A theory for optimal regularization in the finite dimensional case

Consider the matrix problem $\text{Ax = y + ε =}\text{y}\text{?}$ in the case where A is known precisely, the problem is ill conditioned, and ε is a random noise vector. Compute regularized “ridge” estimates,$\text{x}\text{?}{}_{\mathrm{\lambda }}\text{= (A}{}^1\text{A + λI)}{}^{-1}\text{A}{}^1\text{y}\text{?}$,where ¹ denotes matrix transpose. Of great concern is the determination of the value of λ for which x?_λ “best” approximates $\text{x}{}_{\mathrm{<mtext>0</mtext>}}\text{= A}{}^{\mathrm{+}}\text{y}$. Let $\text{Q = 6}\text{x}\text{?}{}_{\mathrm{\lambda }}\text{? x}{}_{\mathrm{<mtext>0</mtext>}}\text{6}{}^2$,and define λ₀ to be the value of λ for which Q is a minimum. We look for λ₀ among solutions of dQ/dλ = 0. Though Q is not computable (since ε is unknown), we can use this approach to study the behavior of λ₀ as a function of y and ε. Theorems involving “noise to signal ratios” determine when λ₀ exists and define the cases λ₀ > 0 and λ₀ = ∞. Estimates for λ₀ and the minimum square error Q₀ = Q(λ₀) are derived.     

Perturbations with several independent parameters

In this paper we discuss the problem of determining a T-periodic solution $\text{x}{}^1\text{(·, λ)}$ of the differential equation x = A(t)x + f(t, x, λ) + b(t), where the perturbation parameter λ is a vector in a parameter-space R^k. The customary approach assumes that λ = λ(?), ??R. One then establishes the existence of an ?₀ > 0 such that the differential equation has a T-periodic solution $\text{x}{}^1\text{(·, λ(?))}$ for all ? satisfying 0 < ? < ?₀. More specifically it is usually assumed that λ(?) has the form λ(?) = ?λ₀ where λ₀ is a fixed vector in R^k. This means that attention is confined in the perturbation procedure to examining the dependence of $\text{x}{}^1\text{(·, λ)}$ on λ as λ varies along a line segment terminating at the origin in the parameter-space R^k. The results established here generalize this previous work by allowing one to study the dependence of $\text{x}{}^1\text{(·, λ)}$ on λ as λ varies through a “conical-horn” whose vertex rests at the origin in R^k. In the process an implicit-function formula is developed which is of some interest in its own right.     

Explicit error terms for asymptotic expansions of Mellin convolutions

Explicit expressions are derived for the error terms associated with the asymptotic expansions of the convolution integral $\text{I(λ) = ∝}{}_0{}^{\mathrm{\infty }}\text{?(t) h(λt) dt}$, where h(t) and $\text{?(t)}$ are algebraically dominated at both 0⁺ and + ∞. Examples included are Fourier, Bessel, generalized Stieltjes, Hilbert and “potential” transforms.     

Diophantine inequalities with mixed powers,II

It is shown that if λ₁, …, λ₅ are non-zero real numbers, not all of the same sign, and at least one of the ratios $\text{λ}{}_{\mathrm{i}}\text{λ}{}_{\mathrm{j}}$ (1 ≤ j ≤ 3) is irrational then the values taken by $\text{λ}{}_1\text{x}{}_1{}^2\text{+ λ}{}_2\text{x}{}_2{}^2\text{+ λ}{}_3\text{x}{}_3{}^2\text{+ λ}{}_4\text{x}{}_4{}^3\text{+ λ}{}_5\text{x}{}_5{}^3$ for integer values of x₁, …, x₅ are everywhere dense on the real line. Similar results are proved for the polynomials $\text{λ}{}_1\text{x}{}_1{}^2\text{+ λ}{}_2\text{x}{}_1{}^2\text{+ λ}{}_3\text{x}{}_3{}^3\text{+ … + λ}{}_6\text{x}{}_6{}^3$ and $\text{λ}{}_1\text{x}{}_1{}^2\text{+ λ}{}_2\text{x}{}_2{}^2\text{+ λ}{}_3\text{x}{}_3{}^3\text{+ λ}{}_4\text{x}{}_4{}^3\text{+ λ}{}_5\text{x}{}_5{}^4\text{+ λ}{}_6\text{x}{}_6{}^4$.     

Diophantine inequalities with mixed powers

It is shown that λ₁, λ₂,…, λ₆, μ are not all of the same sign and at least one ratio $\text{λ}{}_{\mathrm{i}}\text{λ}{}_{\mathrm{j}}$ is irrational then the values taken by $\text{λ}{}_1\text{x}{}_1{}^3\text{+ ? + λ}{}_6\text{x}{}_6{}^3\text{+ μy}{}^3$ for integer values of x₁ ,…, x₆, y are everywhere dense on the real line. A similar result holds for expressions of the form $\text{λ}{}_1\text{x}{}_1{}^3\text{+ ? + λ}{}_4\text{x}{}_4{}^3\text{+ μ}{}_1\text{y}{}_1{}^2\text{+ μ}{}_2\text{y}{}_2{}^3$.     

Sturm—Liouville problems with several parameters

We consider the regular linear Sturm-Liouville problem (second-order linear ordinary differential equation with boundary conditions at two points x = 0 and x = 1, those conditions being separated and homogeneous) with several real parameters λ₁,…,λ_N. Solutions to this problem correspond to eigenvaluesλ = (λ₁,…,λ_N) forming sets $\text{R}$^N determined by the number of zeroes in (0, 1) of solutions. We describe properties of these sets including: boundedness, and when unbounded, asymptotic directions. Using these properties some results are given for the system of N Sturm-Liouville problems which share only the parameters λ. Sharp results are given for the system of two problems sharing two parameters. The eigensurfaces for a single problem are closely related to the cone $\text{K={λ R}{}^{\mathrm{N}}\text{:λ}{}_1\text{a}{}_1\text{(x)+…+λ}{}_{\mathrm{N}}\text{a}{}_{\mathrm{N}}\text{(x)?0 for all x in [0,1]}}$, particularly in questions of boundedness. The cone K and related objects are discussed, and a result is given which relates cones with two oscillation conditions known as “Right-Definiteness” and “Left-Definiteness.”     

Concrete model theory for a class of operators with unitary part

Let m and v_t, 0 ? t ? 2π be measures on T = [0, 2π] with m smooth. Consider the direct integral $\text{H}$ = ⊕L²(v^t) dm(t) and the operator $\text{(L?)(t, λ) = e}{}^{\mathrm{?i\lambda }}\text{?(t, λ) ? 2e}{}^{\mathrm{?i\lambda }}\text{∫}{}_{\mathrm{t}}{}^{\mathrm{2\pi }}\text{∫}{}_{\mathrm{T}}\text{?(s, x) e(s, t) dv}{}_{\mathrm{s}}\text{(x) dm(s)}$ on $\text{H}$, where e(s, t) = exp ∫_s^t ∫_Tdv_λ(θ) dm(λ). Let μ_t be the measure defined by $\text{∫}{}_{\mathrm{T}}\text{?(x) dμ}{}_{\mathrm{t}}\text{(x) = ∫}{}_0{}^{\mathrm{t}}\text{∫}{}_{\mathrm{T}}\text{?(x) dv}{}_{\mathrm{s}}\text{dm(s)}$ for all continuous ?, and let ?_t(z) = exp[?∫ (e^iθ + z)(e^iθ ? z)^?1dμ_t(gq)]. Call {v_t} regular iff for all $\text{t, ¦?}{}_{\mathrm{t}}\text{(e}{}^{\mathrm{i\theta }}\text{)¦ = ¦?}{}_{\mathrm{2\pi }}\text{(e}{}^{\mathrm{i\theta }}\text{)¦}$ for 1 a.e.     

Operator properties of Sobolev imbeddings over unbounded domains

For an open set Ω ? $\text{R}$^N, 1 ? p ? ∞ and λ ∈ $\text{R}$⁺, let $\text{W}\text{?}{}_{\mathrm{p}}{}^{\mathrm{\lambda }}\text{(Ω)}$ denote the Sobolev-Slobodetzkij space obtained by completing $\text{C}{}_0{}^{\mathrm{\infty }}\text{(Ω)}$ in the usual Sobolev-Slobodetzkij norm (cf. A. Pietsch, “r-nukleare Sobol. Einbett. Oper., Ellipt. Dgln. II,” Akademie-Verlag, Berlin, 1971, pp. 203–215). Choose a Banach ideal of operators $\text{U}$, 1 ? p, q ? ∞ and a quasibounded domain Ω ? $\text{R}$^N. Theorem 1 of the note gives sufficient conditions on λ such that the Sobolev-imbedding map $\text{W}\text{?}{}_{\mathrm{p}}{}^{\mathrm{\lambda }}\text{(Ω) λ L}{}_{\mathrm{q}}\text{(Ω)}$ exists and belongs to the given Banach ideal $\text{U}$: Assume the quasibounded domain fulfills condition C_k^l for some l > 0 and 1 ? k ? N. Roughly this means that the distance of any $\text{x ? Ω}$ to the boundary ?Ω tends to zero as $\text{O(¦ x ¦}{}^{\mathrm{?l}}\text{)}$ for $\text{¦ x ¦ → ∞}$, and that the boundary consists of sufficiently smooth ?(N ? k)-dimensional manifolds. Take, furthermore, 1 ? p, q ? ∞, p > k. Then, if μ, ν are real positive numbers with λ = μ + v ∈ $\text{N}$, μ > λ _S($\text{U}$; p,q:N) and v > N/l · λ_D($\text{U}$;p,q), one has that $\text{W}\text{?}{}_{\mathrm{p}}{}^{\mathrm{\lambda }}\text{(Ω) λ L}{}_{\mathrm{q}}\text{(Ω)}$ belongs to the Banach ideal $\text{U}$. Here λ_D($\text{U}$;p,q;N)∈$\text{R}$⁺ and λ_S($\text{U}$;p,q;N)∈$\text{R}$⁺ are the D-limit order and S-limit order of the ideal $\text{U}$, introduced by Pietsch in the above mentioned paper. These limit orders may be computed by estimating the ideal norms of the identity mappings l_pⁿ → l_qⁿ for n → ∞. Theorem 1 in this way generalizes results of R. A. Adams and C. Clark for the ideals of compact resp. Hilbert-Schmidt operators (p = q = 2) as well as results on imbeddings over bounded domains.Similar results over general unbounded domains are indicated for weighted Sobolev spaces.As an application, in Theorem 2 an estimate is given for the rate of growth of the eigenvalues of formally selfadjoint, uniformly strongly elliptic differential operators with Dirichlet boundary conditions in $\text{L}{}_2\text{(Ω)}$, where Ω fulfills condition C₁^l.For an open set Ω in $\text{R}$^N, let $\text{W}\text{?}{}_{\mathrm{p}}{}^{\mathrm{\lambda }}\text{(Ω)}$ denote the Sobolev-Slobodetzkij space obtained by completing $\text{C}{}_0{}^{\mathrm{\infty }}\text{(Ω)}$ in the usual Sobolev-Slobodetzkij norm, see below. Taking a fixed Banach ideal of operators and 1 ? p, q ? ∞, we consider quasibounded domains Ω in $\text{R}$^N and give sufficient conditions on λ such that the Sobolev imbedding operator $\text{W}\text{?}{}_{\mathrm{p}}{}^{\mathrm{\lambda }}\text{(Ω) λ L}{}_{\mathrm{q}}\text{(Ω)}$ exists and belongs to the Banach ideal. This generalizes results of C. Clark and R. A. Adams for compact, respectively, Hilbert-Schmidt operators (p = q = 2) to general Banach ideals of operators, as well as results on imbeddings over bounded domains. Similar results over general unbounded domains may be proved for weighted Sobolev spaces. As an application, we give an estimate for the rate of growth of the eigenvalues of formally selfadjoint, uniformly strongly elliptic differential operators with Dirichlet boundary conditions in $\text{L}{}_2\text{(Ω)}$, where Ω is a quasibounded open set in $\text{R}$^N.     

A trial phase function method for a class of λ-ω reaction-diffusion systems: Reduction to a Schrödinger type problem in an eigen sub-domain

The perturbed central force problem $\text{γ}{}_{\mathrm{xx}}\text{? ?}{}^2\text{γ}\text{a}\text{?}{}^2\text{+ γλ(γ) = 0, γ}\text{a}\text{?}{}_{\mathrm{x}}\text{+ 2}\text{a}\text{?}\text{γ}{}_{\mathrm{x}}\text{+ γ[}\text{}\text{?}\text{gw(γ}{}^1\text{) ?}\text{}\text{?}\text{gw(γ)] = 0}$ arising from the λ-ω system

$\text{α}\text{αt}\text{C}{}_1\text{C}{}_2\text{=}\text{λ}\text{?ω}\text{ω}\text{λ}\text{C}{}_1\text{C}{}_2\text{+}\text{α}\text{αt}\text{C}{}_1\text{C}{}_2$

where $\text{c}{}_1\text{=}\text{γ}\text{(x)cosθ(x,t), c}{}_2\text{=}\text{γ}\text{(x)sinθ(x,t), θ = σt ? ?∝}{}^{\mathrm{x}}\text{a}\text{?}\text{(x′)dx′}\text{and}\text{ω = ?}\text{}\text{?}\text{gw(}\text{γ}\text{)}$ is considered for the class $\text{λ}\text{=}\text{}\text{?}\text{gw}\text{= 1 ? g(}\text{γ}\text{)}$. The resulting linear equation in $\text{γ(x), γ}{}_{\mathrm{xx}}\text{+ 2}\text{a}\text{?}\text{γ}{}_{\mathrm{x}}\text{+}\text{γ[α}{}^2\text{+}\text{a}\text{?}{}_{\mathrm{x}}\text{?}\text{?}{}^2\text{a}\text{?}{}^2\text{] = 0}$ is solved with the aid of a class of trial phase functions $\text{a}\text{?}{}_{\mathrm{T}}\text{(x)}$ generated by the unperturbed central force problem for the case g(γ) = γ². An application of the Liouville-Green approximation procedure reduces the system to a Schrödinger type boundary value problem in an eigen sub-domain. The analytical estimates for α are in reasonable agreement with the results of numerical integration of the nonlinear system. The eigenfunctions γ(x) display expected oscillatory behaviour inside an eigen sub-domain. The higher modes and the span of the most extended centre structure are estimated and interpreted in the context of WKBJ connection formula.     

Concrete model theory for a class of operators

The operator $\text{L?(t, λ) = e}{}^{\mathrm{?i\lambda }}\text{(t, λ) ? 2e}{}^{\mathrm{?i\lambda }}\text{∝}{}_{\mathrm{t}}{}^{\mathrm{2\pi }}\text{∝}{}_{\mathrm{T}}\text{?(s, x) e(s, t) dv}{}_{\mathrm{s}}\text{(x) dm(s)}$ acting on H=∝₀^2π⊕L²(v_t), where m and v_t, 0 ? t ? 2π are measures on [0, 2π] with m smooth and e(s, t) = exp[?∝_t^s∝_Tdv_λ(θ) dm(λ)], satisfies $\text{rank}\text{(I ? LL}{}^1\text{) =}\text{rank}\text{(I ? L}{}^1\text{L) = 1}$. It is, therefore, unitarily equivalent to a scalar Sz.-Nagy-Foia? canonical model. The purpose of this paper is to determine the model explicitly and to give a formula for the unitary equivalence.     

On the stability and asymptotic behavior of solutions of integrodifferential equations in Banach spaces

In this paper asymptotic behavior of solutions of the integrodifferential system is related to that of the differential system $\text{y′(t) = A(t) y(t) + ?(t, y(t))}$. Necessary and sufficient conditions for the uniform asymptotic stability of the trivial solution of the first equation are given.     

On the existence of a periodic solution of nth-order nonlinear differential equations

This note is a generalization of one of a paper by Mehri and Hamedani. Under suitable conditions of ?, the existence of periodic solutions of the nthorder differential equation $\text{x}{}^{\mathrm{(n)}}\text{+ ?(t, x, x′,…, x}{}^{\mathrm{(n\; ?\; 1)}}\text{) = 0}$ is established.     

Two-parameter Turan inequalities for ultraspherical and Laguerre polynomials

It is known that the classical orthogonal polynomials satisfy inequalities of the form U_n²(x) ? U_{n + 1}(x) U_{n ? 1}(x) > 0 when x lies in the spectral interval. These are called Turan inequalities. In this paper we will prove a generalized Turan inequality for ultraspherical and Laguerre polynomials. Specifically if P_n^λ(x) and L_n^α(x) are the ultraspherical and Laguerre polynomials and $\text{F}{}_{\mathrm{n}}{}^{\mathrm{\lambda }}\text{(x) =}\text{P}{}_{\mathrm{n}}{}^{\mathrm{\lambda }}\text{(x)}\text{P}{}_{\mathrm{n}}{}^{\mathrm{\lambda }}\text{(1)}\text{, G}{}_{\mathrm{n}}{}^{\mathrm{\alpha }}\text{(x) =}\text{L}{}_{\mathrm{n}}{}^{\mathrm{\alpha }}\text{(x)}\text{L}{}_{\mathrm{n}}{}^{\mathrm{\alpha }}\text{(0)}\text{,}\text{then}\text{F}{}_{\mathrm{n}}{}^{\mathrm{\alpha }}\text{(x) F}{}_{\mathrm{n}}{}^{\mathrm{\beta }}\text{(x) ? F}{}_{\mathrm{n\; +\; 1}}{}^{\mathrm{\alpha }}\text{(x) F}{}_{\mathrm{n\; ?\; 1}}{}^{\mathrm{\beta }}\text{(x) > 0, ? 1 < x < 1, ?}\text{1}\text{2}\text{< α ? β ? α + 1}\text{and}\text{G}{}_{\mathrm{n}}{}^{\mathrm{\alpha }}\text{(x) G}{}_{\mathrm{n}}{}^{\mathrm{\beta }}\text{(x) ? G}{}_{\mathrm{n\; +\; 1}}{}^{\mathrm{\alpha }}\text{(x) G}{}_{\mathrm{n\; ?\; 1}}{}^{\mathrm{\beta }}\text{(x) > 0, x > 0, 0 < α ? β ? α + 1}$. We also prove the inequality $\text{(n + 1) F}{}_{\mathrm{n}}{}^{\mathrm{\alpha }}\text{(x) F}{}_{\mathrm{n}}{}^{\mathrm{\beta }}\text{(x) ? nF}{}_{\mathrm{n\; +\; 1}}{}^{\mathrm{\alpha }}\text{(x) F}{}_{\mathrm{n\; ?\; 1}}{}^{\mathrm{\beta }}\text{(x) > A}{}_{\mathrm{n}}\text{[F}{}_{\mathrm{n}}{}^{\mathrm{\alpha }}\text{(x)]}{}^2\text{, ?1 < x < 1, ?}\text{1}\text{2}\text{< α ? β < α + 1,}\text{where}\text{A}{}_{\mathrm{n}}$ is a positive constant depending on α and β.     

A maximization problem and its application to canonical correlation

Let Σ be an n × n positive definite matrix with eigenvalues λ₁ ≥ λ₂ ≥ … ≥ λ_n > 0 and let M = {x, y | x?Rⁿ, y?Rⁿ, x ≠ 0, y ≠ 0, x′y = 0}. Then for x, y in M, we have that $\text{x′Σy}\text{(x′Σxy′Σy)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{≤}\text{(λ}{}_1\text{? λ}{}_{\mathrm{n}}\text{)}\text{(λ}{}_1\text{+ λ}{}_{\mathrm{n}}\text{)}$ and the inequality is sharp. If

$\text{∑=}\text{∑}{}_{11}\text{∑}{}_{12}\text{∑}{}_{21}\text{∑}{}_{22}$

is a partitioning of Σ, let θ₁ be the largest canonical correlation coefficient. The above result yields $\text{θ}{}_1\text{≤}\text{(λ}{}_1\text{? λ}{}_{\mathrm{n}}\text{)}\text{(λ}{}_1\text{+ λ}{}_{\mathrm{n}}\text{)}$.     

Szegö limit theorems in quantum mechanics

We prove a Szegö-type theorem for some Schrödinger operators of the form $\text{H =}\text{?1}\text{2Δ}\text{+ V}$ with V smooth, positive and growing like $\text{V}{}_0\text{¦x¦}{}^{\mathrm{k}}\text{, k > 0}$. Namely, let π_λ be the orthogonal projection of L² onto the space of the eigenfunctions of H with eigenvalue ?λ; let A be a 0th order self-adjoint pseudo-differential operator relative to Beals-Fefferman weights $\text{?(x, ξ) = 1, Φ(x, ξ) = (1 + ¦ξ¦}{}^2\text{+ V(x))}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$ and with total symbol a(x, ξ); and let f∈C($\text{R}$). Then

$\text{lim}\text{λ→∞}\text{1}\text{rank}\text{π}{}_{\mathrm{\lambda }}\text{tr}\text{f(π}{}_{\mathrm{\lambda }}\text{Aπ}{}_{\mathrm{\lambda }}\text{)=}\text{lim}\text{λ→∞}\text{1}\text{vol}\text{(H(x, ξ)?λ)}\text{∫}\text{H?λ}\text{f(a(x, ξ))dxdξ}$

(assuming one limit exists).     

Asymptotic properties of general symmetric hyperbolic systems

Results on partition of energy and on energy decay are derived for solutions of the Cauchy problem $\text{?u}\text{?t}\text{+ ∑}{}_{\mathrm{j\; =\; 1}}{}^{\mathrm{n}}\text{A}{}_{\mathrm{j}}\text{?u}\text{?x}{}_{\mathrm{j}}\text{= 0, u(0, x) = ?(x)}$. Here the A_j's are constant, k × k Hermitian matrices, x = (x₁,…, x_n), t represents time, and u = u(t, x) is a k-vector. It is shown that the energy of Mu approaches a limit $\text{E}{}_{\mathrm{M}}\text{(?)}\text{as}\text{¦ t ¦ → ∞}$, where M is an arbitrary matrix; that there exists a sufficiently large subspace of data ?, which is invariant under the solution group U₀(t) and such that $\text{U}{}_0\text{(t)? = 0}\text{for}\text{¦ x ¦ ? a ¦ t ¦ ? R, a}\text{and}\text{R}$ depending on ? and that the local energy of nonstatic solutions decays as $\text{¦ t ¦ → ∞}$. More refined results on energy decay are also given and the existence of wave operators is established, considering a perturbed equation $\text{E(x)}\text{?u}\text{?t}\text{+ ∑}{}_{\mathrm{j\; =\; 1}}{}^{\mathrm{n}}\text{A}{}_{\mathrm{j}}\text{?u}\text{?x}{}_{\mathrm{j}}\text{= 0,}\text{where}\text{¦ E(x) ? I ¦ = O(¦ x ¦}{}^{\mathrm{?1\; ?\; ?}}\text{)}$ at infinity.     

A comparison of different compactness notions in fuzzy topological spaces

Starting from a defining differential equation $\text{(}\text{?}\text{?t}\text{) W(λ, t, u) = (}\text{λ(u ? t)}\text{p(t)}\text{) W(λ, t, u)}$ of the kernel of an exponential operator $\text{S}{}_{\mathrm{\lambda }}\text{(?, t) = ∫}{}_{\mathrm{?\infty }}{}^{\mathrm{\infty }}\text{W(λ, t, u)?(u) du}$ with normalization ∫_?∞^∞W(λ, t, u) du = 1, we determine S_λ for various p(t) including; for example, p(t) a quadratic polynomial, all the known exponential operators are recovered and some new ones are constructed. It is shown that all the exponential operators are approximation operators. Further approximation properties of these operators are discussed. For example, functions satisfying $\text{∥ S}{}_{\mathrm{\lambda }}\text{(?, t) ? ?(t)∥ = O(λ}{}^{\mathrm{?\alpha }}\text{)}$ are characterized. Several results of C. P. May are also improved.     

An abstract nonlinear Volterra integrodifferential equation

We study the nonlinear Volterra equation $\text{u′(t) + Bu(t) + ∫}{}_0{}^{\mathrm{t}}\text{a(t ? s) Au(s) ds ? F(t) (0 < t < ∞) (′ =}\text{d}\text{dt}\text{), u(0) = u}{}_0$, (¹) as well as the corresponding problem with infinite delay $\text{u′(t) + Bu(t) + ∫}{}_{\mathrm{?\infty }}{}^{\mathrm{t}}\text{a(t ? s) Au(s) ds ? ?(t) (0 < t < ∞), u(t) = h(t) (?∞ < t ? 0)}$. (⁷) Under various assumptions on the nonlinear operators A, B and on the given functions a, F, f, h existence theorems are obtained for (¹) and (⁷, followed by results concerning boundedness and asymptotic behaviour of solutions on (0 ? < ∞); two applications of the theory to problems of nonlinear heat flow with “infinite memory” are also discussed.     

Some asymptotic boundary behavior of solutions of non-linear parabolic initial-boundary value problems

For parabolic initial boundary value problems various results such as $\text{lim}{}_{\mathrm{t\; \downarrow \; 0}}\text{{}\text{(}\text{?u}\text{t6x}\text{)(0, t)}\text{(}\text{?u}{}_{\mathrm{\alpha }}\text{?x}\text{)(0, t)}}\text{= 1}$, where u satisfies $\text{?u}\text{?t}\text{= a(u)(}\text{?}{}^2\text{u}\text{?x}{}^2\text{), 0 < x < 1, 0 < t ? T, u(x, 0) = 0, u(0, t) = |}{}_1\text{(t), 0 < t ? T, u(1, t) = |}{}_2\text{(t), 0 < t ? T, u}{}_{\mathrm{\alpha }}\text{satisfies}\text{(}\text{?u}{}_{\mathrm{\alpha }}\text{?t}\text{) = α(}\text{?}{}^2\text{u}{}_{\mathrm{\alpha }}\text{?x}{}^2\text{), 0 < x < 1, 0 < t ? T, u}{}_{\mathrm{\alpha }}\text{(x, 0) = 0, u}{}_{\mathrm{\alpha }}\text{(0, t) = |}{}_1\text{(t), 0 < t ? T, u}{}_{\mathrm{\alpha }}\text{(1, t) = |}{}_2\text{(t), 0 < t ? T,}\text{and}\text{α = a(0)}$, are demonstrated via the maximum principle and potential theoretic estimates.     

Integral inequalities for generalized concave or convex functions

Let ψ be convex with respect to ?, B a convex body in Rⁿ and f a positive concave function on B. A well-known result by Berwald states that $\text{1}\text{¦B¦}\text{∝}{}_{\mathrm{B}}\text{ψ(f(x)) dx ? n ∝}{}_0{}^1\text{ψ(ξt)(1 ? t)}{}^{\mathrm{n\; ?\; 1)}}\text{dt}$ (1) if ξ is chosen such that $\text{1}\text{¦B¦}\text{∝}{}_{\mathrm{B}}\text{?(f(x)) dx = n ∝}{}_0{}^1\text{?(ξt)(1 ? t)}{}^{\mathrm{n\; ?\; 1)}}\text{dt}$.The main purpose in this paper is to characterize those functions f : B → R₊ such that (1) holds.