We study almost gradient pseudo-Ricci solitons on Hopf hypersurfaces of non-flat complex space forms. We prove that such solitons are trivial gradient pseudo-Ricci solitons and that the hypersurface is pseudo-Einstein.
The (multi-)poly-Bernoulli numbers are defined by the generating series using the (multi-)polylogarithm series. Arakawa-Kaneko multiple zeta functions and Kaneko-Tsumura multiple zeta functions are known as generalizations of zeta functions, whose values at non-positive integers interpolate multi-poly-Bernoulli numbers. In this article, as an extension of (multi-)poly-Bernoulli numbers, inspired by Nakasuji-Phuksuwan-Yamazaki [6], we introduce Schur type multi-poly-Bernoulli numbers. Further, we give the expression of Schur type Arakawa-Kaneko multiple zeta functions in terms of Schur type multi-poly-Bernoulli numbers. Moreover, under the restriction to the hook type, we introduce hook type Kaneko-Tsumura multiple zeta functions and obtain expressions in terms of hook type multi-poly-Bernoulli numbers.
Fix a prime number p and an abelian field k. Let Uk∞ (resp. Ck∞) be the projective limit of the semi-local units at p (resp. of the cyclotomic units) of the cyclotomic Zp-extension k∞/k. We prove that the μ-invariant of the Zp[[T]]-module Uk∞/Ck∞ is zero for all p and k. We also show that, for p=2 and any imaginary abelian field k, the μ-invariant of (Uk∞/Ck∞)+ is [k:Q]/2 where (Uk∞/Ck∞)+ denotes the maximal quotient of Uk∞/Ck∞ on which the complex conjugation acts trivially.
In 1990, Gérard-Tahara [4] introduced the Briot-Bouquet type partial differential equation t∂tu=F(t,x,u,∂xu). In [21] the author showed existence of holomorphic and singular solutions of the following type of difference-differential equations tDqu=F(t,x,u,∂xu) when the characteristic exponent ρ(0)≠(qN−1)/(q−1) holds. In this paper the author constructs singular solutions with specific forms whenever ρ(0)=(qN−1)/(q−1).
We construct an infinite family of pairs of distinct imaginary biquadratic fields and pairs of distinct imaginary cyclic quartic fields with the same discriminant and regulator. We also construct an infinite family of imaginary biquadratic fields and imaginary cyclic quartic fields with the same regulator. Moreover, we give examples of a pair of distinct imaginary biquadratic fields and a pair of distinct imaginary cyclic quartic fields with the same discriminant, regulator and class number.
As an analogue to the explicit formula in the stable case, the asymptotic behavior at the origin of the renormalized zero resolvent of one-dimensional Lévy processes is studied under certain regular variation conditions on the Lévy-Khinchin exponent and the Lévy measure.
The present paper is devoted to the problem about the reduction of hyperelliptic functions of genus 3. Our research was motivated by applications to the theory of equations and dynamical systems integrable in hyperelliptic functions. In this paper, we consider a hyperelliptic curve of genus 3 which admits a morphism of degree 2 to an elliptic curve. We express the hyperelliptic functions associated with the curve of genus 3 in terms of the Weierstrass elliptic functions and hyperelliptic functions of genus 2.
The class of semi-hereditary rings is an important class of rings in theories that do not assume the Noetherian condition, such as perfectoid ring theory. We prove several results concerning the structure theory of this class, focusing on the relationship between semi-hereditary rings and the flatness of torsion-free modules. We also consider Shimomoto's problem concerning the flatness of the Frobenius map.
Let E/Q be an elliptic curve. We study the behavior of the Tate–Shafarevich group of E under quadratic extensions Q(D)/Q. By analyzing the cokernel of the restriction map, without assuming the finiteness of the Tate–Shafarevich group, we prove that the ratio #Ш(E/Q(D))[4]#Ш(ED/Q)[2] and #Ш(ED/Q)[2] can, under some conditions on E/Q, grow arbitrarily large simultaneously, where ED denotes the quadratic twist of E by D. For elliptic curves of the form E:y2=x3+px with p≡1mod4 being an odd prime, assuming the finiteness of the relevant Tate–Shafarevich groups, we prove that #Ш(E/Q(D))[2]≤4 and Ш(ED/Q)[2]=0 for infinitely many square-free integers D with −D being a prime number. Additionally, Ш(E/Q(−D))[2]≠0 for all D when p=257.
For the fourth Painlevé transcendents we derive elliptic asymptotic representations, which were announced by late Professor Kapaev without proofs, and an alternative elliptic expression as well. In our calculation WKB analysis is applied to the isomonodromy linear system, and special technical devices are necessary. Then we newly obtain related results on the correction function.
For any integer k, M.Kaneko defined k-th poly-Bernoulli numbers as a kind of generalization of classical Bernoulli numbers using k-th polylogarithm. In case when k is positive, k-th poly-Bernoulli numbers is a sequence of rational numbers as same as classical Bernoulli numbers. On the other hand, in case when k is negative, it is a sequence of positive integers, and many combinatoric and number theoretic properties has been investigated. In the present paper, the negative case is treated, and their congruence and p-adic properties are discussed. Beside of them, application of the results to obtain a congruence property for the number of lonesum matrices is also mentioned.
Knopp explored various properties of rational period functions for group SL2(Z) in relation to modular integrals on SL2(Z). In their study of Hecke-Weil correspondence, Hawkins and Knopp defined two types of modular integrals on a group Gamma 0+(p). When p is an element of{2, 3}, Choi and Kim introduced specific rational period functions for Gamma 0+(p)and established connections with one of the modular integrals on Gamma 0+(p) to obtain some properties of the rational period functions for Gamma 0+(p) which generalize results given by Knopp. In this paper, we define new rational period functions for Gamma 0+(p) that are associated with the other modular integrals on Gamma 0 +(p). Additionally, we discover several properties of these new rational period functions for Gamma 0+ (p).
Let ) is an element of ]0, 1] . We give a new introduction to MacKinney's )-expansions of real numbers in semi-regular continued fractions (1907) and show that Nakada's alpha-expansions are special cases of them. In addition, we define conjugate )-expansions and characterize )-expansions and conjugate )-expansions among semi-regular continued fractions. Then we prove that Lagrange's theorem holds for the )-expansions and conjugate )-expansions of quadratic irrational numbers. Finally, we show that the formula giving the irrationality exponent for regular continued fractions remains valid for )-expansions when ) >= 12.
The logarithmic inequality of Brezis-Gallou & euml;t-Wainger type has been generalized in various ways. In this paper we improve some of its generalizations by changing the involved function. Our results are concerned with the Besov and Triebel-Lizorkin spaces as well as the Besov-Morrey and Triebel-Lizorkin-Morrey spaces.
Let l be a prime less than 10000. We give a method for constructing a quintic cyclic field with ideal class group of l-rank at least 4. We also prove that there are infinitely many such quintic cyclic fields.
A long standing open question is any weak solution of the 3D steady Navier-Stokes equations satisfying the finiteness of the Dirichlet integral is trivial. In this paper, compared with the previous results, we study how the sign of some quantities affects the Liouville problem. In particular, a very novel sign condition is obtained: We give a partial answer the above open question under the non-positive generalized total head pressure by a very elemental method. Furthermore, we give some sign conditions on Liouville type theorems. It is shown that if div(pv) = v del p >= 0 or v del|v| >= 0, or the stretching factor of the vorticity magnitude is non-positive for a.e x is an element of R3, then it holds that v equivalent to 0, where v and p is the velocity vector and the scalar pressure, respectively.
We prove that every compact Sasakian manifold of dimension 2n + 1 whose horizontal sectional curvatures are non-negative and such that 2n is a simple eigenvalue of the Ricci operator admits no non-trivial Codazzi tensors. The same fact is true under suitable pinching assumptions on the phi-sectional curvatures.
We consider the Stark conjecture and Shintani's conjecture for abelian Artin L-functions. We calculate the Stark unit by those of intermediate fields. As an application, we show that Shintani's conjecture holds for C2n 2 C2-isoclinic extensions.
In general, the Hardy-Littlewood maximal operator M is not bounded on BMO( n). In fact, Bennett et al. (Ann. of Math. 1981) showed that for any given f is an element of BMO( n), there are two situations that may happen: either M(f) equivalent to infinity or M(f) is an element of BMO( n), and in the latter case, ||M(f)||BMO <= c||f||BMO. Similar situations come up when M is replaced by the Littlewood-Paley operators. It is known that CMO( n), a proper subspace of BMO( n), is the predual space of the Hardy space H1(n) and plays an important role in the study of the compactness of commutators. In this paper, we study the behavior of the Hardy-Littlewood maximal operator and the Littlewood-Paley operators, acting on CMO( n). A complete answer is given. We prove that if f is an element of CMO( n), either Mf equivalent to infinity or Mf is an element of CMO( n). Similar results are also obtained for Littlewood-Paley operators. It is worth mentioning that several new examples are constructed, which are highly non-trivial.
The real Grassmann manifold Gr4(Rn) of all oriented 4-dimensional subspaces in Rn is known to have the quaternionic K & auml;hler structure. We consider the set H2(Cn) of all 2-dimensional complex subspaces in Cn that are isotropic with respect to the standard complex inner product. H2(Cn) is a complex manifold of dimension 2n - 7 and has the holomorphic contact structure. The natural projection of H2(Cn) onto Gr4(Rn) is a twistor fibration. Through H2(Cn), we show the relation between the geometry regarded as complex Lie sphere geometry and that of totally complex submanifolds of Gr4(Rn). Further we give characterizations of specific Legendrian submanifolds of H2(Cn) by the properties of their curvature spheres.
