Diffraction of light by supersonic waves: The solution of the raman-nath equations—I

In the problem of the diffraction of light by a supersonic wave, at normal incidence of the light, the solution of the system of difference-differential equations of Raman and Nath, for the amplitudes of the diffracted light beams, is reduced to the integration of a partial differential equation. The coefficients of the Laurent expansion of the solution of the latter equation yield the expressions for the amplitudes of the diffracted light waves. The partial differential equation has been integrated for two approximations. (1) Forρ=0, the well-known results of Raman and Nath’s preliminary theory are re-established. (2) Forρ?1 a power series inρ, the terms of which are calculated as far as the third one, leads to the solution of Mertens and Berry obtained by a perturbation method.     

Free quadratic bialgebra

In this paper, we obtain the following main theorem for a free quadratic bialgebraJ:

1. Forp≠0,J is a pointed cosemisimple coalgebra. Forp=0,J is a hyperalgebra.
2. Forp≠0 andq≠0,J has antipodeS iffp·q+2=0 andS(x)=x. Forp=0 orq=0,J has antipode andS(x)=×.
3. All leftJ ^*-modules are rational.

Also, we give some applications in homological theory and algebraicK-theory.     

Compactness and the axiom of choice

In the absence of the axiom of choice four versions of compactness (A-, B-, C-, and D-compactness) are investigated. Typical results:

1. C-compact spaces form the epireflective hull in Haus of A-compact completely regular spaces.
2. Equivalent are:
3. the axiom of choice,
4. A-compactness = D-compactness,
5. B-compactness = D-compactness,
6. C-compactness = D-compactness and complete regularity,
7. products of spaces with finite topologies are A-compact,
8. products of A-compact spaces are A-compact,
9. products of D-compact spaces are D-compact,
10. powers X ^k of 2-point discrete spaces are D-compact,
11. finite products of D-compact spaces are D-compact,
12. finite coproducts of D-compact spaces are D-compact,
13. D-compact Hausdorff spaces form an epireflective subcategory of Haus,
14. spaces with finite topologies are D-compact.

1. Equivalent are:
2. the Boolean prime ideal theorem,
3. A-compactness = B-compactness,
4. A-compactness and complete regularity = C-compactness,
5. products of spaces with finite underlying sets are A-compact,
6. products of A-compact Hausdorff spaces are A-compact,
7. powers X ^k of 2-point discrete spaces are A-compact,
8. A-compact Hausdorff spaces form an epireflective subcategory of Haus.

1. Equivalent are:
2. either the axiom of choice holds or every ultrafilter is fixed,
3. products of B-compact spaces are B-compact.

1. Equivalent are:
2. Dedekind-finite sets are finite,
3. every set carries some D-compact Hausdorff topology,
4. every T ₁-space has a T ₁-D-compactification,
5. Alexandroff-compactifications of discrete spaces and D-compact.

     

The problem of differentiating an asymptotic expansion in real powers. Part II: Factorizational theory

In Part II of our work we approach the problem discussed in Part I from the new viewpoint of canonical factorizations of a certain nth order differential operator L. The main results include:

1. characterizations of the set of relations $$ f^{(k)} (x) = P^{(k)} (x) + o^{(k)} (x^{\alpha _n - k} ),x \to + \infty ,0 \leqslant k \leqslant n - 1, $$ where $$ P(x) = a_1 x^{\alpha _1 } + \cdots + a_n x^{\alpha _n } and \alpha _1 > \alpha _2 > \cdots > \alpha _n , $$ by means of suitable integral conditions
2. formal differentiation of a real-power asymptotic expansion under a Tauberian condition involving the order of growth of L
3. remarkable properties of asymptotic expansions of generalized convex functions.

     

Multidimensional symmetric stable processes

This paper surveys recent remarkable progress in the study of potential theory for symmetric stable processes. It also contains new results on the two-sided estimates for Green functions, Poisson kernels and Martin kernels of discontinuous symmetric α-stable process in boundedC ^1,1 open sets. The new results give explicit information on how the comparing constants depend on parameter α and consequently recover the Green function and Poisson kernel estimates for Brownian motion by passing α ↑ 2. In addition to these new estimates, this paper surveys recent progress in the study of notions of harmonicity, integral representation of harmonic functions, boundary Harnack inequality, conditional gauge and intrinsic ultracontractivity for symmetric stable processes. Here is a table of contents.

1. Introduction
2. Green function and Poisson kernel estimates

1. Estimates on balls
2. Estimates on boundedC ^1,1 domains
3. Estimates on boundedC ^1,1 open sets

1. Harmonic functions and integral representation
2. Two notions of harmonicity
3. Martin kernel and Martin boundary
4. Integral representation and uniqueness
5. Boundary Harnack principle
6. Conditional process and its limiting behavior
7. Conditional gauge and intrinsic ultracontractivity

     

Diffraction of light by supersonic waves: The solution of the raman-nath equations

The partial differential equation associated with the system of difference-differential equations of Raman-Nath for the amplitudes of the diffracted light-waves is solved exactly by the method of the separation of the variables. The solution is presented as a double infinite series containing the Fourier coefficients of the even periodic Mathieu functions with periodπ and the corresponding eigenvalues. Considering this solution as a Laurent series in one of the variables, the Laurent coefficients immediately give the exact expressions for the amplitudes of the diffracted light-waves, from which the formulae for the intensities are calculated. The connection between the Raman-Nath method and Brillouin’s Mathieu function method has thus been achieved.     

The asymptotics of interlacing sequences and the growth of continual young diagrams

We obtain a series of concrete results establishing a somewhat unexpected connection between the asymptotic representation theory of symmetric groups and the classical results for one-dimensional problems of mathematical physics and function theory. In particular:

1. The universal character of the division of roots for a wide class of orthogonal polynomials is shown.
2. A connection between the Plancherel measure of the infinite symmetric group and Markov's moment problem is established.
3. Asymptotics of the Plancherel measure turns out to be connected with the soliton-like solution of the simplest quasilinear equation, R′_t+RR′_x=0. Bibliography: 14 titles.

     

On the geometry of the characteristic vector of an lcQS-manifold

We study conditions under which the characteristic vector of a normal lcQS-manifold is a torsion-forming or even a concircular vector field. We prove that the following assertions are equivalent:

• An lcQS-structure is normal, and its characteristic vector is a torsion-forming vector field.
• An lcQS-structure is normal, and its characteristic vector is a concircular vector field.
• An lcQS-structure is locally conformally cosymplectic and has a closed contact form.

     

A Remark on non-separable solutions to the Schroeder-Bernstein Problem for Banach spaces

We show that the geometric structure of Banach spaces which are solutions to the Schroeder-Bernstein Problem is very complex. More precisely, we prove that there exists a non-separable solution E to this problem such that

1. E is isomorphic to each one of its finite codimensional subspaces.
2. E has no complemented Hereditarily Indecomposable subspace.
3. E has no complemented subspace isomorphic to its square.
4. E has no non-trivial divisor.

     

E-free objects inE-varieties of inverse rings

We show that for any regular ring (R, +, -), the following conditions are equivalent:

1. (R, -) is inverse.
2. (R, -) isE-solid.
3. (R, -) is locally inverse.
4. (R, -) is locallyE-solid.

We also show that there is ane-free object in eache-variety of inverse rings.     

Analysis of bounded sets in a topological tensor product

The aim of this paper is to investigate the nature of bounded sets in a topological ∈-tensor product EX_∈* F of any two locally convex topological vector spaces E and F over the same scalar field K. Next, we apply the results of this investigation to the study of each of the following:

1. Totally summable families in EX_∈*F;
2. ∈-tensor product of DF-spaces;
3. Topological nature of the dual of E X_∈*F, where E and F are strong duals of Banach spaces;
4. Properties of bounded sets in an ∈-tensor product of metrizable spaces.

Forπ-tensor product, the result corresponding to (b) is well known (see Grothendieck¹) that if E and F are DF-spaces then EX_π* F and EX_π* F are DF-spaces and that the strong topology on the topological dual (EX_π*F)′, which equals the space of continuous bilinear forms on EXF, coincides with the bibounded topology. We study each of the problems from (a) to (d) for ∈-tensor products. For terminology, notations and the well-known results in the theory of topological vector spaces and the topological tensor products we refer to [1–11]. However, for convenience in presentation of the results of our investigation we give a brief survey of notations and fundamental theorems which are needed throughout this paper.     

States on bounded commutative residuated lattices

We define states on bounded commutative residuated lattices and consider their property. We show that, for a bounded commutative residuated lattice X,

1. If s is a state, then X/ker(s) is an MV-algebra.
2. If s is a state-morphism, then X/ker(s) is a linearly ordered locally finite MV-algebra.

Moreover we show that for a state s on X, the following statements are equivalent:

1. s is a state-morphism on X.
2. ker(s) is a maximal filter of X.
3. s is extremal on X.

     

Stability criteria for linear second order delay differential system

Consider second order delay differential system where r is a positive constant and all coefficients are real constants.Our main results are as follows:(1) The maximal length of the delay for which the stability of system (*) is maintained is given in the case where the zero solution of system (*) is asymptotically stable in the absence of delay.(2) The necessary and sufficient criteria for judging that asymptetical stability of system (*) is preserved for an arbitrary large delay are obtained.     

On the density of image of differential operators generated by polynomials

The main result of this work is the following theorem: LetE denote the ring of entire functions on C². LetP∈C[x, y]. Forf∈E, set 1 $$D_P (f) = \frac{{\partial P}}{{\partial x}}\frac{{\partial f}}{{\partial y}} - \frac{{\partial P}}{{\partial y}}\frac{{\partial f}}{{\partial x}}$$ .Theorem.The image of D _p is dense in E if and only ifP=σ(x), where σ is an automorphism of C[x, y].     

On the number of non-isomorphic subgraphs

Let $\mathcal{K}$ be the family of graphs on ω₁ without cliques or independent subsets of sizew ₁. We prove that

1. it is consistent with CH that everyGε $\mathcal{K}$ has 2^ω many pairwise non-isomorphic subgraphs,
2. the following proposition holds in L: (*)there is a Gε $\mathcal{K}$ such that for each partition (A, B) of ω₁ either G?G[A] orG?G[B],
3. the failure of (*) is consistent with ZFC.

     

Some results on entire functions of finite lower order

Letf(z) be an entire function of order λ and of finite lower order μ. If the zeros off(z) accumulate in the vicinity of a finite number of rays, then

1. λ is finite;
2. for every arbitrary numberk ₁>1, there existsk ₂>1 such thatT(k ₁ r,f)≤k ₂ T(r,f) for allr≥r ₀. Applying the above results, we prove that iff(z) is extremal for Yang's inequalityp=g/2, then
3. every deficient values off(z) is also its asymptotic value;
4. every asymptotic value off(z) is also its deficient value;
5. λ=μ;
6. $\sum\limits_{a \ne \infty } {\delta (a,f) \leqslant 1 - k(\mu ).} $

     

Complex Banach spaces whose duals areL 1-spaces

We prove that for a complex Banach spaceA the following properties are equivalent:

1. A ^* is isometric to anL ₁(μ)-space;
2. every family of 4 balls inA with the weak intersection property has a non-empty intersection;
3. every family of 4 balls inA such that any 3 of them have a non-empty intersection, has a non-empty intersection.

     

Regular sets of matrices and applications

SupposeA ₁,...,A _s are (1, - 1) matrices of order m satisfying 1 $$A_i A_j = J, i,j \in \left\{ {1,...s} \right\}$$ 2 $$A_i^T A_j = A_j^T A_i = J, i \ne j, i,j \in \left\{ {1,...,s} \right\}$$ 3 $$\sum\limits_{i = 1}^s {(A_i A_i^T = A_i^T A_i ) = 2smI_m } $$ 4 $$JA_i = A_i J = aJ, i \in \left\{ {1,...,s} \right\}, a constant$$ Call A₁,…,A _s ,a regular s- set of matrices of order m if Eq. 1-3 are satisfied and a regular s-set of regular matrices if Eq. 4 is also satisfied, these matrices were first discovered by J. Seberry and A.L. Whiteman in “New Hadamard matrices and conference matrices obtained via Mathon’s construction”, Graphs and Combinatorics, 4(1988), 355-377. In this paper, we prove that

1. if there exist a regular s-set of order m and a regulart-set of order n there exists a regulars-set of ordermn whent =sm
2. if there exist a regular s-set of order m and a regulart-set of order n there exists a regulars-set of ordermn when 2t = sm (m is odd)
3. if there exist a regulars-set of order m and a regulart-set of ordern there exists a regular 2s-set of ordermn whent = 2sm As applications, we prove that if there exist a regulars-set of order m there exists
4. an Hadamard matrices of order4hm whenever there exists an Hadamard matrix of order4h ands =2h
5. Williamson type matrices of ordernm whenever there exists Williamson type matrices of ordern and s = 2n
6. anOD(4mp;ms₁,…,ms_u whenever anOD (4p;s₁,…,s_u)exists and s = 2p
7. a complex Hadamard matrix of order 2cm whenever there exists a complex Hadamard matrix of order 2c ands = 2c

This paper extends and improves results of Seberry and Whiteman giving new classes of Hadamard matrices, Williamson type matrices, orthogonal designs and complex Hadamard matrices.     

Some results on the nonstationary ideal

The strength of precipitousness, presaturatedness and saturatedness of NS_κ and NS _κ ^λ is studied. In particular, it is shown that:

1. The exact strength of “ $NS_{\mu ^ + }^\lambda $ for a regularμ > max(λ, ?₁)” is a (ω,μ)-repeat point.
2. The exact strength of “NS_κ is presaturated over inaccessible κ” is an up-repeat point.
3. “NS_κ is saturated over inaccessible κ” implies an inner model with ?αo(α) =α ⁺⁺.

     

On the Optimality of the Assumptions Used to Prove the Existence and Symmetry of Minimizers of Some Fractional Constrained Variational Problems

In this paper, we discuss the optimality of the assumptions used, in a previous paper, to prove existence and symmetry of minimizers of the fractional constrained variational problem: $$\inf \;\left\{\frac{1}{2} \int|\nabla_s u|^2 - \int F(|x|, u):\;u\in H^s (\mathbb{R}^N) \mbox{ and } \int u^2 = c^2\right\},$$ where c is a prescribed number. More precisely, we will show that if one of the conditions, used to prove that all minimizers of the above constrained variational problem, are radial and radially decreasing for all c, do not hold true, then there are several interesting situations:

There is no minimizer at all.

The infimum is achieved but no minimizer is radial.

For some values of c there is no minimizer. For large values, the minimizer is radial and radially decreasing.

In the fractional setting, such a study is more subtle than in the classical one. We take advantage of some brilliant results obtained recently in Cabre and Sire (Anal. PDEs, 2012), Dyda (Fractional calculus for power functions, 2012) and Hajaiej et al. (Necessary and sufficient conditions for the fractional Gagliardo–Nirenberg inequalities and applications to Navier–Stokes and generalized Boson equations, 2012).