these cases the limit process is the multiple Stratonovich integral of the function f.     

The (b; c)-inverse in semigroups and rings with involution

We first prove that if a is both left (b; c)-invertible and left (c; b)- invertible, then a is both (b; c)-invertible and (c; b)-invertible in a *-monoid, which generalizes the recent result about the inverse along an element by L. Wang and D. Mosić [Linear Multilinear Algebra, Doi.org/10.1080/03081087. 2019.1679073], under the conditions (ab)* = ab and (ac)* = ac: In addition, we consider that ba is (c; b)-invertible, and at the same time ca is (b; c)-invertible under the same conditions, which extend the related results about Moore- Penrose inverses studied by J. Chen, H. Zou, H. Zhu, and P. Patrício [Mediterr J. Math., 2017, 14: 208] to (b; c)-inverses. As applications, we obtain that under condition (a²)* = a²; a is an EP element if and only if a is one-sided core invertible, if and only if a is group invertible.     

Stable limits of empirical processes of moving averages with infinite variance

The paper obtains a functional limit theorem for the empirical process of a stationary moving average process X_t with i.i.d. innovations belonging to the domain of attraction of a symmetric -stable law, 1<<2, with weights b_j decaying as j^−β, 1<β<2/. We show that the empirical process (normalized by N^1/β) weakly converges, as the sample size N increases, to the process c_x⁺L⁺+c_x⁻L⁻, where L⁺,L⁻ are independent totally skewed β-stable random variables, and c_x⁺,c_x⁻ are some deterministic functions. We also show that, for any bounded function H, the weak limit of suitably normalized partial sums of H(X_s) is an β-stable Lévy process with independent increments. This limiting behavior is quite different from the behavior of the corresponding empirical processes in the parameter regions 1/<β<1 and 2/<β studied in Koul and Surgailis (Stochastic Process. Appl. 91 (2001) 309) and Hsing (Ann. Probab. 27 (1999) 1579), respectively.     

A two-sided estimate in the Hsu-Robbins-Erdös law of large numbers

Let X₁, X₂, … be independent identically distributed random variables. Then, Hsu and Robbins (1947) together with Erdös (1949, 1950) have proved that

,

if and only if E[X²₁] < ∞ and E[X₁] = 0. We prove that there are absolute constants C₁, C₂ (0, ∞) such that if X₁, X₂, … are independent identically distributed mean zero random variables, then

c₁λ⁻² E[X₁²·1_{|X1|λ}]S(λ)C₂λ⁻² E[X₁²·1_{|X1|λ}]

,

for every λ > 0.     

On the weak Freese-Nation property of complete Boolean algebras

The following results are proved:

(a) In a model obtained by adding ₂ Cohen reals, there is always a c.c.c. complete Boolean algebra without the weak Freese-Nation property. (b) Modulo the consistency strength of a supercompact cardinal, the existence of a c.c.c. complete Boolean algebra without the weak Freese-Nation property is consistent with GCH. (c) If a weak form of □_μ and cof([μ]^₀,)=μ⁺ hold for each μ>cf(μ)=ω, then the weak Freese-Nation property of is equivalent to the weak Freese-Nation property of any of or for uncountable κ. (d) Modulo the consistency of (_ω+1,_ω)(₁,₀), it is consistent with GCH that does not have the weak Freese-Nation property and hence the assertion in (c) does not hold, and also that adding _ω Cohen reals destroys the weak Freese-Nation property of .

These results solve all of the problems except Problem 1 in S. Fuchino, L. Soukup, Fundament. Math. 154 (1997) 159–176, and some other problems posed by Geschke.     

Strong continuity of the relative rearrangement maps and application to a Galerkin approach for nonlocal problems

We study the strong continuity of the map u   (b_*u, b_*u(| > u(·)|)). Here,

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for σ]0 means Ω[, u_* (respectively, (b|_{u=u*(σ)})*) denotes the decreasing rearrangement of u (respectively b restricted to the set {u = u_*(σ)}) and |E| denotes the Lebesgue measure of a set E included in a domain Ω. The results are useful for solving plasmas physics equations or any nonlocal problems involving the monotone rearrangement, its inverse or its derivatives.     

Subsets of an interval whose product is a power

We form squares from the product of integers in a short interval [n, n + t_n], where we include n in the product. If p is prime, p|n, and (₂^p) > n, we prove that p is the minimum t_n. If no such prime exists, we prove t_n √5n when n> 32. If n = p(2p − 1) and both p and 2p − 1 are primes, then t_n = 3p> 3 √n/2. For n(n + u) a square > n², we conjecture that a and b exist where n < a < b < n + u and nab is a square (except n = 8 and N = 392). Let g₂(n) be minimal such that a square can be formed as the product of distinct integers from [n, g₂(n)] so that no pair of consecutive integers is omitted. We prove that g₂(n) 3n − 3, and list or conjecture the values of g₂(n) for all n. We describe the generalization to kth powers and conjecture the values for large n.     

Large 2P3-free graphs with bounded degree

Let ex^* (D;H) be the maximum number of edges in a connected graph with maximum degree D and no induced subgraph H; this is finite if and only if H is a disjoint union of paths. If the largest component of such an H has order m, then ex^*(D; H) = O(D²ex^*(D; P_m)). Constructively, ex^*(D;qP_m) = Θ(gD²ex^*(D;P_m)) if q>1 and m> 2(Θ(gD²) if m = 2). For H = 2P₃ (and D 8), the maximum number of edges is if D is even and

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if D is odd, achieved by a unique extremal graph.     

On some properties of the series ∑k=0 kx and the Stirling numbers of the second kind

We partially characterize the rational numbers x and integers n 0 for which the sum ∑_k=0^∞ kⁿx^k assumes integers. We prove that if ∑_k=0^∞ kⁿx^k is an integer for x = 1 − a/b with a, b> 0 integers and gcd(a,b) = 1, then a = 1 or 2. Partial results and conjectures are given which indicate for which b and n it is an integer if a = 2. The proof is based on lower bounds on the multiplicities of factors of the Stirling number of the second kind, S(n,k). More specifically, we obtain for all integers k, 2 k n, and a 3, provided a is odd or divisible by 4, where v_a(m) denotes the exponent of the highest power of a which divides m, for m and a> 1 integers.

New identities are also derived for the Stirling numbers, e.g., we show that ∑_k=0²ⁿk! S(2n, k) , for all integers n 1.