Adiabatic limits of η-invariants and the Meyer functions

We construct a function on the orbifold fundamental group of the moduli space of smooth theta divisors, which we call the Meyer function for smooth theta divisors. In the construction, we use the adiabatic limits of the η-invariants of the mapping torus of theta divisors. We shall prove that the Meyer function for smooth theta divisors cobounds the signature cocycle, and we determine the values of the Meyer function for the Dehn twists. In particular, we give an analytic construction of the Meyer function of genus two.     

Solvable extensions of naturally graded quasi-filiform Leibniz algebras of second type ℒ1 and ℒ3

For sequences of naturally graded quasi-filiform Leibniz algebras of second type ?¹ and ?³ introduced by Camacho et al., all possible right and left solvable indecomposable extensions over the field ? are constructed so that these algebras serve as the nilradicals of the corresponding solvable algebras. The construction continues Winternitz’ and colleagues’ program established to classify solvable Lie algebras using special properties rather than trying to extend one dimension at a time.     

λ-invariants and Γ-transforms

In this paper we give the relationship between the -invariants of two power series which are naturally associated to the -transform of ap-adic measure. Since thep-adicL-functions over arise as such -transforms, we obtain information about the minus part of Iwasawa's -invariant for the basic _p -extension of an abelian CM-field.I would like to thank W. Sinnott, R. Gold and T. Dowling for their comments and suggestions.     

σk-Scalar curvature and eigenvalues of the Dirac operator

On a four-dimensional closed spin manifold (M ⁴, g), the eigenvalues of the Dirac operator can be estimated from below by the total σ₂-scalar curvature of M ⁴ as follows: Equality implies that (M ⁴, g) is a round sphere and the corresponding eigenspinors are Killing spinors.Dedicated to Professor Wang Guangyin on the occasion of his 80th birthday.     

On eigenvalues of Seidel matrices and Haemers’ conjecture

For a graph G, let S(G) be the Seidel matrix of G and \({\theta }_1(G),\ldots ,{\theta }_n(G)\) be the eigenvalues of S(G). The Seidel energy of G is defined as \(|{\theta }_1(G)|+\cdots +|{\theta }_n(G)|\). Willem Haemers conjectured that the Seidel energy of any graph with n vertices is at least \(2n-2\), the Seidel energy of the complete graph with n vertices. Motivated by this conjecture, we prove that for any \(\alpha \) with \(0<\alpha <2,|{\theta }_1(G)|^\alpha +\cdots +|{\theta }_n(G)|^\alpha \geqslant (n-1)^\alpha +n-1\) if and only if \(|\hbox {det}\,S(G)|\geqslant n-1\). This, in particular, implies the Haemers’ conjecture for all graphs G with \(|\hbox {det}\,S(G)|\geqslant n-1\). A computation on the fraction of graphs with \(|\hbox {det}\,S(G)|<n-1\) is reported. Motivated by that, we conjecture that almost all graphs G of order n satisfy \(|\hbox {det}\,S(G)|\geqslant n-1\). In connection with this conjecture, we note that almost all graphs of order n have a Seidel energy of order \(\Theta (n^{3/2})\). Finally, we prove that self-complementary graphs G of order \(n\equiv 1\pmod 4\) have \(\det S(G)=0\).     

On the second-order Fréchet derivatives of eigenvalues of Sturm–Liouville problems in potentials

Archiv der Mathematik - The works of V.A. Vinokurov have shown that eigenvalues and normalized eigenfunctions of Sturm–Liouville problems are analytic in potentials, considered as mappings...     

Fluctuations of eigenvalues and second order Poincaré inequalities

Linear statistics of eigenvalues in many familiar classes of random matrices are known to obey gaussian central limit theorems. The proofs of such results are usually rather difficult, involving hard computations specific to the model in question. In this article we attempt to formulate a unified technique for deriving such results via relatively soft arguments. In the process, we introduce a notion of ‘second order Poincaré inequalities’: just as ordinary Poincaré inequalities give variance bounds, second order Poincaré inequalities give central limit theorems. The proof of the main result employs Stein’s method of normal approximation. A number of examples are worked out, some of which are new. One of the new results is a CLT for the spectrum of gaussian Toeplitz matrices.     

Continuous variation of eigenvalues and Gärding's inequality

Let D_t be a smooth 1-parameter family of elliptic self-adjoint partial differential operators on a compact manifold without boundary. We use Gärding's inequality to give a simple proof that the kth eigenvalue of D_t is continuous in t.     

Derivatives of Frobenius and Derivatives of Hodge–Tate Weights

In this paper we study the derivatives of Frobenius and the derivatives of Hodge–Tate weights for families of Galois representations with triangulations. We generalize the Fontaine–Mazur L-invariant and use it to build a formula which is a generalization of the Colmez–Greenberg–Stevens formula. For the purpose of proving this formula we show two auxiliary results called projection vanishing property and "projection vanishing implying L-invariants" property.     

Convergence of dynamics and the Perron–Frobenius operator

We complete the picture how the asymptotic behavior of a dynamical system is reflected by properties of the associated Perron–Frobenius operator. Our main result states that strong convergence of the powers of the Perron–Frobenius operator is equivalent to setwise convergence of the underlying dynamic in the measure algebra. This situation is furthermore characterized by uniform mixing-like properties of the system.     

Generalization of Mirsky’s theorem on diagonals and eigenvalues of matrices

Mirsky proved that, for the existence of a complex matrix with given eigenvalues and diagonal entries, the obvious necessary condition is also sufficient. We generalize this theorem to matrices over any field and provide a short proof. Moreover, we show that there is a unique companion-matrix-type solution for this problem.     

Directional approximation of functions of two variables in the spaces ℒ p (R2) and ℒ p (R + 2 )

Let r ∈ N, α, t ∈ R, x ∈ R ², f: R ² → C, and denote $ \Delta _{t,\alpha }^r (f,x) = \sum\limits_{k = 0}^r {( - 1)^{r - k} c_r^k f(x_1 + kt\cos \alpha ,x_2 + kt\sin \alpha ).} $ In this paper, we investigate the relation between the behavior of the quantity $ \left\| {\int\limits_E {\Delta _{t,\alpha }^r (f, \cdot )\Psi _n (t)dt} } \right\|_{p,G} , $ as n → ∞ (here, E ? R, G ∈ {R ², R ₊ ² }, and ψ _n ∈ L ₁(E) is a positive kernel) and structural properties of function f. These structural properties are characterized by its “directional” moduli of continuity: $ \omega _{r,\alpha } (f,h)_{p,G} = \mathop {\sup }\limits_{0 \leqslant t \leqslant h} \left\| {\Delta _{t,\alpha }^r (f)} \right\|_{p,G} . $ Here is one of the results obtained. Theorem 1. Let E and A be intervals in R ₊ such that A ? E, f ∈ L _p (G), α ∈ [0, 2π] when G =R ² and α ∈ [0, π/2] when G = R ₊ ² Denote Δ _{n, k} = ∫ _A t ^k ψ _n (t)dt. If there exists an r ∈ N such that, for any m ∈ N, we have Δ _{m, r} > 0, Δ _{m, r + 1} < ∞, and $ \mathop {\lim }\limits_{n \to \infty } \frac{{\Delta _{n,r + 1} }} {{\Delta _{n,r} }} = 0,\mathop {\lim }\limits_{n \to \infty } \Delta _{n,r}^{ - 1} \int\limits_{E\backslash A} {\Psi _n = 0} , $ then the relations $ \mathop {\lim }\limits_{n \to \infty } \Delta _{n,r}^{ - 1} \left\| {\int\limits_E {\Delta _{t,\alpha }^r (f, \cdot )\Psi _n dt} } \right\|_{p,G} \leqslant K, \mathop {\sup }\limits_{t \in (0,\infty )} t^r \omega _{r,\alpha } (f,t)_{p,G} \leqslant K $ are equivalent. Particular methods of approximation are considered. We establish Corollary 1. Let p, G, α, and f be the same as in Theorem 1, and $ \sigma _{n,\alpha } (f,x) = \frac{2} {{\pi n}}\int\limits_{R_ + } {\Delta _{t,\alpha }^1 (f,x)} \left( {\frac{{\sin \frac{{nt}} {2}}} {t}} \right)^2 dt. $ Then the relations $ \mathop {\underline {\lim } }\limits_{n \to \infty } \frac{{\pi n}} {{\ln n}}\left\| {\sigma _{n,\alpha } (f)} \right\|_{p,G} \leqslant K Let r ∈ N, α, t ∈ R, x ∈ R ², f: R ² → C, and denote In this paper, we investigate the relation between the behavior of the quantity as n → ∞ (here, E ⊂ R, G ∈ {R ², R ₊²}, and ψ _n ∈ L ₁(E) is a positive kernel) and structural properties of function f. These structural properties are characterized by its “directional” moduli of continuity: Here is one of the results obtained. Theorem 1. Let E and A be intervals in R ₊ such that A ⊂ E, f ∈ L _p (G), α ∈ [0, 2π] when G =R ² and α ∈ [0, π/2] when G = R ₊² Denote Δ _{n, k} = ∫ _A t ^k ψ _n (t)dt. If there exists an r ∈ N such that, for any m ∈ N, we have Δ _{m, r} > 0, Δ _{m, r + 1} < ∞, and then the relations are equivalent. Particular methods of approximation are considered. We establish Corollary 1. Let p, G, α, and f be the same as in Theorem 1, and Then the relations and are equivalent. Original Russian Text ? N.Yu. Dodonov, V.V. Zhuk, 2008, published in Vestnik Sankt-Peterburgskogo Universiteta. Seriya 1. Matematika, Mekhanika, Astronomiya, 2008, No. 2, pp. 23–33.     

Riemannian submersions, δ-invariants, and optimal inequality

In this study, we establish a sharp relation between δ-invariants and Riemannian submersions with totally geodesic fibers. By using this relationship, we establish an optimal inequality involving δ-invariants for submanifolds of the complex projective space CP ^m (4) via Hopf’s fibration ${\pi:S^{2m+1}\to CP^{m}(4)}$ . Moreover, we completely classify submanifolds of complex projective space which satisfy the equality case of the inequality.     

On Iwasawa λ−-invariants of imaginary abelian fields

The value distribution of λ⁻-invariants for imaginary abelian fields (and for a fixed prime number p) will be studied.     

Frobenius Partitions and the Combinatorics of Ramanujan's 1ψ1 Summation

We examine the combinatorial significance of Ramanujan's famous summation. In particular, we prove bijectively a partition theoretic identity which implies the summation formula.     

Isomorphisms and projections for quotient-spaces of ℒ1-spaces by their reflexive subspaces

Let Z₁=X₁/E₁ and Z₂=X₂/E₂, where X₁ and X₂ are ₁-spaces, E₁cX₁, E₂c X₂. In this paper we study the following questions: 1) under what conditions are Z₁ and Z₂ isomorphic; 2) under what conditions is Z₁ isomorphic to a complemented subspace of Z₂. Some results: (a) if E₁ and E₂ are reflexive and Z₁. is isomorphic to Z₂, then one of the spaces E₁ E₂ is isomorphic to the product of the other by a finite-dimensional space; (b) if is a circle), E₁=H and E₂ is reflexive and X₂=Y^* for some Y, then it is impossible to imbed Z₁ in Z₂ as a complemented subspace.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 73, pp. 91–101, 1977.     

A note on Iwasawa µ-invariants of elliptic curves

Suppose that E ₁ and E ₂ are elliptic curves defined over ℚ and p is an odd prime where E ₁ and E ₂ have good ordinary reduction. In this paper, we generalize a theorem of Greenberg and Vatsal [3] and prove that if E ₁[p ⁱ ] and E ₂[p ⁱ ] are isomorphic as Galois modules for i = μ(E ₁), then μ(E ₁) ≤ μ(E ₂). If the isomorphism holds for i = μ(E ₁) + 1, then both the curves have same μ-invariants. We also discuss one numerical example.