Modular Invariance, Tauberian Theorems, and Microcanonical Entropy

We analyze modular invariance drawing inspiration from tauberian theorems. Given a modular invariant partition function with a positive spectral density, we derive lower and upper bounds on the number of operators within a given energy interval. They are most revealing at high energies. In this limit we rigorously derive the Cardy formula for the microcanonical entropy together with optimal error estimates for various widths of the averaging energy shell. We identify a new universal contribution to the microcanonical entropy controlled by the central charge and the width of the shell. We derive an upper bound on the spacings between Virasoro primaries. Analogous results are obtained in holographic 2d CFTs. We also study partition functions with a UV cutoff. Control over error estimates allows us to probe operators beyond the unity in the modularity condition. We check our results in the 2d Ising model and the Monster CFT and find perfect agreement.Comment: 39 pages, 9 figure

Factorization and complex couplings in SYK and in Matrix Models

We consider the factorization problem in toy models of holography, in SYK and in Matrix Models. In a theory with fixed couplings, we introduce a fictitious ensemble averaging by inserting a projector onto fixed couplings. We compute the squared partition function and find that at large $N$ for a typical choice of the fixed couplings it can be approximated by two terms: a "wormhole" plus a "pair of linked half-wormholes". This resolves the factorization problem. We find that the second, half-wormhole, term can be thought of as averaging over the imaginary part of the couplings. In SYK, this reproduces known results from a different perspective. In a matrix model with an arbitrary potential, we propose the form of the "pair of linked half-wormholes" contribution. In GUE, we check that errors are indeed small for a typical choice of the hamiltonian. Our computation relies on a result by Brezin and Zee for a correlator of resolvents in a "deterministic plus random" ensemble of matrices.Comment: 26 pages, 1 figur

Representation of grossone-based arithmetic in simulink for scientific computing

AbstractNumerical computing is a key part of the traditional computer architecture. Almost all traditional computers implement the IEEE 754-1985 binary floating point standard to represent and work with numbers. The architectural limitations of traditional computers make impossible to work with infinite and infinitesimal quantities numerically. This paper is dedicated to the Infinity Computer, a new kind of a supercomputer that allows one to perform numerical computations with finite, infinite, and infinitesimal numbers. The already available software simulator of the Infinity Computer is used in different research domains for solving important real-world problems, where precision represents a key aspect. However, the software simulator is not suitable for solving problems in control theory and dynamics, where visual programming tools like Simulink are used frequently. In this context, the paper presents an innovative solution that allows one to use the Infinity Computer arithmetic within the Simulink environment. It is shown that the proposed solution is user-friendly, general purpose, and domain independent

Computation of higher order Lie derivatives on the Infinity Computer

In this paper, we deal with the computation of Lie derivatives, which are required, for example, in some numerical methods for the solution of differential equations. One common way for computing them is to use symbolic computation. Computer algebra software, however, might fail if the function is complicated, and cannot be even performed if an explicit formulation of the function is not available, but we have only an algorithm for its computation. An alternative way to address the problem is to use automatic differentiation. In this case, we only need the implementation of the algorithm that evaluates the function in terms of its analytic expression in a programming language, but we cannot use this if we have only a compiled version of the function. In this paper, we present a novel approach for calculating the Lie derivative of a function, even in the case where its analytical expression is not available, that is based on the Infinity Computer arithmetic. A comparison with symbolic and automatic differentiation shows the potentiality of the proposed technique

Numerical methods for solving initial value problems on the Infinity Computer

New algorithms for the numerical solution of Ordinary Differential Equations (ODEs) with initial condition are proposed. They are designed for work on a new kind of a supercomputer – the Infinity Computer, – that is able to deal numerically with finite, infinite and infinitesimal numbers. Due to this fact, the Infinity Computer allows one to calculate the exact derivatives of functions using infinitesimal values of the stepsize. As a consequence, the new methods described in this paper are able to work with the exact values of the derivatives, instead of their approximations

Краниоспинальная менингиома вентролатерального расположения: клиническое наблюдение и обзор литературы

In this clinical case of surgical treatment of craniovertebral junction meningioma (CVJ-area) of ventrolateral location. Meningiomas of CVJ-region are relatively rare compared with intracranial meningiomas at other location. About 90% meningiomas of CVJ-region has ventral and ventrolateral location. Surgical treatment of tumors of this area is associated with a high risk of neurological complications. At the same time, surgical outcomes and postoperative projections depend on the degree of removal of these radical benign tumors. For the best exposure of the tumor and control of vital structures (major vessels, cranial and spinal nerves, the brain stem) requires the use of appropriate approaches to the pathological process. In this case, an example of the total removal of the ventrolateral craniospinal meningioma of the extreme lateral (far-lateral) access to almost complete regression of neurological symptoms in the postoperative period.Менингиомы области краниовертебрального перехода встречаются относительно редко в сравнении с интракраниальными менингиомами другой локализации. Около 90% менингиом данной локализации расположено вентрально и вентролатерально. Хирургическое лечение опухолей этой области сопряжено с высокими рисками неврологических осложнений. В то же время исходы хирургического лечения и послеоперационные прогнозы зависят от степени радикальности удаления этих доброкачественных опухолей. Для наилучшей экспозиции опухоли и контроля жизненно важных структур (магистральных сосудов, черепных и спинальных нервов, ствола головного мозга) требуется использование адекватных подходов к патологическому процессу. В представленном клиническом наблюдении приведен пример тотального удаления вентролатеральной кранио спинальной менингиомы из крайнелатерального (far-lateral) доступа с практически полным регрессом неврологической симптоматики в послеоперационном периоде

Minimal Liouville gravity correlation numbers from Douglas string equation

We continue the study of (q, p) Minimal Liouville Gravity with the help of Douglas string equation. We generalize the results of [1,2], where Lee-Yang series (2, 2s + 1) was studied, to (3, 3s + p 0) Minimal Liouville Gravity, where p 0 = 1, 2. We demonstrate that there exist such coordinates \u3c4 m,n on the space of the perturbed Minimal Liouville Gravity theories, in which the partition function of the theory is determined by the Douglas string equation. The coordinates \u3c4 m,n are related in a non-linear fashion to the natural coupling constants \u3bb m,n of the perturbations of Minimal Lioville Gravity by the physical operators O m,n . We find this relation from the requirement that the correlation numbers in Minimal Liouville Gravity must satisfy the conformal and fusion selection rules. After fixing this relation we compute three- and four-point correlation numbers when they are not zero. The results are in agreement with the direct calculations in Minimal Liouville Gravity available in the literature [3-5]. \ua9 2014 The Author(s)