Geometric interpretations of two branching theorems of D.E. Littlewood

D.E. Littlewood proved two branching theorems for decomposing the restriction of an irreducible finite-dimensional representation of a unitary group to a symmetric subgroup. One is for restriction of a representation of U(n) to the rotation group SO(n) when the given representation τ_λ of U(n) has nonnegative highest weight λ of depth n/2. It says that the multiplicity in τ_λ|_SO(n) of an irreducible representation of SO(n) of highest weight ν is the sum over μ of the multiplicities of τ_λ in the U(n) tensor product τ_μτ_ν, the allowable μ's being all even nonnegative highest weights for U(n). Littlewood's proof is character-theoretic. The present paper gives a geometric interpretation of this theorem involving the tensor products τ_μτ_ν explicitly. The geometric interpretation has an application to the construction of small infinite-dimensional unitary representations of indefinite orthogonal groups and, for each of these representations, to the determination of its restriction to a maximal compact subgroup. The other Littlewood branching theorem is for restriction from U(2r) to the rank-r quaternion unitary group Sp(r). It concerns nonnegative highest weights for U(2r) of depth r, and its statement is of the same general kind. The present paper finds an analogous geometric interpretation for this theorem also.     

Theory of the hypersonic viscous shock layer at high Reynolds numbers and intensive injection of foreign gases : PMM vol. 38, n 6, 1974, pp. 1015–1024

The hypersonic flow around smooth blunted bodies in the presence of intensive injection from the surface of these is considered. Using the method of external and internal expansions the asymptotics of the Navier-Stokes equations is constructed for high Reynolds numbers determined by parameters of the oncoming stream and of the injected gas. The flow in the shock layer falls into three characteristic regions. In regions adjacent to the body surface and the shock wave the effects associated with molecular transport are insignificant, while in the intermediate region they predominate. In the derivation of solution in the first two regions the surface of contact discontinuity is substituted for the region of molecular transport (external problem). An analytic solution of the external problem is obtained for small values of parameters ₁ = ρ_∞/ρs^* and δ = ρ_ω^*1/2ν_ω^*/ρ_∞1/2ν_∞, in the form of corresponding series expansions in these parameters. Asymptotic formulas are presented for velocity profiles, temperatures, and constituent concentration across the shock layer and, also, the shape of the contact discontinuity and of shock wave separation. The derived solution is compared with numerical solutions obtained by other authors. The flow in the region of molecular transport is defined by equations of the boundary layer with asymptotic conditions at plus and minus infinity, determined by the external solution (internal problem). A numerical solution of the internal problem is obtained taking into consideration multicomponent diffusion and heat exchange. The problem of multicomponent gas flow in the shock layer close to the stagnation line was previously considered in [1] with the use of simplified Navier-6tokes equations.The supersonic flow of a homogeneous inviscid and non-heat-conducting gas around blunted bodies in the presence of subsonic injection was considered in [2–7] using Euler's equations. An analytic solution, based on the classic solution obtained by Hill for a spherical vortex, was derived in [2] for a sphere on the assumption of constant but different densities in the layers between the shock wave and the contact discontinuity and between the latter and the body. Certain results of a numerical solution of the problem of intensive injection at the surface of axisymmetric bodies of various forms, obtained by Godunov's method [3], are presented. Telenin's method was used in [4] for numerical investigation of flow around a sphere; the problem was solved in two formulations: in the first, flow parameters were determined for the whole of the shock layer, while in the second this was done for the sutface of contact discontinuity, which was not known prior to the solution of the problem, with the pressure specified by Newton's formula and flow parameters determined only in the layer of injected gases. The flow with injection over blunted cones was numerically investigated in [5] by the approximate method proposed by Maslen. The flow in the shock layer in the neighborhood of the stagnation line was considered in [6, 8], and intensive injection was investigated by methods of the boundary layer theory in [8–12].     

Trace cocharacters and the Kronecker products of Schur functions

It follows from the theory of trace identities developed by Procesi and Razmyslov that the trace cocharacters arising from the trace identities of the algebra M_r(F) of r×r matrices over a field F of characteristic zero are given by TC_r,n=∑_λΛr(n)χ^λχ^λ where χ^λχ^λ denotes the Kronecker product of the irreducible characters of the symmetric group associated with the partition λ with itself and Λ_r(n) denotes the set of partitions of n with r or fewer parts, i.e. the set of partitions λ=(λ₁λ_k) with kr. We study the behavior of the sequence of trace cocharacters TC_r,n. In particular, we study the behavior of the coefficient of χ^(ν,n−m) in TC_r,n as a function of n where ν=(ν₁ν_k) is some fixed partition of m and n−mν_k. Our main result shows that such coefficients always grow as a polynomial in n of degree r−1.     

On the behavior of solutions of equations for double waves in the neighborhood of the quiescent region : PMM vol. 39, n 6, 1975, pp. 1043–1050

The structure of solutions of gasdynamic equations is investigated in the case of unsteady double waves in the neighborhood of the quiescent region. A general concept of double waves is presented in the form of special series with logarithmic terms. Results of numerical computations are given.The problem of determining the flow of plane and three-dimensional waves separated from the quiescent region by a weak discontinuity was considered in [1–3], where approximate solutions were derived for that neighborhood, and the formulation of boundary value problems required for solving the equation for the analog of the velocity potential in the hodograph plane was investigated.The more general problem (without the assumption of the degeneration of motion) of arbitrary potential flows of polytropic gas adjacent to the quiescent region and separated by a weak discontinuity was considerd in [4–8]. Solution of that problem was obtained in the form of special series in powers of the mo dulus of the velocity vector r in the space of the time hodograph. The value r = 0 corresponds to the surface of weak discontinuity that separates the perturbed motion region from that at rest. Some applications of derived solutions to problems such as the motion of a convex piston and the propagation of weak shock waves were also investigated in those papers. Convergence in the small of obtained series was proved in [9]. However the attempts of constructing series in powers of r, which were used in [4–8] for the presentation of equations of double waves in the neighborhood of the quiescent region, proved to be unsuccessful.Although parts of expansions in series in powers of r (accurate to within 0 (r²)), were constructed in [1–3], it was found that the coefficient at r⁸ in equations for double waves cannot be determined owing to the insolvability of its equation. This is related to the fact that the surface r = 0in the case of equations for double waves is simultaneously a line of parabolic degeneration and a characteristic.The object of the present note is the formulation of solutions of equations for plane unsteady double waves in the neighborhood of the quiescent region. Parts of the derived series, which generally are nonanalytic functions of r, can be used for defining flows at small r in particular those downstream of two-dimensional normal detonation waves [10] or in problems of angular pistons [11]. The method used for the derivation of series can be also applied in investigations of threedimensional self-similar flows with variables x₁/x₃ and x₂/x₃ (steady flows) or x₁/t, x₂/t and x₃/t (unsteady flows). However it was not possible to obtain in such cases regular series in powers of r.     

Strong approximation of Fourier series and embedding theorems

Summary In this paper, embedding results are considered which arise in the strong approximation by Fourier series. We prove several theorems on the interrelation between the classes W^r H_β^ω and H (λ,p,r,ω), the latter being defined by L. Leindler. Previous related results in Leindler’s book [2] and paper [5] are particular cases of our results.     

On the Maximal Multiplicity of Parts in a Random Integer Partition

We study the asymptotic behavior of the maximal multiplicity μ_n = μ_n(λ) of the parts in a partition λ of the positive integer n, assuming that λ is chosen uniformly at random from the set of all such partitions. We prove that πμ_n/(6n)^1/2 converges weakly to max _jX_j/j as n→∞, where X₁, X₂, … are independent and exponentially distributed random variables with common mean equal to 1.2000 Mathematics Subject Classification: Primary—05A17; Secondary—11P82, 60C05, 60F05     

Existence of three solutions for a class of elliptic eigenvalue problems

In this paper, we consider a problem of the type −Δu = λ(f(u) + μg(u)) in Ω, u¦∂Ω = 0, where Ω Rⁿ is an open-bounded set, f, g are continuous real functions on R, and λ, μ ε R. As an application of a new approach to nonlinear eigenvalues problems, we prove that, under suitable hypotheses, if ¦μ¦ is small enough, then there is some λ > 0 such that the above problem has at least three distinct weak solutions in W₀^1,2(Ω).     

Orthogonal exponentials, translations, and Bohr completions

We are concerned with an harmonic analysis in Hilbert spaces L²(μ), where μ is a probability measure on . The unifying question is the presence of families of orthogonal (complex) exponentials e_λ(x)=exp(2πiλx) in L²(μ). This question in turn is connected to the existence of a natural embedding of L²(μ) into an L²-space of Bohr almost periodic functions on . In particular we explore when L²(μ) contains an orthogonal basis of e_λ functions, for λ in a suitable discrete subset in ; i.e, when the measure μ is spectral. We give a new characterization of finite spectral sets in terms of the existence of a group of local translation. We also consider measures μ that arise as fixed points (in the sense of Hutchinson) of iterated function systems (IFSs), and we specialize to the case when the function system in the IFS consists of affine and contractive mappings in . We show in this case that if μ is then assumed spectral then its partitions induced by the IFS at hand have zero overlap measured in μ. This solves part of the Łaba–Wang conjecture. As an application of the new non-overlap result, we solve the spectral-pair problem for Bernoulli convolutions advancing in this way a theorem of Ka-Sing Lau. In addition we present a new perspective on spectral measures and orthogonal Fourier exponentials via the Bohr compactification.     

Uniform Estimates for a Class of Evolution Equations

L^∞ estimates are derived for the oscillatory integral ∫⁺₀^∞e^{−i(xλ + (1/m) tλm)}a(λ) dλ, where 2 ≤ m and (x, t) × ₊. The amplitude a(λ) can be oscillatory, e.g., a(λ) = e^{it (λ)} with (λ) a polynomial of degree ≤ m − 1, or it can be of polynomial type, e.g., a(λ) = (1 + λ)^k with 0 ≤ k ≤ (m − 2). The estimates are applied to the study of solutions of certain linear pseudodifferential equations, of the generalized Schrödinger or Airy type, and of associated semilinear equations.     

Continuity in G

For a discrete group G, we consider βG, the Stone– ech compactification of G, as a right topological semigroup, and G^*=βGG as a subsemigroup of βG. We study the mappings λ_p^* :G^*→G^*and μ^* :G^*→G^*, the restrictions to G^* of the mappings λ_p :βG→βG and μ :βG→βG, defined by the rules λ_p(q)=pq, μ(q)=qq. Under some assumptions, we prove that the continuity of λ_p^* or μ^* at some point of G^* implies the existence of a P-point in ω^*.     

A one-parameter family of unimodal, concave, polynomial maps of the interval exhibiting multiplicities

This paper shows there exists a polynomial map, p, of the interval [0, 1] onto itself that is concave, symmetric about the point and such that, when parameterized {μp}, 0 ≤ μ ≤ 1, there exist three distinct values of the parameter μ₀ < μ₁ < μ₂ such that RLR³C = K(μ₀p) ≠ K(μ₁p) ≠ K(μ₂p) = RLR³C. There is also given an explicit construction of a C¹ family with the same properties.     

Summability of orthogonal expansions of several variables

Summability of spherical h-harmonic expansions with respect to the weight function ∏_j=1^d |x_j|^2κ_j (κ_j0) on the unit sphere S^d−1 is studied. The main result characterizes the critical index of summability of the Cesàro (C,δ) means of the h-harmonic expansion; it is proved that the (C,δ) means of any continuous function converge uniformly in the norm of C(S^d−1) if and only if δ>(d−2)/2+∑_j=1^d κ_j−min_1jd κ_j. Moreover, it is shown that for each point not on the great circles defined by the intersection of the coordinate planes and S^d−1, the (C,δ) means of the h-harmonic expansion of a continuous function f converges pointwisely to f if δ>(d−2)/2. Similar results are established for the orthogonal expansions with respect to the weight functions ∏_j=1^d |x_j|^2κ_j(1−|x|²)^μ−1/2 on the unit ball B^d and ∏_j=1^d x_j^κ_j−1/2(1−|x|₁)^μ−1/2 on the simplex T^d. As a related result, the Cesàro summability of the generalized Gegenbauer expansions associated to the weight function |t|^2μ(1−t²)^λ−1/2 on [−1,1] is studied, which is of interest in itself.     

On the existence of a field of stress rates in a hardening elastic-plastic medium : PMM vol. 38, n 6, 1974, pp. 1114–1121

The boundary value problem for the stress rates and rates of change fields in the quasi-static motion of a volume V of an elastic-plastic medium [1] consists of finding the pairs σ_ij^., _ij^. related by the governing equations of an appropriate model; here the σ_ij^. should be statically admissible i.e. should satisfy the equations and boundary conditions σ_ij=−X/._i; /.σ_ijn_j|S_p=p_i and _ij should be kinematically admissible i.e. 2/._ij = v_ij + v_ji, where v_i|S_u = u_io Here S_p and S_u are nonintersecting parts of the boundary of the volume V, X_i, p_i, u_i/.o are specified functions. The question of the existence of a solution of this problem reduces to the question of the functional reaching the lower bound in a set of kinematically admissible /._ij^o and statically admissible σ_ij/.^/*. However, its lower bound may not be reached if in the minimization we limit ourselves only to smooth fields. It is proposed to augment the set of admissible fields σ_ij/.^/*,_ij/.^o by closing them in the norm L₂ (for v_i^o this corresponds to closure in the norm II¹). Some properties of the functional I (σ_ij^*,_ij/.) are considered in the augmented set of admissible fields. It is shown that the equivalence of the two problems is conserved, where I (σ_ij^*,_ij⁰ can be minimized in σ_ij^/*,_ij^o or in σ_ij^/*,_ij/.^o, The lower bound is reached in each of three cases, at a single point. From the fact that u_i^o belongs to the Sobolev space W₂⁽¹⁾, there results the absence of surfaces of velocity discontinuity. Variational principles have been used in plasticity theory to construct models [2] and to investigate the existence and properties of solutions [1, 3].     

Boundary effects on convergence rates for Tikhonov regularization

We consider the Tikhonov regularizer f_λ of a smooth function f ε H^2m[0, 1], defined as the solution (see [1]) to We prove that if f^(j)(0) = f^(j)(1) = 0, J = m, …, k < 2m − 1, then ¦f − f_λ¦_j² Rλ^{(2k − 2j + 3)/2m}, J = 0, …, m. A detailed analysis is given of the effect of the boundary on convergence rates.     

The approximate identity kernels of product type for the Walsh system

At present there are only a few approximate identity kernels for the Walsh system, for example, the p^N-truncated Dirichlet kernel D_{p^N − 1}(t) = ∑_{j = 0}^{pN − 1} w_j(t) [6]; the Abel-Poisson kernel λ_γ(t) = ∑_{k = 0}^∞ γ^kw_k(t) [3], and so on. In [6], Zheng has introduced a new kind of approximate identity kernels for the Walsh system—the kernels of product type. In the present paper we discuss the approximation properties of such product type kernels. Estimates of their moments as well as a direct approximation theorem are obtained. Then, to establish an inverse approximation theorem, we need the p-adic derivative of product type kernels and we estimate this derivative in L¹-norm.     

Maximum likelihood estimation of a change-point in the distribution of independent random variables: General multiparameter case

In a sequence ofn independent random variables the pdf changes fromf(x, 0) tof(x, 0 + δv_n⁻¹) after the firstnλ variables. The problem is to estimateλ (0, 1 ), where 0 and δ are unknownd-dim parameters andv_n → ∞ slower thann^1/2. Let_n denote the maximum likelihood estimator (mle) ofλ. Analyzing the local behavior of the likelihood function near the true parameter values it is shown under regularity conditions that ifn_n²(− λ) is bounded in probability asn → ∞, then it converges in law to the timeT_(δjδ)^1/2 at which a two-sided Brownian motion (B.M.) with drift1/2(δ′Jδ)^1/2ton(−∞, ∞) attains its a.s. unique minimum, whereJ denotes the Fisher-information matrix. This generalizes the result for small change in mean of univariate normal random variables obtained by Bhattacharya and Brockwell (1976,Z. Warsch. Verw. Gebiete37, 51–75) who also derived the distribution ofT_μ forμ > 0. For the general case an alternative estimator is constructed by a three-step procedure which is shown to have the above asymptotic distribution. In the important case of multiparameter exponential families, the construction of this estimator is considerably simplified.     

Multi-Resolution Analysis of Multiplicity d: Applications to Dyadic Interpolation

This paper studies the Multi-Resolution Analyses of multiplicity d (d *), that is, the families (V_n)_n of closed subspaces in ²( ) such that V_n V_{n + 1}, V_{n + 1} = DV_n, where Dƒ(x) = ƒ(2x), and such that there exists a Riesz basis for V₀ of the form {φ_i(· − k), i = 1, . . . , d,k }, with φ₁, . . . , φ_d V₀. Using the Fourier transform, we prove that (λ) = ^t[ ₁(λ), . . . , _d(λ)] = H(λ/2) (λ/2), where H is in the set _d of continuous 1-periodic functions taking values in (d, ). If d = 1, the definition corresponds to the standard Multi-Resolution Analyses, and one can characterize the regular 1-periodic complex-valued functions H (called, then, scaling filters) which yield a Multi-Resolution Analysis. In this paper, we generalize this study to d ≥ 2 by giving conditions on H _d so that there exists = ^t[ ₁, . . . , _d] in ²( , ^d) solution of (λ) = H(λ/2) (λ/2), and so that the integer translates of φ₁, . . . , φ_d form a Riesz family. Then, the latter span the space V₀ of a Multi-Resolution Analysis of multiplicity d. We show that the conditions on H focus on the zeros of det H(·) and on simple spectral hypotheses for the operator P_H defined on _d by P_HF(λ) = H(λ/2)F(λ/2)H(λ/2)* + H(λ/2 + 1/2)F(λ/2 + 1/2)H(λ/2 + 1/2)*. Finally, we explore connections with the order r dyadic interpolation schemes, where r *.     

Borel ideals vs. Borel sets of countable relations and trees

For every μ < ω₁, let I_μ be the ideal of all sets S ω^μ whose order type is <ω^μ. If μ = 1, then I₁ is simply the ideal of all finite subsets of ω, which is known to be Σ⁰₂-complete. We show that for every μ < ω₁, I_μ is Σ⁰_2μ-complete. As corollaries to this theorem, we prove that the set WO_ωμ of well orderings Rω × ω of order type <ω^μ is Σ⁰_2μ-complete, the set LP_μ of linear orderings R ω × ω that have a μ-limit point is Σ⁰_2μ+1-complete. Similarly, we determine the exact complexity of the set LT_μ of trees T ^<ωω of Luzin height <μ, the set WR_μ of well-founded partial orderings of height <μ, the set LR_μ of partial orderings of Luzin height <μ, the set WF_μ of well-founded trees T ^<ωω of height <μ(the latter is an old theorem of Luzin). The proofs use the notions of Wadge reducibility and Wadge games. We also present a short proof to a theorem of Luzin and Garland about the relation between the height of ‘the shortest tree’ representing a Borel set and the complexity of the set.     

The Bernstein Constant and Polynomial Interpolation at the Chebyshev Nodes

In this paper, we establish new asymptotic relations for the errors of approximation in L_p[−1,1], 0<p∞, of x^λ, λ>0, by the Lagrange interpolation polynomials at the Chebyshev nodes of the first and second kind. As a corollary, we show that the Bernstein constant

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is finite for λ>0 and .     

Second-Order Properties of a Two-Stage Fixed-Size Confidence Region for the Mean Vector of a Multivariate Normal Distribution

We consider the classical fixed-size confidence region estimation problem for the mean vectorμin theN_p(μ, Σ) population where Σ is unknown but positive definite. We writeλ₁for the largest characteristic root of Σ and assume thatλ₁is simple. Moreover, we suppose that, in many practical applications, we will often have available a numberλ*(>0) and that we can assumeλ₁>λ*. Given this addi- tional, and yet very minimal, knowledge regardingλ₁, the two-stage procedure of Chatterjee (Calcutta Statist. Assoc. Bull.8(1959a), 121–148;9(1959b), 20–28;11(1962), 144–159) is revised appropriately. The highlight in this paper involves the verification ofsecond-order propertiesassociated with such revised two-stage estimation techniques, along with the maintenance of the nominal confidence coefficient.