arXiv subjects

N. Uday Kiran

Publications and source records attributed to N. Uday Kiran.

17 recordsLinked to original sources

arXiv2026

On Some Generalisations of Gauss Sequences

In this paper, we introduce integer sequences satisfying new congruence properties inspired by the Euler and Gauss congruences, which we call Euler--Gauss sequences. Noting that every Gauss sequence is an Euler--Gauss sequence, we compare them with certain generalisations of Gauss sequences and provide several counterexamples. Unlike Gauss sequences, Euler--Gauss sequences include sequences based on distinct prime factors, such as the Smallest Prime Factor and Greatest Prime Factor sequences (suitably defined at $1$). Moreover, we show that the prime-divisor subclass of Gauss sequences, given by $\sum_{p\mid n} pg_p$ for an integer sequence $(g_n)$, admits a natural extension to Euler--Gauss sequences of the form $\sum_{p\mid n} pf_p(\operatorname{rad}(n))$, where, for each prime $p$, $f_p$ is an integer-valued function and $\operatorname{rad}(n)$ denotes the square-free kernel of $n$. Further, we obtain $q$-analogs of the Euler--Gauss sequences, fill gaps in the literature on $q$-Gauss sequences, and conjecture a divisibility criterion for $q$-Euler--Gauss sequences, which we have verified computationally. We also show that not only do our $q$-Euler--Gauss sequences satisfy the Cyclic Sieving Phenomenon (CSP) exhibited by the $q$-Gauss sequences, but we also derive a new CSP condition for the SPF and GPF sequences, not hitherto known in the literature.

math.NT↗

arXiv2026

Some Generalizations of Totient Function with Elementary Symmetric Sums

We generalize certain totient functions using elementary symmetric polynomials and derive explicit product forms for the totient functions involving the second elementary symmetric sum. This work follows from the work of Toth [The Ramanujan Journal, 2022] where the totient function was generalized using the first and the kth elementary symmetric polynomial. We also provide some observations on the behavior of the totient function with an arbitrary jth elementary symmetric polynomial. We then outline a method for solving a certain the restricted linear congruence problem with a greatest common divisor constraint on a quadratic form, illustrated by a concrete example. Most importantly, we demonstrate the equivalence between obtaining product forms for generalized totient functions, counting zeros of specific polynomials over finite fields, and resolving a broad class of restricted linear congruence problems .

math.NT↗

arXiv2025

An Algebraic Generalization of the Ramanujan Sum

Ramanujan sums have attracted significant attention in both mathematical and engineering disciplines due to their diverse applications. In this paper, we introduce an algebraic generalization of Ramanujan sums, derived through polynomial remaindering. This generalization is motivated by its applications in Restricted Partition Theory and Coding Theory. Our investigation focuses on the properties of these sums and expresses them as finite trigonometric sums subject to a coprime condition. Interestingly, these finite trigonometric sums with a coprime condition, which arise naturally in our context, were recently introduced as an analogue of Ramanujan sums by Berndt, Kim, and Zahaescu. Furthermore, we provide an explicit formula for the size of Levenshtein codes with an additional parity condition (also known as Shifted Varshamov-Tenengolts deletion correction codes), which have found many interesting applications in studying the Little-Offord problem, DNA-based data storage and distributed synchronization. Specifically, we present an explicit formula for a particularly important open case $\text{SVT}_{t,b}(s \pm δ, 2s + 1)$ for $s$ or $s+1$ are divisible by $4$ and for small values of $δ$.

math.NT↗

arXiv2025

Further Generalization of Ramanujan Sums with Regular A-Functions

In the study of Ramanujan sums, the so-called regular $A$-function is a set-valued multiplicative function that tracks certain subsets of the divisor sets of natural numbers. McCarthy provided a generalization of the Ramanujan sum using these regular $A$-function based arithmetic convolutions. This approach has recently attracted considerable interest from several researchers. In this paper, we extend McCarthy's generalization by introducing two regular $A$-functions corresponding to both parameters in the Ramanujan sum. Fortunately, these sums exhibit several properties of the Ramanujan sums. We also generalize the greatest common divisor (GCD) function and the Von Sterneck formula. Our introduction of two regular $A$-functions into these expressions enables us to explore a novel perspective on the connection between these expressions and the order relation between the two regular $A$-functions. In particular, we establish the necessary and sufficient conditions for orthogonality and Dedekind-Hölder's identity (i.e., Ramanujan sum = Von Sterneck function) to hold. Our primary motivation for this further generalization proposed in this paper is expansions of arithmetic functions based on arbitrary regular $A$-functions. To the best of our knowledge, the expansions of arbitrary $A$-functions discussed here are new in the literature.

math.GM↗

arXiv2024

A Framework for Fluid Motion Estimation using a Constraint-Based Refinement Approach

Physics-based optical flow models have been successful in capturing the deformities in fluid motion arising from digital imagery. However, a common theoretical framework analyzing several physics-based models is missing. In this regard, we formulate a general framework for fluid motion estimation using a constraint-based refinement approach. We demonstrate that for a particular choice of constraint, our results closely approximate the classical continuity equation-based method for fluid flow. This closeness is theoretically justified by augmented Lagrangian method in a novel way. The convergence of Uzawa iterates is shown using a modified bounded constraint algorithm. The mathematical wellposedness is studied in a Hilbert space setting. Further, we observe a surprising connection to the Cauchy-Riemann operator that diagonalizes the system leading to a diffusive phenomenon involving the divergence and the curl of the flow. Several numerical experiments are performed and the results are shown on different datasets. Additionally, we demonstrate that a flow-driven refinement process involving the curl of the flow outperforms the classical physics-based optical flow method without any additional assumptions on the image data.

cs.CV↗

arXiv2023

Global Wellposedness of a Class of Weakly Hyperbolic Cauchy Problems with Variable Multiplicities on $\mathbb{R}^d$

We study a class of weakly hyperbolic Cauchy problems on $\mathbb{R}^d$, involving linear operators with characteristics of variable multiplicities, whose coefficients are unbounded in the space variable. The behaviour in the time variable is governed by a suitable "shape function". We develop a parameter-dependent symbolic calculus, corresponding to an appropriate subdivision of the phase space. By means of such calculus, a parametrix can be constructed, in terms of (generalized) Fourier integral operators naturally associated with the employed symbol class. Further, employing the parametrix, we prove $\mathscr{S}(\mathbb{R}^{d})$-wellposedness and give results about the global decay and regularity of the solution, within a scale of weighted Sobolev space.

math.AP↗

arXiv2023

The Restricted Partition and q-Partial Fractions

The restricted partition function $p_{N}(n)$ counts the partitions of $n$ into at most $N$ parts. In the nineteenth century Sylvester showed that these partitions can be expressed as a sum of $k$-periodic quasi-polynomials ($1\leq k\leq N$) which he termed as Waves. It is now well-known that one can easily perform a wave decomposition using a special type of partial fraction decomposition (the so-called $q$-partial fractions) of the generating function of $p_{N}(n)$. In this paper we show that the coefficients of these $q$-partial fractions can be expressed as a linear combination of the Ramanujan sums. In particular, we show, for the first time, an appearance of the degenerate Bernoulli numbers, the degenerate Euler numbers and a special generalization of the Ramanujan sums, which we term as a Gaussian-Ramanujan sum, in the formulae for certain waves. These coefficients not only provide a good approximation of $p_{N}(n)$ but they can also be used for obtaining good bounds. Further, we provide a combinatorial meaning to these sums. Our approach for partial fractions is based on a projection operator on the $I$-adic completion of the ring of polynomials, where $I$ is an ideal generated by the Cyclotomic polynomial.

math.NT↗

arXiv2022

On an Edge-Preserving Variational Model for Optical Flow Estimation

It is well known that classical formulations resembling the Horn and Schunck model are still largely competitive due to the modern implementation practices. In most cases, these models outperform many modern flow estimation methods. In view of this, we propose an effective implementation design for an edge-preserving $L^1$ regularization approach to optical flow. The mathematical well-posedness of our proposed model is studied in the space of functions of bounded variations $BV(Ω,\mathbb{R}^2)$. The implementation scheme is designed in multiple steps. The flow field is computed using the robust Chambolle-Pock primal-dual algorithm. Motivated by the recent studies of Castro and Donoho we extend the heuristic of iterated median filtering to our flow estimation. Further, to refine the flow edges we use the weighted median filter established by Li and Osher as a post-processing step. Our experiments on the Middlebury dataset show that the proposed method achieves the best average angular and end-point errors compared to some of the state-of-the-art Horn and Schunck based variational methods.

cs.CV↗

arXiv2022

Global Well-Posedness of a Class of Hyperbolic Cauchy Problems with Coefficients Sublogarithmic in Time

The goal of this paper is to study global well-posedness, cone of dependence and loss of regularity of the solutions to a class of strictly hyperbolic equations with coefficients displaying "mild" blow-up of sublogarithmic order - $|\ln t|^γ,γ\in(0,1).$ The problems we study are of strictly hyperbolic type with respect to a generic weight and a metric on the phase space. The coefficients are polynomially bound in $x$ with their $x$-derivatives and $t$-derivative of order $O(t^{-δ}),δ\in [0,1),$ and $O(t^{-1}|\ln t|^{γ-1}), γ\in(0,1),$ respectively. We employ the Planck function associated with the metric to subdivide the extended phase space and define appropriate generalized parameter dependent symbol classes. To arrive at an energy estimate, we perform a conjugation by a pseudodifferential operator. This operator explains the loss of regularity by linking it to the metric on the phase space and the singular behavior. We call the conjugating operator as loss operator. We report that the solution experiences an arbitrarily small loss in relation to the initial datum defined in the Sobolev space tailored to the loss operator.

math.AP↗

arXiv2022

Nonlinear Evolutionary PDE-Based Refinement of Optical Flow

The goal of this paper is to propose two nonlinear variational models for obtaining a refined motion estimation from an image sequence. Both the proposed models can be considered as a part of a generalized framework for an accurate estimation of physics-based flow fields such as rotational and fluid flow. The first model is novel in the sense that it is divided into two phases: the first phase obtains a crude estimate of the optical flow and then the second phase refines this estimate using additional constraints. The correctness of this model is proved using an evolutionary PDE approach. The second model achieves the same refinement as the first model, but in a standard manner, using a single functional. A special feature of our models is that they permit us to provide efficient numerical implementations through the first-order primal-dual Chambolle-Pock scheme. Both the models are compared in the context of accurate estimation of angle by performing an anisotropic regularization of the divergence and curl of the flow respectively. We observe that, although both the models obtain the same level of accuracy, the two-phase model is more efficient. In fact, we empirically demonstrate that the single-phase and the two-phase models have convergence rates of order $O(1/N^2)$ and $O(1/N)$ respectively.

cs.CV↗

arXiv2021

Global Well-posedness of a Class of Singular Hyperbolic Cauchy Problems

The goal of this paper is to establish a global well-posedness, cone condition and loss of regularity for singular hyperbolic equations with coefficients in { $L^1((0,T];C^\infty(\mathbb{R}^n)) \cap C^1((0,T];C^\infty(\mathbb{R}^n))$} and Cauchy data in an appropriate Sobolev space tailored to a metric on the phase space. The coefficients are unbounded near the singular hyperplane $t=0$ and polynomially growing as $|x| \to \infty.$ The singular behavior is characterized by the blow-up rate of the coefficients and their first $t$-derivatives near $t=0.$ In order to study the interplay of the singularity in $t$ and unboundedness in $x$, we consider a class of metrics on the phase space. Our methodology relies on the use of the Planck function associated to the metric to subdivide the extended phase and to define an infinite order pseudodifferential operator for the conjugation. We also give some counterexamples.

math.AP↗

arXiv2021

Energy Estimates and Global Well-posedness for a Broad Class of Strictly Hyperbolic Cauchy Problems with Coefficients Singular in Time

The goal of this paper is to establish a global well-posedness for a broad class of strictly hyperbolic Cauchy problems with coefficients in $C^2((0,T];C^\infty(\mathbb{R}^n))$ growing polynomially in $x$ and singular in $t$. The problems we study are of strictly hyperbolic type with respect to a generic weight and a metric on the phase space. The singular behavior is captured by the blow-up of the first and second $t$-derivatives of the coefficients which allows the coefficients to be either logarithmic-type or oscillatory-type near $t=0$. To arrive at an energy estimate, we perform a conjugation by a pseudodifferential operator of the form $e^{ν(t)Θ(x,D_x)},$ where $Θ(x,D_x)$ explains the quantity of the loss by linking it to the metric on the phase space and the singular behavior while $ν(t)$ gives a scale for the loss. We call the conjugating operator as {\itshape{loss operator}} and depending on its order we report that the solution experiences zero, arbitrarily small, finite or infinite loss in relation to the initial datum. We also provide a counterexample and derive the anisotropic cone conditions in our setting.

math.AP↗

arXiv2021

An Algebraic Approach to q-Partial Fractions and Sylvester Denumerants

In 1857 Sylvester established an elegant theory that certain counting functions (which he termed denumerants) are quasi-polynomials by decomposing them into periodic and non-periodic parts. Each component of the decomposition, called a Sylvester wave, corresponds to a root of unity. Recently several researchers, using either combinatorial arguments or complex analytic techniques, obtained explicit formulas for the waves. In this work, we develop an algebraic approach to the Sylvester's theory. Our methodology essentially relies on deriving $q$-partial fractions of the generating functions of the denumerants, and thereby obtain new explicit formulas for the waves. The formulas we obtain are expressed in terms of the degenerate Bernoulli numbers and a generalization of the Fourier-Dedekind sum. Further, we also prove certain reciprocity theorems of the generalized Fourier-Dedekind sums and a structure result on the top-order terms of the waves. The proofs rely on our evaluation operator and our far-reaching generalization of the Heaviside's cover-up method for partial fractions. SageMath code for this work is available in the public domain.

math.NT↗

arXiv2021

An Algebraic Approach to Degenerate Bernoulli Numbers

In this work we study the properties of a new algebraic variant of the degenerate Bernoulli polynomial $\tildeβ_{k}(m,x)$ and study the corresponding degenerate Bernoulli number $\tildeβ_{k}(m,1)=m^{k}β_{k}(1/m)$, where $β_{k}(λ), λ\neq 0$ is the standard degenerate Bernoulli number. Our approach relies on a new algebraic framework for generating functions and the action of a symbolic evaluation function on powers of polynomials. We show that $\tildeβ_{k}(m,x)$ displays surprising links with other mathematical objects (such as Circulant matrices and Galois fields) and enjoys many interesting algebraic, symmetric and dynamical properties that could be deployed to perform efficient computations.

math.NT↗

arXiv2021

Strictly Hyperbolic Cauchy Problems on $\mathbb{R}^n$ with Unbounded and Singular Coefficients

We investigate the behavior of the solutions of a class of certain strictly hyperbolic equations defined on $(0,T]\times \mathbb{R}^n$ in relation to a class of metrics on the phase space. In particular, we study the global regularity and decay issues of the solution to an equation with coefficients polynomially bound in $x$ with their $x$-derivatives and $t$-derivative of order ${O}(t^{-δ}),δ\in [0,1),$ and ${O}(t^{-1})$ respectively. This type of singular behavior allows coefficients to be either oscillatory or logarithmically bounded at $t=0$. We use the Planck function associated with the metric to subdivide the extended phase space and define an appropriate generalized parameter dependent symbol class. We report that the solution not only experiences a finite loss of derivatives but also a decay in relation to the initial datum defined in a Sobolev space tailored to the metric. Our analysis suggests that an infinite loss is quite expected when the order of singularity of the first time derivative of the leading coefficients exceeds $O(t^{-1})$. We confirm this by providing counterexamples. Further, using the $L^1$ integrability of the logarithmic singularity in $t$ and the global properties of the operator with respect to $x$, we derive the anisotropic cone conditions in our setting.

math.AP↗

arXiv2021

Global Well-Posedness of a Class of Strictly Hyperbolic Cauchy Problems with Coefficients Non-Absolutely Continuous in Time

We investigate the behavior of the solutions of a class of certain strictly hyperbolic equations defined on $[0,T]\times \R^n$ in relation to a class of metrics on the phase space. In particular, we study the global regularity and decay issues of the solution to an equation with coefficients polynomially bound in $x$ and with their $t$-derivative of order $\textnormal{O}(t^{-q}),$ where $q \in \big[1,\frac{3}{2}\big)$. For this purpose, an appropriate generalized symbol class based on the metric is defined and the associated Planck function is used to define an infinite order operator to perform conjugation. We demonstrate that the solution not only experiences a loss of regularity (usually observed for the case of coefficients bounded in $x$) but also a decay in relation to the initial datum defined in a Sobolev space tailored to the generalized symbol class. Further, we observe that a precise behavior of the solution could be obtained by making an optimal choice of the metric in relation to the coefficients of the given equation. We also derive the cone conditions in the global setting.

math.AP↗

arXiv2020

Some Partial Fraction Identities associated with the Cyclotomic Polynomials

We establish some partial fraction identities for rational functions whose denominators are implicit products of the cyclotomic polynomials. To achieve this, we first develop a general algebraic approach for partial fraction decomposition inspired by the Heaviside's cover-up method. We thus call our method the Extended Cover-Up Method. Using our method we obtain direct formulas for $q$-partial fractions for certain generating functions. As a direct consequence of our formulas one can compute the Sylvester denumerants, the Frobenius number and the Ehrhart polynomials in pseudo-polynomial time. Further, we provide a framework for a generalization of the Fourier-Dedekind sum and their associated Rademacher reciprocity theorem extending the results of Carlitz, Zagier and Gessel. By performing a Fourier analysis we demonstrate that our extended cover-up method explains in simple terms the mechanism behind the reciprocity law.

math.NT↗