In [4], a framework for the definition of Donaldson-Thomas type invariants for Calabi-Yau categories endowed with a stability structure is developed. One of the key features of this setup is a wall-crossing formula for these invariants, describing their behaviour under a change of stability structure in terms of a factorization formula for automorphisms of certain Poisson algebras defined using...

A method for analysing a class of divergent series is developed from the Euler– Maclaurin summation formula. The conditions that the summand must satisfy are explored, and a significant simplification is obtained for cases where the summation ranges over all integers. As an example, we consider the Ewald representation for Schlömilch series, and show that this includes Twersky’s dual series for...

The main goal of this paper is to prove a formula that expresses the limit behaviour of Dedekind zeta functions for Re s > 1/2 in families of number fields, assuming that the Generalized Riemann Hypothesis holds. This result can be viewed as a generalization of the Brauer–Siegel theorem. As an application we obtain a limit formula for Euler–Kronecker constants in families of number fields.

We study a trace formula for tamely ramified abelian varieties A over a complete discretely valued field, which expresses the Euler characteristic of the special fiber of the Néron model of A in terms of the Galois action on the l-adic cohomology of A. If A has purely additive reduction, the trace formula yields a cohomological interpretation for the number of connected components of the specia...

Romberg extrapolation is a long-known method to improve the convergence rate of the trapezoidal rule on intervals. For simple regions such as the cube [Formula: see text] it is directly transferable to cubature in q dimensions. In this paper, we formulate Romberg extrapolation for Euler summation-based cubature on arbitrary q-dimensional regular regions [Formula: see text] and derive an explici...

Abstract. We study multivariable (bilateral) basic hypergeometric series associated with (type A) Macdonald polynomials. We derive several transformation and summation properties for such series, including analogues of Heine’s 2φ1 transformation, the q-Pfaff-Kummer and Euler transformations, the q-Saalschütz summation formula, and Sear’s transformation for terminating, balanced 4φ3 series. For ...

We present several formulas for the intersection multiplicity numbers considered in [3]. These numbers are associated to any dimensional cell of the common intersection of the tropical hypersurfaces. The first formula involves a sum of generalized mixed volumes, and turns out to be equal to the absolute value of the Euler characteristic of some complex toric complete intersection. Another formu...

These notes provide a self-contained account of the combinatorial theory of intrinsic volumes for polyhedral cones. Streamlined derivations of the General Steiner formula, the conic analogues of the Brianchon-Gram-Euler and the Gauss-Bonnet relations, and the Principal Kinematic Formula are given. In addition, a connection between the characteristic polynomial of a hyperplane arrangement and th...

We revisit old conjectures of Fermat and Euler regarding representation of integers by binary quadratic form x2 + 5y2. Making use of Ramanujan’s 1ψ1 summation formula we establish a new Lambert series identity for ∑∞ n,m=−∞ q n2+5m2 . Conjectures of Fermat and Euler are shown to follow easily from this new formula. But we don’t stop there. Employing various formulas found in Ramanujan’s noteboo...

A notion of curvature is introduced in multivariable operator theory. The curvature invariant of a Hilbert module over C[z(1),., z(d)] is a nonnegative real number which has significant extremal properties, which tends to be an integer, and which is hard to compute directly. It is shown that for graded Hilbert modules, the curvature agrees with the Euler characteristic of a certain finitely gen...