Respect Tox Research Articles

Residual Division Graph of Lattice Modules

Let L be a multiplicative lattice and M be a lattice module over L. In this paper, we assign a graph to M called residual division graph RG(M) in which the element N ∈ M is a vertex if there exists 0M ≠ P ∈ M such that NP = 0M and two vertices N1, N2 are adjacent if N1N2 = 0M (where N1N2 = (N1 : IM)(N2 : IM)IM). It is proved that such a graph with the greatest element IM which does not belong to the vertex set is nonempty if and only if M is a prime lattice module. Also, we provide conditions such that RG(M) is isomorphic to a subgraph of Zariski topology graph ĢX(M) with respect to X.

Existence and Multiplicity of Solutions for Sublinear Schrödinger Equations with Coercive Potentials

In this study, we consider the following sublinear Schrödinger equations−Δu+Vxu=fx,u,for x∈ℝN,ux⟶0,asu⟶∞,wherefx,usatisfies some sublinear growth conditions with respect touand is not required to be integrable with respect tox. Moreover,Vis assumed to be coercive to guarantee the compactness of the embedding from working space toLpℝNfor allp∈1,2∗. We show that the abovementioned problem admits at least one solution by using the linking theorem, and there are infinitely many solutions whenfx,uis odd inuby using the variant fountain theorem.

Accelerating indefinite hypergeometric summation algorithms

Let K be a field of characteristic zero, x an independent variable, E the shift operator with respect to x, i.e., Ef ( x ) = f ( x + 1) for an arbitrary f ( x ). Recall that a nonzero expression F ( x ) is called a hypergeometric term over K if there exists a rational function r ( x ) ∈ K( x ) such that F ( x + 1)/ F ( x ) = r ( x ). Usually r ( x ) is called the rational certificate of F ( x ). The problem of indefinite hypergeometric summation (anti-differencing) is: given a hypergeometric term F ( x ), find a hypergeometric term G ( x ) which satisfies the first order linear difference equation ( E − 1) G ( x ) = F ( x ). (1) If found, write Σ x F ( x ) = G ( x ) + c , where c is an arbitrary constant.

On computation of a power series root with arbitrary degree of convergence

Given a bivariate polynomial f(x, y), let ☎i(y) be a power series root of f(x, y) = 0 with respect tox, i.e., ☎i(y) is a function ofy such thatf(☎i(y),,y) = 0. If ☎i(y) is analytic aty = 0, then we have its power series expansion (1) $$\phi (y) = \alpha _0 + \alpha _1 y + \alpha _2 y^2 + \cdots + \alpha _r y^r + \cdots .$$ Let ☎i(k)(y) denote ☎i(y) truncated aty k , i.e., (2) $$\phi ^{(k)} (y) = \alpha _0 + \alpha _1 y + \alpha _2 y^2 + \cdots + \alpha _k y^k .$$ Then, it is well known that, given initial value ☎i(0)(y) = α0 ∈ C, the symbolic Newton’s method with the formula (3) $$\phi ^{(2^m - 1)} (y) \leftarrow \phi ^{(2^{m - 1} - 1)} (y) - \frac{{f(\phi ^{(2^{m - 1} - 1)} (y),y)}}{{\frac{{\partial f}}{{\partial x}}(\phi ^{(2^{m - 1} - 1)} (y),y)}} (mod y^{2m} )$$ computes $$\phi ^{(2^m - 1)} (y) (1 \le m)$$ in (2) with quadratic convergence (the roots are computed in the order $$\phi ^{(0)} (y) \to \phi ^{(2^1 - 1)} (y) \to \phi ^{(2^2 - 1)} (y) \to \cdots \to \phi ^{(2^m - 1)} (y))$$ .

Sequential experimental design and response optimisation

We consider the situation where one wants to maximise a functionf(θ,x) with respect tox, with θ unknown and estimated from observationsy k . This may correspond to the case of a regression model, where one observesy k =f(θ,x k )+ε k , with ε k some random error, or to the Bernoulli case wherey k ∈{0, 1}, with Pr[y k =1|θ,x k |=f(θ,x k ). Special attention is given to sequences given by\(x_{k + 1} = \arg \max _x f(\hat \theta ^k ,x) + \alpha _k d_k (x)\), with\(\hat \theta ^k \) an estimated value of θ obtained from (x1, y1),...,(x k ,y k ) andd k (x) a penalty for poor estimation. Approximately optimal rules are suggested in the linear regression case with a finite horizon, where one wants to maximize ∑ N i=1 w i f(θ, x i ) with {w i } a weighting sequence. Various examples are presented, with a comparison with a Polya urn design and an up-and-down method for a binary response problem.

Tight Distance-regular Graphs and the Subconstituent Algebra

We consider a distance-regular graph Γ with diameter D≥ 3, intersection numbers ai, bi, ciand eigenvalues k=θ0>θ1>⋯>θD. Let X denote the vertex set of Γ and fix x∈X. Let T=T(x) denote the subalgebra of Mat X(C) generated by A, E0*, E1*,⋯ , ED*, where A denotes the adjacency matrix of Γ and Ei*denotes the projection onto the i th subconstituent of Γ with respect to x. T is called the subconstituent algebra (or Terwilliger algebra) of Γ with respect to x. An irreducible T -module W is said to be thin whenever dimEi*W≤ 1 for 0 ≤i≤D. By the endpoint of W we mean min{ i | Ei*W≠= 0}. Let W denote a thin irreducible T -module with endpoint 1. Observe E1*W is a one-dimensional eigenspace for E1*AE1*; let η denote the corresponding eigenvalue. We call η the local eigenvalue of W. It is known yeujc5970x.gif θ1≤η≤ yeujc5971x.gif θDwhere yeujc5972x.gif θ1=−1 −b1(1 +θ1)−1and yeujc5973x.gif θD=−1 −b1(1 +θD)−1. Let n= 1 or n=D and assume η= yeujc5974x.gif θn. We show the dimension of W is D− 1. Let v denote a nonzero vector in E1*W. We show W has a basis Eiv(1 ≤i≤D, i≠=n), where Eidenotes the primitive idempotent of A associated with θi. We show this basis is orthogonal (with respect to the Hermitean dot product) and we compute the square norm of each basis vector. We show W has a basis Ei+1*Aiv(0 ≤i≤D− 2), where Aidenotes the i th distance matrix for Γ. We find the matrix representing A with respect to this basis. We show this basis is orthogonal and we compute the square norm of each basis vector. We find the transition matrix relating our two bases for W. For notational convenience, we say Γ is 1- thin with respect tox whenever every irreducible T -module with endpoint 1 is thin. Similarly, we say Γ is tight with respect tox whenever every irreducible T -module with endpoint 1 is thin with local eigenvalue yeujc5975x.gif θ1or yeujc5976x.gif θD. In [ J. Algebr. Comb., 12,(2000), 163–197] Jurišić, Koolen and Terwilliger showed (θ1+ka1+ 1) (θD+ka1+ 1) ≥−ka1b1(a1+ 1)2.They defined Γ to be tight whenever Γ is nonbipartite and equality holds above. We show the following are equivalent: (i) Γ is tight; (ii) Γ is tight with respect to each vertex; (iii) Γ is tight with respect to at least one vertex. We show the following are equivalent: (i) Γ is tight; (ii) Γ is nonbipartite, aD= 0, and Γ is 1-thin with respect to each vertex; (iii) Γ is nonbipartite, aD= 0, and Γ is 1-thin with respect to at least one vertex.

The crystal and magnetic structure relationship in Cu(W1-xMox)O4 compoundswith wolframite-type structure

The magnetic structures of Cu(W1-xMox)O4 compounds withwolframite-type structure at 1.5 K have been determined by neutronpowder diffraction for average composition ⟨x⟩ = 0.15, 0.25 and 0.35. For⟨x⟩ = 0.15the magnetic structure is antiferromagnetic with a magnetic unit cell doubled along thea-axis, k⃗ = (1/2, 0, 0),i.e. the same magnetic structure as for CuWO4. For ⟨x⟩ = 0.25 and 0.35two magnetic structures are observed: one is identical to that for ⟨x⟩ = 0.15,while the other is doubled with respect to thec-axis, k⃗ = (0, 0, 1/2),i.e. the same magnetic structure as for the high-pressure modificationCuMoO4 III. The coexistence of these two magnetic arrangements isinterpreted as reflecting a slightly inhomogeneous contribution of Mo andW in different crystallites together with a sharp transition between thestability ranges of the two types of magnetic structure with respect tox. Thespecific Mo:W distributions in the grains of the powdered samples were deduced froma profile analysis based on high-resolution synchrotron powder diffraction data.No additional, intermediate magnetic phase with k⃗ = (1/2, 1/2, 0)was found in Cu(W0.75Mo0.25)O4, in contrast to predictions in the frameworkof extended Hückel calculations based on the precise crystal structure.

Hilbert’s irreducibility theorem for prime degree and general polynomials

Letf (X, t)eℚ[X, t] be an irreducible polynomial. Hilbert’s irreducibility theorem asserts that there are infinitely manyt0eℤ such thatf (X, t0) is still irreducible. We say thatf (X, t) isgeneral if the Galois group off (X, t) over ℚ(t) is the symmetric group in its natural action. We show that if the degree off with respect toX is a prime ≠ 5 or iff is general of degree ≠ 5, thenf (X, t0) is irreducible for all but finitely manyt0eℤ unless the curve given byf (X, t)=0 has infinitely many points (x0,t0) withx0eℚ,t0eℤ. The proof makes use of Siegel’s theorem about integral points on algebraic curves, and classical results about finite groups, going back to Burnside, Schur, Wielandt, and others.

Boundary stabilization of a hybrid Euler—Bernoulli beam

We consider a problem of boundary stabilization of small flexural vibrations of a flexible structure modeled by an Euler-Bernoulli beam which is held by a rigid hub at one end and totally free at the other. The hub dynamics leads to a hybrid system of equations. By incorporating a condition of small rate of change of the deflection with respect tox as well ast, over the length of the beam, for appropriate initial conditions, uniform exponential decay of energy is established when a viscous boundary damping is present at the hub end.

Convergence of the Nested Multivariate Padé Approximants

The nested multivariate Padé approximants were recently introduced. In the case of two variablesxandy, they consist in applying the Padé approximation with respect to y to the coefficients of the Padé approximation with respect tox. The principal advantage of the method is that the computation only involves univariate Padé approximation. This allows us to obtain uniform convergence where the classical multivariate Padé approximants fail to converge.

A SIMPLE AND DIRECT PERIODIC SOLUTION OF THE KORTEWEG-DE VRIES EQUATIONS

The solutionq(x, t) of one of the KdV hierarchy is assumed to be a potential in the Schrodinger equation as usual. We differentiate this equation with respect to the time variable and solve it with the aid of the Green function. The obtained equation relatesw t (x, t, λ)=φ t (x + c, x, t, λ) withq t (x, t). The functionφ(x, x 0,t, λ) obeys the Schrodinger equation and the boundary conditionsφ(x 0,x 0,t, λ)=0,φ x (x 0,x 0;t, λ)=1. The shiftingc is equal to the period. We differentiatew t (x, t, λ) three times with respect to thex coordinate and obtain the time derivative of the Milne equation. The integration of this equation with respect tox allows to solve simply the inverse problem. The reconstructed periodic potential is given by means of the well known formula for the root functions ofw(x, t, λ). The time behaviour of this function, i.e. the solution of the KdV equation, is obtained when one replacesq t (x, t) by an expression of the KdV hiearchy in the relation betweenq t (x, t) andw t (x, t, λ) and transforms it. We estimated also the limit, whenc → ∞, i.e. the possible relation of the periodic solutions with the soliton ones.

The girth of a thin distance-regular graph

Let Γ be a distance-regular graph of diameterd?3. For each vertexx of Γ, letT(x) denote the Terwilliger algebra for Γ with respect tox. An irreducibleT(x)-moduleW is said to bethin if dimE i * (x)W≤1 for 0≤i≤d, whereE i * (x) is theith dual idempotent for Γ with respect tox. The graph Γ isthin if for each vertexx of Γ, every irreducibleT(x)-module is thin. Aregular generalized quadrangle is a bipartite distance-regular graph with girth 8 and diameter 4. Our main results are as follows: Theorem. Let Γ=(X,R) be a distance-regular graph with diameter d?3 and valency k?3. Then the following are equivalent: Corollary. Let Γ=(X,R) be a thin distance-regular graph with diameter d?3 and valency k?3. Then Γ has girth 3, 4, 6, or 8. Then girth of Γ is 8 exactly when Γ is a regular generalized quadrangle.

Asymptotics of the first correction in the perturbation of theN-soliton solution to the KdV equation

We consider a triple Fourier-type integral that represents a solution to the KdV equation linearized on anN-soliton potential. Assuming that the parameters of the potential depend on the slow timeet, we construct an asymptotics of this integral ase → 0 uniform with respect tox, t up to large timet ∼ e−1.

Positive Lyapunov exponent for generic one-parameter families of unimodal maps

Letfaa∈A be a C2 one-parameter family of non-flat unimodal maps of an interval into itself anda* a parameter value such that (a) fa* satisfies the Misiurewicz Condition, (b) fa* satisfies a backward Collet-Eckmann-like condition, (c) the partial derivatives with respect tox anda of fan(x), respectively at the critical value and ata*, are comparable for largen.

L 1-convergence of double cosine- and Walsh-Fourier series

We prove the convergence inL 1([−gp, π)2)-norm of the double Fourier series of an integrable functionf(x, y) which is periodic and even with respect tox andy, with coefficientsa jk satisfying certain conditions of Hardy-Karamata kind, and such thata jk logj logk→0 asj, k→∞. These sufficient conditions become quite natural in particular cases. Then we extend these results to the convergence of double Walsh-Fourier series inL 1 (0, 1)2)- norm. As a by-product, we obtain Tauberian conditions ensuring the convergence of a double numerical series provided it is Cesaro summable.

Finite-gap solutions of two-dimensional isotropic ferromagnet with fixed boundary conditions

For the model of the two-dimensional stationary Heisenberg magnet, real finite-gap solutions that describe the main structures that are almost periodic with respect tox andy are constructed. Some solutions of boundary-value problems are considered.

The intersection local time of the Westwater process

In this paper, self-intersection properties of the Westwater process are investigated. As a result, we obtain that the Westwater process has an intersection local time\(\bar \alpha \)(x, [0,s] × [t, 1]) which is Holder continuous with respect tox, s, t)∈R3×[0,1/2]×[1/2,1], and the Hausdorff dimension of the double time set is 1/2, as for Brownian motion inR3.

Second-order necessary and sufficient optimality conditions for minimizing a sup-type function

In this paper, we give second-order necessary and sufficient optimality conditions for a minimization problem of a sup-type functionS(x)=sup{f(x,t);te T}, whereT is a compact set in a metric space and f is a function defined on ℝn ×T. Our conditions are stated in terms of the first and second derivatives of f(x, t) with respect tox, and involve an extra term besides the second derivative of the ordinary Lagrange function. The extra term is essential when {f(x,t)} t forms an envelope. We study the relationship between our results, Wetterling [14], and Hettich and Jongen [6].

Inductive inference of monotonic formal systems from positive data

A formal system is a finite set of expressions, such as a grammar or a Prolog program. A semantic mapping from formal systems to concepts is said to be monotonic if it maps larger formal systems to larger concepts. A formal system Γ is said to be reduced with respect to a finite setX if the concept defined by Γ containsX but the concepts defined by any proper subset Γ′ of Γ cannot contain some part ofX. Assume a semantic mapping is monotonic and formal systems consisting of at mostn expressions that are reduced with respect toX can define only finitely many concepts for any finite setX and anyn. Then, the class of concepts defined by formal systems consisting of at mostn expressions is shown to be inferable from positive data. As corollaries, the class of languages defined by length-bounded elementary formal systems consisting of at most,n axioms, the class of languages generated by context-sensitive grammars consisting of at mostn productions, and the class of minimal models of linear Prolog programs consisting of at mostn definite clauses are all shown to be inferable from positive data.

An exact solution for two-dimensional frictionless motion in the atmosphere

A solution of the nonlinear problem for determining the wind velocity in frictionless atmosphere (the gradient wind) under given geopotential (pressure) field is proposed. The approach is analytical and is based on quadratic polynomial approximation of the geopotential field and linear approximation of the wind velocity field with respect tox andy, the coefficients of the expansions being functions of the timet. The derived system of ordinary nonlinear differential equations is analyzed as a dynamical system. Exact analytical solutions are found for some particular cases. Some of their properties bear a resemblance to those or really existing atmospheric vortices (cyclones and anticyclones).