Propagation of sound pulses in a semiinfinite stratified medium

We consider the propagation of sound pulses due to a line source in the inhomogeneous, semi-infinite mediumy ≥ 0 with the boundary conditionφ=0 or?φ/(?y)=0 aty=0, whereφ is the acoustic velocity potential. We suppose that the velocity of wave propagation,c, is given byc ^?2=p ?qe ^?ay , wherep, q, α are real and positive andp>q. The method of dual integral transformation is used. The solution in terms of pulse propagation modes yields the diffracted pulse and the method of steepest descents gives the geometrical acoustic field.     

Temperature dependence of the photoelastic behaviour of crystals—Part I

A modified form of Filon’s spectrometer method is used to study the variation of the stress-optical constants (q ₁₁-q ₁₂) andq ₄₄ of KCl, KBr, KI, LiF, MgO and NaCl in the temperature range 30° C. to 400° C. It is found that (q ₁₁-q ₁₂) andq ₄₄ generally increase numerically with the increase of temperature except those of NaCl which show a decrease. In KBr and KI,q ₄₄ first increases and then decreases numerically. The potassium halides show an interesting gradation of variation of these constants. In all the crystals studied, which are of NaCl type, the variation in (q ₁₁-q ₁₂) is greater than that inq ₄₄. Mueller’s ultrasonic method is used for measuring the ratio of strain-optical constantsp ₁₂/p ₁₁, in the temperature range 30°C. to 250°C. Combining the results obtained by these two methods, the absolute strain-optical constantsp ₁₁ andp ₁₂ have been evaluated at different temperatures. Curves are given showing the variation ofp ₁₁ andp ₁₂ with temperature. The variation ofp ₁₁ andp ₁₂ with temperature is discussed in terms of the contributions of the various factors considered by Mueller in his theory of photoelastic effect in cubic crystals.     

Brochette percolation

We study bond percolation on the square lattice with one-dimensional inhomogeneities. Inhomogeneities are introduced in the following way: A vertical column on the square lattice is the set of vertical edges that project to the same vertex on Z. Select vertical columns at random independently with a given positive probability. Keep (respectively remove) vertical edges in the selected columns, with probability p (respectively 1?p). All horizontal edges and vertical edges lying in unselected columns are kept (respectively removed) with probability q (respectively 1 ? q). We show that, if p > p_c(Z²) (the critical point for homogeneous Bernoulli bond percolation), then q can be taken strictly smaller than p_c(Z²) in such a way that the probability that the origin percolates is still positive.     

Liouville type results for a <Emphasis Type="Italic">p</Emphasis>-Laplace equation with negative exponent

Positive entire solutions of the equation \(\Delta _p u = u^{ - q} in \mathbb{R}^N (N \geqslant 2)\) where 1 < p ≤ N, q > 0, are classified via their Morse indices. It is seen that there is a critical power q = q _c such that this equation has no positive radial entire solution that has finite Morse index when q > q _c but it admits a family of stable positive radial entire solutions when 0 < q ≤ q _c. Proof of the stability of positive radial entire solutions of the equation when 1 < p < 2 and 0 < q ≤ q _c relies on Caffarelli–Kohn–Nirenberg’s inequality. Similar Liouville type result still holds for general positive entire solutions when 2 < p ≤ N and q > q _c. The case of 1 < p < 2 is still open. Our main results imply that the structure of positive entire solutions of the equation is similar to that of the equation with p = 2 obtained previously. Some new ideas are introduced to overcome the technical difficulties arising from the p-Laplace operator.     

Propagation of sound pulses in a homogeneous layer under an inhomogeneous half space

We consider the propagation of two-dimensional sound pulses in a homogeneous layer ?y ₁?y?0. It is bounded by a plane stratified inhomogeneous half spacey?0 on one side and a perfectly reflecting boundary on the other. A line source is situated in the layer. The boundary condition isφ=0 or?φ/?y=0 aty=?y ₁, whereφ is the acoustic velocity potential. We suppose that the velocity of wave propagationc is given byc ^?2=p?qe ^?αy iny>0, wherep, q, α are real and positive andp>q. It is equal to C′ in the layer where C′ is a constant. The method of dual integral transformation is used and the velocity potentialφ is obtained after using asymptotic expressions for some of the functions which are in the integrand. We obtain the incident, reflected, multiply-reflected and diffracted pulses in the layer.     

Seifert manifolds and (1, 1)-knots

The aim of this paper is to investigate the relations between Seifert manifolds and (1, 1)-knots. In particular, we prove that each orientable Seifert manifold with invariants

$\{ Oo,0| - 1;\underbrace {(p,q),...,(p,q)}_{n times},(l,l - 1)\} $

has the fundamental group cyclically presented by G _n ((x ₁ ^q ...x _n ^{q l} x _n ^?p ) and, moreover, it is the n-fold strongly-cyclic covering of the lens space L(|nlq ? p|, q) which is branched over the (1, 1)-knot K(q, q(nl ? 2), p ? 2q, p ? q) if p ≥ 2q and over the (1, 1)-knot K(p? q, 2q ? p, q(nl ? 2), p ? q) if p< 2q.

     

The influence of isobar production on charge ratio and transverse momentum of secondary particles in high energy collisions

An isobar model in which collision between two particles leads to the creation of only two bodies which by subsequent decay give rise to the observed secondaries has been considered. On the basis of such a model, the charge ratios of pions, kaons andΣ-hyperons inp?p andπ?p collisions have been computed and compared with the available experimental data. Some features of transverse momentum of pions and protons in 24 GeV/cp?p collisions have also been studied. The main conclusions can be summarised as follows:

1. (1)
   The observed positive excess among pions produced in high energyp?p collisions leading toπ ⁺/π ^? andπ ⁺/π ⁰ ratios of ～3 and 1·6 respectively for high momentum pions can be explained on the basis of the isobar model. Further, the fast increase of K⁺/K^? ratio as the kaon momentum increases, the high ratio (～4) ofΣ ⁺/Σ ^? in 24 GeV/cp?p collisions and the existence of a strong positive (negative) excess amongΣ-hyperons produced inπ ⁺?p(π ^??p) collisions at various primary energies result, in a natural way, from such a model. The agreement results mainly from the restriction of only two bodies in the final states and does not critically depend on the isospins of produced isobars.     
   
   

Linear natural operators lifting p-vectors to tensors of type (q, 0) on Weil bundles

We give a classification of all linear natural operators transforming p-vectors (i.e., skew-symmetric tensor fields of type (p, 0)) on n-dimensional manifolds M to tensor fields of type (q, 0) on T^AM, where T^A is a Weil bundle, under the condition that p ≥ 1, n ≥ p and n ≥ q. The main result of the paper states that, roughly speaking, each linear natural operator lifting p-vectors to tensor fields of type (q, 0) on T^A is a sum of operators obtained by permuting the indices of the tensor products of linear natural operators lifting p-vectors to tensor fields of type (p, 0) on T^A and canonical tensor fields of type (q ? p, 0) on T^A.     

Potential operators associated with Hankel and Hankel-Dunkl transforms

We study Riesz and Bessel potentials in the settings of Hankel transform, modified Hankel transform and Hankel-Dunkl transform. We prove sharp or qualitatively sharp pointwise estimates of the corresponding potential kernels. Then we characterize those 1 ≤ p, q≤∞, for which the potential operators satisfy L ^p -L ^q estimates. In case of the Riesz potentials, we also characterize those 1 ≤ p, q ≤ ∞, for which two-weight L ^p -L ^q estimates, with power weights involved, hold. As a special case of our results, we obtain a full characterization of two power-weight L ^p -L ^q bounds for the classical Riesz potentials in the radial case. This complements an old result of Rubin and its recent reinvestigations by De Nápoli, Drelichman and Durán, and Duoandikoetxea.     

A Classification of Spectrum-determined Circulant Digraphs

Some researchers have proved that ádám’s conjecture is wrong. However, under special conditions, it is right. Let Z_n be a cyclic group of order n and C_n(S) be the circulant digraph of Z_n with respect to S ? Z_n\{0}. In the literature, some people have used a spectral method to solve the isomorphism for the circulants of prime-power order. In this paper, we also use the spectral method to characterize the circulants of order p^aq^bw^c(where p, q and w are all distinct primes), and to make ádám’s conjecture right.     

Blow-up of <Emphasis Type="Italic">p</Emphasis>-Laplacian evolution equations with variable source power

We study the blow-up and/or global existence of the following p-Laplacian evolution equation with variable source power

$${s_j} = {\beta _j} + \overline {{\beta _{n - j}}}p$$

where Ω is either a bounded domain or the whole space ? ^N , q(x) is a positive and continuous function defined in Ω with 0 < q _? = inf q(x) ? q(x) ? sup q(x) = q+ < ∞. It is demonstrated that the equation with variable source power has much richer dynamics with interesting phenomena which depends on the interplay of q(x) and the structure of spatial domain Ω, compared with the case of constant source power. For the case that Ω is a bounded domain, the exponent p ? 1 plays a crucial role. If q+ > p ? 1, there exist blow-up solutions, while if q ₊ < p ? 1, all the solutions are global. If q _? > p ? 1, there exist global solutions, while for given q _? < p ? 1 < q ₊, there exist some function q(x) and Ω such that all nontrivial solutions will blow up, which is called the Fujita phenomenon. For the case Ω = ? ^N , the Fujita phenomenon occurs if 1 < q _? ? q ₊ ? p ? 1 + p/N, while if q _? > p ? 1 + p/N, there exist global solutions.

     

Representation and character varieties of the Baumslag-Solitar groups

Representation and character varieties of the Baumslag–Solitar groups BS(p, q) are analyzed. Irreducible components of these varieties are found, and their dimension is calculated. It is proved that all irreducible components of the representation variety R_n(BS(p, q)) are rational varieties of dimension n², and each irreducible component of the character variety X_n(BS(p, q)) is a rational variety of dimension k ≤ n. The smoothness of irreducible components of the variety R_n^s (BS(p, q)) of irreducible representations is established, and it is proved that all irreducible components of the variety R_n^s (BS(p, q)) are isomorphic to A¹ {0}.     

A new characterization for some extensions of PSL(2, <Emphasis Type="Italic">q)</Emphasis> for some <Emphasis Type="Italic">q</Emphasis> by some character degrees

In [22] (Tong-Viet H P, Simple classical groups of Lie type are determined by their character degrees, J. Algebra, 357 (2012) 61–68), the following question arose: Which groups can be uniquely determined by the structure of their complex group algebras? The authors in [12] (Khosravi B et al., Some extensions of PSL(2,p²) are uniquely determined by their complex group algebras, Comm. Algebra, 43(8) (2015) 3330–3341) proved that each extension of PSL(2,p²) of order 2|PSL(2,p²)| is uniquely determined by its complex group algebra. In this paper we continue this work. Let p be an odd prime number and q = p or q = p³. Let M be a finite group such that |M| = h|PSL(2,q), where h is a divisor of |Out(PSL(2,q))|. Also suppose that M has an irreducible character of degree q and 2p does not divide the degree of any irreducible character of M. As the main result of this paper we prove that M has a unique nonabelian composition factor which is isomorphic to PSL(2,q). As a consequence of our result we prove that M is uniquely determined by its order and some information on its character degrees which implies that M is uniquely determined by the structure of its complex group algebra.     

On the polyhedral method of solving problems of control strategy synthesis

Based on the method of nonlinear capacity, we study the nonexistence of nonnegative monotonic solutions for the quasilinear elliptic inequality of the form ?Δ_pu ≥ u^q in a half-space in terms of the parameters p and q.     

Exact Solutions of a Second-Order Nonlinear Partial Differential Equation

The equation ?²u/?t?x + u^p?u/?x = u^q describing a nonstationary process in semiconductors, with parameters p and q that are a nonnegative integer and a positive integer, respectively, and satisfy p + q ≥ 2, is considered in the half-plane (x, t) ∈ ? × (0,∞). All in all, fourteen families of its exact solutions are constructed for various parameter values, and qualitative properties of these solutions are noted. One of these families is defined for all parameter values indicated above.     

On Schrödinger Propagator for the Special Hermite Operator

We establish a Strichartz type estimate for the Schrödinger propagator e ^it? for the special Hermite operator ? on ? ⁿ . Our method relies on a regularization technique. We show that no admissibility condition is required on (q,p) when 1≤q≤2.     

Improved bounds on the Hadwiger–Debrunner numbers

Let HD _d (p, q) denote the minimal size of a transversal that can always be guaranteed for a family of compact convex sets in Rd which satisfy the (p, q)-property (p ≥ q ≥ d + 1). In a celebrated proof of the Hadwiger–Debrunner conjecture, Alon and Kleitman proved that HD _d (p, q) exists for all p ≥ q ≥ d + 1. Specifically, they prove that \(H{D_d}(p,d + 1)is\tilde O({p^{{d^2} + d}})\).We present several improved bounds: (i) For any \(q \geqslant d + 1,H{D_d}(p,d) = \tilde O({p^{d(\frac{{q - 1}}{{q - d}})}})\). (ii) For q ≥ log p, \(H{D_d}(p,q) = \tilde O(p + {(p/q)^d})\). (iii) For every ? > 0 there exists a p₀ = p₀(?) such that for every p ≥ p₀ and for every \(q \geqslant {p^{\frac{{d - 1}}{d} + \in }}\) we have p ? q + 1 ≤ HD _d (p, q) ≤ p ? q + 2. The latter is the first near tight estimate of HD _d (p, q) for an extended range of values of (p, q) since the 1957 Hadwiger–Debrunner theorem.We also prove a (p, 2)-theorem for families in R² with union complexity below a specific quadratic bound.     

Asymptotics for the sum of powers of distances between power residues modulo a prime

For fixed q ∈ (0, 4), prime p → ∞, and \(d \leqslant \exp \left( {c\sqrt {\ln p} } \right)\), where c > 0 is a constant, we obtain the asymptotics for the sum of qth powers of distances between neighboring residues of degree d modulo p.