We first extensively investigate the solvability of the Diophantine equation as in the title, where c > 1 is a square-free integer, p(1), p(2), ... , p(r) are distinct odd primes, and x, y, m(1), m(2), ... , m(r) are unknown positive integers with gcd(x, y) = 1. We then use this result to describe all the positive integer solutions of some of its variants.
In a two-dimensional region ohm, a Brownian particle released from a fixed location z0 will travel randomly until it reaches the region's boundary partial derivative ohm. For every value of r, we are interested in the probability that it will strike the boundary somewhere within the distance r from the starting point z0. The function h(r), also known as the harmonic-measure distribution function or h-function of ohm with respect to z0, is obtained by summing these probabilities for all values of r. This h-function conveys details regarding the form of the boundary of ohm. In this paper, we compute the h-functions of several new planar simply connected unbounded regions using the method of conformal maps. In addition, we describe the specific features of these h-functions, including the asymptotic behavior. Moreover, we cross-check our h-function formulae through different methods.
Inspired from [5], we generalize and introduce a family of polyhedral & ell;1-predual spaces with an extreme point that does not contain a subspace isometric to c. This class of & ell;(1)-predual spaces disproves some previously known facts about polyhedral Lindenstrauss spaces appeared in the literature. Moreover, these spaces are not isometric to a quotient of any C(alpha). We also present a necessary condition on a representing matrix of this special family of polyhedral & ell;1-predual spaces.
In this paper, we study the inverse problem of finding a time-dependent multiplier of the right-hand side of a one-dimensional fractional time diffusion-wave equation with variable coefficients. The direct problem is an initial-boundary value one with the usual Cauchy conditions, homogeneous Dirichlet boundary conditions for this equation. The overdetermination condition has the form of an integral over a spatial segment from the solution of the direct problem, in which the weight function is a spatially dependent known factor of the right-hand side of the equation. This made it possible to construct a solution to the inverse problem in explicit form and prove its correctness in the class of regular solutions. Although we consider only the one-dimensional model, the analysis and computation in this part can be extended into the general multi-dimensional case, upon suitable modifications.
Let f be a continuous linear map between Banach algebras A and B satisfying f (a o b) = f (a) o f(b) for all a, b is an element of A with ab = ba = 0. In this paper, under special hypothesis we prove that f is a Jordan homomorphism. We also characterize derivable maps that preserves commutative zero products. Finally, we determine continuous linear maps which preserves idempotents products, that is, f (ab) = f(a)f(b) for all a, b is an element of A with ab = p, where p is an idempotent in A.
Let K = Q(theta) be an algebraic number field with theta a root of an irreducible quadrinomial f (x) = x(6) + ax(m) + bx + c is an element of Z[x] with m is an element of {2, 3, 4, 5}. In the present paper, we give some explicit conditions involving only a, b, c and m for which K is non-monogenic. Moreover, in each case we determine the highest powers of the primes 2 and 3 dividing the index of the field K. In particular, we provide a partial answer to the Problem 22 of Narkiewicz [10] for these number fields. Finally, we illustrate our results through examples.
Assuming that the Rankin-Cohen bracket of two complex-valued holomorphic functions defined on H & times; C-2 is a Hermitian Jacobi form, we prove that both functions have to be Hermitian Jacobi forms, if at least one of them is. We also study the relation between Rankin-Cohen brackets of Hermitian Jacobi forms and Hermitian Jacobi Poincar & eacute; series. Finally, we construct certain Hermitian Jacobi cusp forms by computing the adjoint of higher-order heat operator for Hermitian Jacobi forms with respect to the Petersson scalar product.
Let X be an m & times; n matrix of distinct indeterminates over a field K, where m n. Set the polynomial ring K[X] := K[X-ij: 1 i m, 1 j n]. Let 1 k < l n be such that l-k + 1 m. Consider the submatrix Y-kl of consecutive columns of X from kth column to lth column. Let J(kl) be the ideal generated by 'diagonal monomials' of all m & times; m submatrices of Y-kl, where the diagonal monomial of a square matrix means product of its main diagonal entries. We show that J(k1l1) J(k2l2) & centerdot;& centerdot;& centerdot;J(ksls )has a linear free resolution, where k(1) k(2) & centerdot; & centerdot; & centerdot; k(s )and l(1) l(2) & centerdot; & centerdot; & centerdot; l(s). This result is a variation of a theorem due to Bruns and Conca. Moreover, our proof is self-contained, elementary and combinatorial.
Let u be an even integer not divisible by 3. For each z = (z1, z2, z3) E Z(3) satisfying z(2) = 0 mod 2 and Qu(z) = z(1)(2)-z(2)(2)-((u(6)-1)/3)z23 = 0, we find integral binary quadratic forms sz, tz such that sz(x, y)(3) + tz(x, y)(3) = sz(y, x)(3) + tz(y, x)(3). The coefficients of sz, tz are integer linear combinations of (z1, z2, z3). This yields a three parameter family of 4-tuples of binary quadratic forms spanning infinitely many GL2(Z) orbits, whose cubes add up to zero. The first couple of examples go back to Ramanujan.
The notion of virtual global generation (VGG) for a vector bundle has multiple possible generalization from the case of curves to higher dimensional normal projective varieties. We study relationship between these notions. All these notions agree for curves but in higher dimension we show that this is not the case.
Let K be a complete discretely valued field whose residue field has characteristic different from 2. Let (D, sigma) be a K-division algebra with involution of the first kind, and h be a K-anisotropic & varepsilon;-hermitian form over (D, sigma). By a theorem due to Larmour, there is a decomposition h = h(0) perpendicular to h(1) such that the elements in a diagonalization of h(0) are units, the elements in a diagonalization of h(1) are uniformizers, and h(0), h(1) are determined uniquely up to K-isometry. In this paper, we give an explicit description of the elements in the diagonalization of h(0) and h(1) in the case of quaternion algebras. Then we derive explicit formulas for Larmour's isomorphism of Witt groups.
In this paper, we examine specific submonoids within the singular twisted virtual braid monoid STVB_n. Notably, we establish that the singular twisted virtual pure braid monoid STVP_n serves as the kernel of an epimorphism from STVB_n onto the symmetric group S_n. We identify the generators and defining relations for STVP_n. Additionally, we construct other epimorphisms from STVB_n onto S_n, whose kernels are analogous to STVP_n, and determine their respective generators and defining relations. Furthermore, we demonstrate the embedding of the monoid STVB_n into a group. Also, we provide the extension of the representation of the twisted virtual braid group to the representation of the singular twisted virtual braid monoid.
In a 2022 paper, Dawsey, Just and the present author prove that the set of integer partitions, taken as a monoid under a partition multiplication operation I defined in my Ph.D. work, is isomorphic to the positive integers as a monoid under integer multiplication. In this note, I extend partition multiplication to the set of overpartitions, which are of much interest in partition theory. I prove the overpartitions form an Abelian group under partition multiplication. Moreover, the overpartitions and the positive rational numbers are isomorphic as multiplicative groups. I then prove further overpartition isomorphisms and discuss approaches to a ring theory of overpartitions.
In this paper we address the question of non-vanishing of L '(0, f) where f is an algebraic valued periodic function. In 2011, Gun, Murty and Rath studied the nature of special values of the derivatives of even Dirichlet-type functions and proved that it can be either zero or transcendental. Here for some special cases we characterize the set of functions for which L '(0, f) is zero or transcendental. Using a theorem of Ramachandra about multiplicative independence of cyclotomic units we also provide some non-trivial examples of functions where L '(0, f) is zero. Finally, assuming Schanuel's conjecture we derive the algebraic independence of special values of derivatives of L-functions. 2010 Mathematics Subject Classification: Primary 11J81, 11J86, 11M06; Secondary 11J91.
We produce a large class of hyperbolic homology 3-spheres admitting arbitrarily many distinct tight contact structures. We also produce a sub-class admitting arbitrarily many distinct tight contact structures within the same homotopy class of oriented plane distributions. As a corollary, we give a recipe to construct hyperbolic L-spaces admitting arbitrarily many distinct tight contact structures. We also introduce a notion of geometric limits of contact structures compatible with geometric limits of hyperbolic manifolds and study the behavior of the tight contact structures we construct under geometric limits.
Minakshisundaram and Pleijel gave an asymptotic formula for the sum of squares of the pointwise values of the eigenfunctions of the Laplace-Beltrami operator on a compact Riemannian manifold, with eigenvalues less than a fixed number. Zelditch later extended this result by replacing the pointwise values with the Fourier coefficients of a smooth measure supported on a compact submanifold. Zelditch's result is very general, and his proof relies on the theory of Fourier integral operators. Here we give a proof based on methods of Riemannian geometry.
We discuss the shuffle product of the Schur multiple zeta values, which are the special values of Schur multiple zeta functions. We first define 2-labeled Schur posets to generalize Yamamoto’s integral expression of the multiple zeta values and consider the product of hook-type Schur multiple zeta values by using these posets. Then, for the derived terms, we introduce a modified Hurwitz-type Schur multiple zeta function of hook type, named an elementary factorial Schur multiple zeta function. Furthermore, we generalize 2-labeled Schur posets to consider the shuffle product of the elementary factorial Schur multiple zeta values and obtain an explicit formula for their shuffle product.
We present the results of our search for the orders of Tate-Shafarevich groups for the quadratic twists of elliptic curves. We formulate a general conjecture, giving for a fixed elliptic curve $E$ over $\Bbb Q$ and positive integer $k$, an asymptotic formula for the number of quadratic twists $E_d$, $d$ positive square-free integers less than $X$, with finite group $E_d(\Bbb Q)$ and $|\Sha(E_d(\Bbb Q))| = k^2$. This paper continues the authors previous investigations concerning orders of Tate-Shafarevich groups in quadratic twists of the curve $X_0(49)$. In section 8 we exhibit $88$ examples of rank zero elliptic curves with $|\Sha(E)| > 63408^2$, which was the largest previously known value for any explicit curve. Our record is an elliptic curve $E$ with $|\Sha(E)| = 1029212^2$.
Lucas's Theorem is about finding the result of a binomial coefficient modulo a prime p efficiently. The result is expressed as a product of binomial coefficients involving the base p expansions of the parameters of the original binomial coefficient. We give an elementary proof of Lucas's Theorem by deriving an analogous Vander-monde identity modulo a prime number.
The present study extends the geometric property, called quasi-polyhedrality (QP) associated with normed linear spaces to more general locally convex topological vector spaces (LCTVS). In LCTVS, we analyze these quasi-polyhedral convex bodies with the aid of generated cones and investigate the stability of QP-property under the product topology. Furthermore, we characterize the QP-points in dual normed linear spaces using the quotient space and an unbounded sequence of closed balls.
