Theoretical Verification of Formula for Charge Function in Time q = c * v in RC Circuit for Charging/Discharging of Fractional & Ideal Capacitor

Objective of this paper is verification of newly developed formula of charge storage in capacitor as q = c*v, in RC circuit, to get validation for ideal loss less capacitor as well as fractional order capacitors for charging and discharging cases. This new formula is different to usual and conventional way of writing capacitance multiplied by voltage to get charge stored in a capacitor i.e. q = cv. We use this new formulation i.e. q = c*v in RC circuits to verify the results that are obtained via classical circuit theory, for a case of classical loss less capacitor as well as fractional capacitor. The use of this formulation is suited for super-capacitors, as they show fractional order in their behavior. This new formula is used to get the ‘memory effect’ that is observed in self-discharging phenomena of super-capacitors-that memorizes its history of charging profile. Special emphasis is given to detailed derivational steps in order to clarity in usage of this new formula in the RC circuit examples. This paper validates the new formula of charge storage in capacitor i.e. q = c*v.

The spectral decomposition of shifted convolution sums over number fields

Letis obtained for any nonzero fractional ideal

On error distributions in ring-based LWE

Since its introduction in 2010 by Lyubashevsky, Peikert and Regev, the ring learning with errors problem (ring-LWE) has become a popular building block for cryptographic primitives, due to its great versatility and its hardness proof consisting of a (quantum) reduction from ideal lattice problems. But, for a given modulus$q$and degree$n$number field$K$, generating ring-LWE samples can be perceived as cumbersome, because the secret keys have to be taken from the reduction mod$q$of a certain fractional ideal${\mathcal{O}}_{K}^{\vee }\subset K$called the codifferent or ‘dual’, rather than from the ring of integers${\mathcal{O}}_{K}$itself. This has led to various non-dual variants of ring-LWE, in which one compensates for the non-duality by scaling up the errors. We give a comparison of these versions, and revisit some unfortunate choices that have been made in the recent literature, one of which is scaling up by${|\unicode[STIX]{x1D6E5}_{K}|}^{1/2n}$with$\unicode[STIX]{x1D6E5}_{K}$the discriminant of$K$. As a main result, we provide, for any$\unicode[STIX]{x1D700}>0$, a family of number fields$K$for which this variant of ring-LWE can be broken easily as soon as the errors are scaled up by${|\unicode[STIX]{x1D6E5}_{K}|}^{(1-\unicode[STIX]{x1D700})/n}$.

LOCAL ANALYSIS OF GRAUERT–REMMERT-TYPE NORMALIZATION ALGORITHMS

Normalization is a fundamental ring-theoretic operation; geometrically it resolves singularities in codimension one. Existing algorithmic methods for computing the normalization rely on a common recipe: successively enlarge the given ring in form of an endomorphism ring of a certain (fractional) ideal until the process becomes stationary. While Vasconcelos' method uses the dual Jacobian ideal, Grauert–Remmert-type algorithms rely on so-called test ideals. For algebraic varieties, one can apply such normalization algorithms globally, locally, or formal analytically at all points of the variety. In this paper, we relate the number of iterations for global Grauert–Remmert-type normalization algorithms to that of its local descendants. We complement our results by a study of ADE singularities. All intermediate singularities occurring in the normalization process are determined explicitly. Besides ADE singularities the process yields simple space curve singularities from the list of Frühbis-Krüger.

On the gcd of local Rankin–Selberg integrals for even orthogonal groups

We study the Rankin–Selberg integral for a pair of representations of ${\rm SO}_{2l}\times {\rm GL}_{n}$, where ${\rm SO}_{2l}$ is defined over a local non-Archimedean field and is either split or quasi-split. The integrals span a fractional ideal, and its unique generator, which contains any pole which appears in the integrals, is called the greatest common divisor (gcd) of the integrals. We describe the properties of the gcd and establish upper and lower bounds for the poles. In the tempered case we can relate it to the $L$-function of the representations defined by Shahidi. Results of this work may lead to a gcd definition for the $L$-function.

ON ⋆-COLON-MULTIPLICATION DOMAINS

Let ⋆ be a star operation on an integral domain R. An ideal A is a ⋆-colon-multiplication ideal if A⋆ = (B(A : B))⋆ for all fractional ideal B of R. We prove that every maximal ideal of R is a ⋆-colon-multiplication ideal if and only if R is a ⋆-CICD or R is a local ⋆-MTP domain. It is also shown that every ideal of R is ⋆-colon-multiplication if and only if R is a ⋆-CICD.

THE v-OPERATION IN EXTENSIONS OF INTEGRAL DOMAINS

An extension D ⊆ R of integral domains is strongly t-compatible (respectively, t-compatible) if (IR)-1 = (I-1R)v (respectively, (IR)v = (IvR)v) for every nonzero finitely generated fractional ideal I of D. We show that strongly t-compatible implies t-compatible and give examples to show that the converse does not hold. We also indicate situations where strong t-compatibility and its variants show up naturally. In addition, we study integral domains D such that D ⊆ R is strongly t-compatible (respectively, t-compatible) for every overring R of D.

Note on Colon-Multiplication Domains

LetRbe an integral domain with quotient fieldL.Call a nonzero (fractional) idealAofRa colon-multiplication ideal any idealA, such thatB(A:B)=Afor every nonzero (fractional) idealBofR.In this note, we characterize integral domains for which every maximal ideal (resp., every nonzero ideal) is a colon-multiplication ideal. It turns that this notion unifies Dedekind andMTPdomains.