We have observed that this method (the general shifted Jacobimatrix method) is very eﬃcient for numerical approximation of the general high order linear PDEs with variable coeﬃcients. Also, during the running of pro- grams, we found out the run time of the general shifted Jacobimatrix scheme is 2.015 second and the run time of the collocation method and Taylor approximation is 10.047 and 11.018 seconds, respectively. Thus in the case of the numerical solution of linear PDEs with variable coeﬃcients, we prefer the general shifted Jacobimatrix scheme to the collocation method and Taylor approximation. This is the excellent advantage on the application of the general shifted Jacobimatrix scheme to the linear PDEs with variable coeﬃcients. Also, closer look at the results of the general shifted Jacobimatrix scheme reveals that this method of solution is also stable.
prove that the monic Jacobimatrix associated with x 2 U can be obtained from the monic Jacobimatrix associated with U applying two consecutive Darboux transfor- mations without parameter with shifts − and , and taking limits when tends to zero. This problem was considered by Buhmann and Iserles  in a more general context. They considered a positive definite linear functional L and the symmetric Jacobimatrix associated with the orthonormal sequence of polynomials associated with L, and proved that one step of the QR method applied to the Jacobimatrix corre- sponds to finding the Jacobimatrix of the orthonormal polynomial system associated with x 2 L.
Let L be a quasi-definite linear functional defined on the linear space of polynomials with real coefficients. In the literature, three canonical transformations of this functional are stud- ied: xL, L + Cδ(x) and 1 x L + Cδ(x) where δ(x) denotes the linear functional (δ(x))(x k ) = δ k,0 , and δ k,0 is the Kronecker symbol. Let us consider the sequence of monic polynomials orthogonal with respect to L. This sequence satisfies a three-term recurrence relation whose coefficients are the entries of the so-called monic Jacobimatrix. In this paper we show how to find the monic Jacobimatrix associated with the three canonical perturbations in terms of the monic Jacobimatrix associated with L. The main tools are Darboux transformations. In the case that the LU factorization of the monic Jacobimatrix associated with xL does not exist and Darboux transformation does not work, we show how to obtain the monic Jacobimatrix associated with x 2 L as a limit case. We also study perturbations of the functional L that are obtained by combining the canonical cases. Finally, we present explicit algebraic relations between the polynomials orthogonal with respect to L and orthogonal with respect to the perturbed functionals.
Abstract—The purpose of this paper is to present a new interpretation of Darboux transforms in the context of 2− orthogonal polynomials and find conditions in order for any Darboux transforms to yield a new set of 2− orthogonal polynomials. We also introduce the LU and U L factorizations of the monic Jacobimatrix J associated with a quasi-definite linear functional Γ defined on the linear space of polynomials with real coefficients, as well as the Darboux transforms without parameters.
positive deﬁnite sequence given by Ahiezer and Krein , we establish an explicit formula for recovering the Jacobimatrix A via a given set of spectral data and a di- agonal matrix B. We use the usual approach of dealing with this type of problems, that is, the orthogonal polynomials. In this paper, we only discuss the inﬁnite dimen- sional version of GIEP. The ﬁnite dimensional case of this problem has been studied by many authors, see for example  for the classical case Ax = λx, and [2, 3] for the generalized case Ax = λBx.
of orthogonal polynomials and the Verblunsky coeﬃcients. The increasing attention to the analysis of the zeros of OPUC, according to their extensive applications, makes the spectral study of the multiplication operator of special interest. The recurrence relation of the orthogonal polynomials on the unit circle (unlike the scalar case) does not conclude a spectral representation for their zeros. In fact, such a representation can be obtained by computing the matrix corresponding to the multiplication operator in the linear space of complex polynomials when orthogonal polynomials are chosen as a basis, but the result is the irreducible Hessenberg matrix (2.13) with much more complicated structure than the Jacobimatrix on the real line.
Diphtheria became an epidemic; consequently, the majority of Jacobi’s writings were in this field. Iron- ically, he lost his 7-year-old son to diphtheria. For- tunately diphtheria antitoxin was on the way. Jacobi had performed hundreds of tracheostomies on diph- theritic children who were strangling. In 1882, a phy- sician named O’Dwyer invented a tube for intuba- tion that saved many infants from diphtheritic croup and obviated many tracheostomies.
A major application of the DLP is to design cryptographic protocols whose secu- rity depends on the difficulty of solving the DLP. A cryptosystem has to be secure and fast. Hence we have to consider groups with an efficient arithmetic, a compact representation of their elements and where the DLP is intractable. To this end, in 1985 Miller  and Koblitz  independently introduced elliptic curve cryptog- raphy based on the DLP in the group formed by rational points of an elliptic curve defined over a finite field. This particular problem is denoted ECDLP. More re- cently, some curve representations such as twisted Edwards [5, 4, 18] and twisted Jacobi intersections [9, 29] have been widely studied by the cryptology community for their efficient arithmetic. A few years after the introduction of elliptic curve cryptography, it has been proposed to use the divisor class group of a hyperelliptic curve over a finite field , in this case we note the discrete logarithm problem HCDLP.
Let E a : y 2 = x 4 + 2 ax 2 + 1 ( a 2 ≠ 1 ) be a Jacobi quartic curve defined over a finite field F q with characteristic of F q is greater than 3. Note that the j-invariant of E a is 16( a 2 + 12) / ( 3 a 2 − 4) 2 . Recall the Legendre elliptic curve are of the form
In the ﬁrst part of this note, it was shown that a Jacobi form of odd prime index is entirely determined by any one of its associated vector-valued components and the above calculation showed how to generate the other components. There are still open questions. For example, the above construction of a Jacobi form was biased in that a priori, it was known that there was an associated Jacobi form. Given a function with some modular properties ((4.11), (4.12) for example), is there an attached Jacobi form? Is the above construction possible by starting with the h 0 component? How will this
The necessary condition of the above problem can be written as a Hamilton-Jacobi equation or a Hamiltonian system.Here, we assume that optimal control inputs exist.We define the minimum value of the cost function (2) for some initial condition as follows:
In this paper, we study mixed Jacobi-like forms of several variables associated to equi- variant maps of the Poincar´e upper half-plane in connection with usual Jacobi-like forms, Hilbert modular forms, and mixed automorphic forms. We also construct a lifting of a mixed automorphic form to such a mixed Jacobi-like form.
It is natural to study the development of a theory of viscosity solutions and their numerical approxi- mation to first order equations on evolving surfaces which may be useful in the modelling of transport on moving surfaces, for example in material science and cell biology. In this paper we are concerned with the existence, uniqueness and numerical approximation of Hamilton–Jacobi equations on moving hypersur- faces. Let Γ(t), t ∈ [0, T ] be a family of smooth, closed, connected and oriented hypersurfaces in R 3 and
mials and to their derivatives. We call these the nonsymmetric Jacobi polynomials. We establish also a new orthogonality relations for these polynomials and we give integral representation for the nonsymmetric Jacobi function. In section 3, we study the finite continuous nonsymmetric Jacobi transform and we give for this transform an inversion formula. In section 4, sampling theorem associated with the differential-difference operator is investigated.
In this paper we describe a new inversion and Plancherel formula for Fourier-Jacobi wavelet transform. Also we construct a Calderon’s reproducing formula for Fourier- Jacobi wavelet transform. Some applications associated with Calderon’s reproducing formula for Fourier-Jacobi convolution are also given.
Shohat  translated Szeg˝o’s theory from the unit circle to the real line and was able to identify all Jacobi matrices which lead to measures with no mass points outside [−2, 2] and have Z(J ) < ∞. The strongest result of this type, so far, is the following theorem from Killip-Simon [13, Theorem 4 0 ] (the methods of Nevai  can prove the