Double scaling limit for the O(N)3-invariant tensor model

We study the double scaling limit of the O(N)3-invariant tensor model, initially introduced in Carrozza and Tanasa, Lett. Math. Phys. (2016). This model has an interacting part containing two types of quartic invariants, the tetrahedric and the pillow one. For the 2-point function, we rewrite the sum over Feynman graphs at each order in the 1/N expansion as a finite sum, where the summand is a function of the generating series of melons and chains (a.k.a. ladders). The graphs which are the most singular in the continuum limit are characterized at each order in the 1/N expansion. This leads to a double scaling limit which picks up contributions from all orders in the 1/N expansion. In contrast with matrix models, but similarly to previous double scaling limits in tensor models, this double scaling limit is summable. The tools used in order to prove our results are combinatorial, namely a thorough diagrammatic analysis of the Feynman graphs, as well as an analytic analysis of the singularities of the relevant generating series.

A study on product-sum of triangular fuzzy numbers

We study the problem: if a˜i, i ∈ N are fuzzy numbers of triangular form, then what is the membership function of the infinite (or finite) sum -˜a1 + a˜2 + · · · (defined via the sub-product-norm convolution)

FINITE SUM OF COMPOSITION OPERATORS ON FOCK SPACE

We investigate unbounded, linear operators arising from a finite sum of composition operators on Fock space. Real symmetry and complex symmetry of these operators are characterised.

The inflationary wavefunction from analyticity and factorization

We study the analytic properties of tree-level wavefunction coefficients in quasi-de Sitter space. We focus on theories which spontaneously break dS boost symmetries and can produce significant non-Gaussianities. The corresponding inflationary correlators are (approximately) scale invariant, but are not invariant under the full conformal group. We derive cutting rules and dispersion formulas for the late-time wavefunction coefficients by using factorization and analyticity properties of the dS bulk-to-bulk propagator. This gives a unitarity method which is valid at tree-level for general n-point functions and for fields of arbitrary mass. Using the cutting rules and dispersion formulas, we are able to compute n-point functions by gluing together lower-point functions. As an application, we study general four-point, scalar exchange diagrams in the EFT of inflation. We show that exchange diagrams constructed from boost-breaking interactions can be written as a finite sum over residues. Finally, we explain how the dS identities used in this work are related by analytic continuation to analogous identities in Anti-de Sitter space.

Finite Sums Involving Reciprocals of the Binomial and Central Binomial Coefficients and Harmonic Numbers

We prove some finite sum identities involving reciprocals of the binomial and central binomial coefficients, as well as harmonic, Fibonacci and Lucas numbers, some of which recover previously known results, while the others are new.

Cyclostationary Approach to the Analysis of the Power in Electric Circuits under Periodic Excitations

This paper proposes a cyclostationary based approach to power analysis carried out for electric circuits under arbitrary periodic excitation. Instantaneous power is considered to be a particular case of the two-dimensional cross correlation function (CCF) of the voltage across, and current through, an element in the electric circuit. The cyclostationary notation is used for deriving the frequency domain counterpart of CCF—voltage–current cross spectrum correlation function (CSCF). Not only does the latter exhibit the complete representation of voltage–current interaction in the element, but it can be systematically exploited for evaluating all commonly used power measures, including instantaneous power, in the form of Fourier series expansion. Simulation examples, which are given for the parallel resonant circuit excited by the periodic currents expressed as a finite sum of sinusoids and periodic train of pulses with distorted edges, numerically illustrate the components of voltage–current CSCF and the characteristics derived from it. In addition, the generalization of Tellegen’s theorem, suggested in the paper, leads to the immediate formulation of the power conservation law for each CSCF component separately.

Katyusha Acceleration for Convex Finite-Sum Compositional Optimization

Structured optimization problems arise in many applications. To efficiently solve these problems, it is important to leverage the structure information in the algorithmic design. This paper focuses on convex problems with a finite-sum compositional structure. Finite-sum problems appear as the sample average approximation of a stochastic optimization problem and also arise in machine learning with a huge amount of training data. One popularly used numerical approach for finite-sum problems is the stochastic gradient method (SGM). However, the additional compositional structure prohibits easy access to unbiased stochastic approximation of the gradient, so directly applying the SGM to a finite-sum compositional optimization problem (COP) is often inefficient. We design new algorithms for solving strongly convex and also convex two-level finite-sum COPs. Our design incorporates the Katyusha acceleration technique and adopts the mini-batch sampling from both outer-level and inner-level finite-sum. We first analyze the algorithm for strongly convex finite-sum COPs. Similar to a few existing works, we obtain linear convergence rate in terms of the expected objective error; from the convergence rate result, we then establish complexity results of the algorithm to produce an ε-solution. Our complexity results have the same dependence on the number of component functions as existing works. However, because of the use of Katyusha acceleration, our results have better dependence on the condition number κ and improve to [Formula: see text] from the best-known [Formula: see text]. Finally, we analyze the algorithm for convex finite-sum COPs, which uses as a subroutine the algorithm for strongly convex finite-sum COPs. Again, we obtain better complexity results than existing works in terms of the dependence on ε, improving to [Formula: see text] from the best-known [Formula: see text].

On \phi prime submodules

In this paper, we study some properties of φ-prime submodules andwe give another charactrization for it. For given submodules N and K of a moduleM with K ⊆ N, we prove that N is φ-prime submodule if and only if N/Kis φ_K-prime submodule. Finally, we show that any finite sum of φ-prime submodules isφ-prime.