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R. Balasubramanian

Publications and source records attributed to R. Balasubramanian.

At least 19 recordsLinked to original sources
arXiv2025

The $m$-th Element of a Sidon Set

We prove that if $A=\{a_1,\dots ,a_{|A|}\}\subset \{1,2,\dots ,n\}$ is a Sidon set so that $|A|=n^{1/2}-L^\prime$, then $$a_m = m\cdot n^{1/2} + \mathcal O\left( n^{7/8}\right) + \mathcal O\left(L^{1/2}\cdot n^{3/4}\right)$$ where $L=\max\{0,L^\prime\}$. As an application of this, we give easy proofs of some previously derived results. We proceed on to proving that for a dense Sidon set $S$ and for any $\varepsilon >0$, we have $$\sum_{a\in S} a = \frac 12 n^{3/2} + \mathcal O \left (n^{11/8} \right )$$ for all $n\le N$ but at most $\mathcal O_{\varepsilon} \left (N^{\frac 45 + \varepsilon} \right )$ exceptions.

math.NT↗
arXiv2023

Soft Soil Gait Planning and Control for Biped Robot using Deep Deterministic Policy Gradient Approach

Biped robots have plenty of benefits over wheeled, quadruped, or hexapod robots due to their ability to behave like human beings in tough and non-flat environments. Deformable terrain is another challenge for biped robots as it has to deal with sinkage and maintain stability without falling. In this study, we are proposing a Deep Deterministic Policy Gradient (DDPG) approach for motion control of a flat-foot biped robot walking on deformable terrain. We have considered a 7-link biped robot for our proposed approach. For soft soil terrain modeling, we have considered triangular Mesh to describe its geometry, where mesh parameters determine the softness of soil. All simulations have been performed on PyChrono, which can handle soft soil environments.

cs.RO↗
arXiv2023

Programmatic Innovations that Accord with the Retention of Women in STEM Careers

Gender representation in the physical sciences remains inequitable and continues to lag behind other fields. Even though there exists adequate documentation regarding programmatic postures and innovations, difficulties persist within the physics discipline. In this paper, we present innovative, programmatic implementations over an eight-year period at an undergraduate, liberal arts physics program. Some of these innovations accord with nationally-published, `best practices' for undergraduate physics programs, while others are novel to the program's context. Within this eight-year period, alterations are separated into curricular, co-curricular, and experiential elements. Innovations are introduced in some detail, and data are presented before, during, and after their introduction. While it is currently impossible to say which elements had the greatest impact, the synergistic combination did have a positive effect on the program. Not only did the number of total majors and graduates increase, there was a 200% increase of women degree recipients compared to the previous ten years, which boosted average graduation rates above the national average (30% > 20%). Women were retained within the undergraduate physics major at a higher percentage during this time period when compared to men in the program. Lastly, these women physics majors maintained careers in science advancement fields at a rate of ~90% after $\leq5$ years post-graduation. While this paper presents a singular case study, the purpose is two-fold: a) to validate quantitatively the work of national physics organizations within the context of a liberal arts institution, and b) to suggest that a multi-level approach is most efficacious when considering programmatic innovations.

physics.ed-ph↗
arXiv2021

Cycloidal Trajectory Realization on Staircase based on Neural Network Temporal Quantized Lagrange Dynamics (NNTQLD) with Ant Colony Optimization for a 9-Link Bipedal Robot

In this paper, a novel optimal technique for joint angles trajectory tracking control with energy optimization for a biped robot with toe foot is proposed. For the task of climbing stairs by a 9-link biped model, a cycloid trajectory for swing phase is proposed in such a way that the cycloid variables depend on the staircase dimensions. Zero Moment Point(ZMP) criteria is taken for satisfying stability constraint. This paper mainly can be divided into 3 steps: 1) Planning stable cycloid trajectory for initial step and subsequent step for climbing upstairs and Inverse Kinematics using an unsupervised artificial neural network with knot shifting procedure for jerk minimization. 2) Modeling Dynamics for Toe foot biped model using Lagrange Dynamics along with contact modeling using spring-damper system followed by developing Neural Network Temporal Quantized Lagrange Dynamics which takes inverse kinematics output from neural network as its inputs. 3) Using Ant Colony Optimization to tune PD (Proportional Derivative) controller parameters and torso angle with the objective to minimize joint space trajectory errors and total energy consumed. Three cases with variable staircase dimensions have been taken and a brief comparison is done to verify the effectiveness of our proposed work Generated patterns have been simulated in MATLAB .

cs.RO↗
arXiv2020

Planning Brachistochrone Hip Trajectory for a Toe-Foot Bipedal Robot going Downstairs

A novel efficient downstairs trajectory is proposed for a 9 link biped robot model with toe-foot. Brachistochrone is the fastest descent trajectory for a particle moving only under the influence of gravity. In most situations, while climbing downstairs, human hip also follow brachistochrone trajectory for a more responsive motion. Here, an adaptive trajectory planning algorithm is developed so that biped robots of varying link lengths, masses can climb down on varying staircase dimensions. We assume that the center of gravity (COG) of the biped concerned lies on the hip. Zero Moment Point (ZMP) based COG trajectory is considered and its stability is ensured. Cycloidal trajectory is considered for ankle of the swing leg. Parameters of both cycloid and brachistochrone depends on dimensions of staircase steps. Hence this paper can be broadly divided into 4 steps 1) Developing ZMP based brachistochrone trajectory for hip 2) Cycloidal trajectory planning for ankle by taking proper collision constraints 3) Solving Inverse kinematics using unsupervised artificial neural network (ANN) 4) Comparison between the proposed, a circular arc and a virtual slope based hip trajectory. The proposed algorithms have been implemented using MATLAB.

cs.RO↗
arXiv2020

Beyond the extended Selberg class: $1<d_F< 2$

We show that a class of Dirichlet series ${\mathfrak{A}}^{\#}$ that is much larger than the extended Selberg class ${\mathscr{S}}^{\#}$, and also contains the standard as well as the tensor product, exterior square and symmetric square $L$-functions of automorphic $L$-functions of $GL_n$ over number fields, does not have any elements of degrees between $1$ and $2$. The proof of our more general theorem is very different from the proof of Kaczorowski and Perelli for the class ${\mathscr{S}}^{\#}$, and is much shorter and simpler even in that case.

math.NT↗
arXiv2017

On Freiman's 3k-4 theorem

One of the many theorems Freiman proved, in the second half of the twentieth century, in the subject which later came to be known as "structure theory of set addition", was 'Freiman's $3k-4$ theorem' for subsets of $\Z$. In this article we introduce concept of a new `structure' on finite subsets of integers. Sets with this structure are quite useful in additive number theory in some contexts. Also we give some criterion for subsets to posses this structure. Then this is used to establish an analog of Freiman's $3k-4$ theorem for the groups $\Z \times G,$ where $G$ is any abelian group.

math.CO↗
arXiv2017

Text Recognition in Scene Image and Video Frame using Color Channel Selection

In recent years, recognition of text from natural scene image and video frame has got increased attention among the researchers due to its various complexities and challenges. Because of low resolution, blurring effect, complex background, different fonts, color and variant alignment of text within images and video frames, etc., text recognition in such scenario is difficult. Most of the current approaches usually apply a binarization algorithm to convert them into binary images and next OCR is applied to get the recognition result. In this paper, we present a novel approach based on color channel selection for text recognition from scene images and video frames. In the approach, at first, a color channel is automatically selected and then selected color channel is considered for text recognition. Our text recognition framework is based on Hidden Markov Model (HMM) which uses Pyramidal Histogram of Oriented Gradient features extracted from selected color channel. From each sliding window of a color channel our color-channel selection approach analyzes the image properties from the sliding window and then a multi-label Support Vector Machine (SVM) classifier is applied to select the color channel that will provide the best recognition results in the sliding window. This color channel selection for each sliding window has been found to be more fruitful than considering a single color channel for the whole word image. Five different features have been analyzed for multi-label SVM based color channel selection where wavelet transform based feature outperforms others. Our framework has been tested on different publicly available scene/video text image datasets. For Devanagari script, we collected our own data dataset. The performances obtained from experimental results are encouraging and show the advantage of the proposed method.

cs.CV↗
arXiv2016

On correlations of certain multiplicative functions

In this paper, we study sums of shifted products $\sum\limits_{n \leq x} F(n) G(n-h)$ for any $|h| \leq x/2$ and arithmetic functions $F=f*1$ and $G=g*1$, with $f$ and $g$ small. We obtain asymptotic formula for different orders of magnitude of $f$ and $g$. We also provide asymptotic formula for sums of the type $\sum\limits_{n \leq x} μ^2(n) G(n-h)$, where $G=g*1$ and $g$ is small. For small order of magnitudes of $f$ and $g$, we improve the error terms and make them independent of $h$.

math.NT↗
arXiv2016

On the number of factorizations of an integer

Let $f(n)$ denote the number of unordered factorizations of a positive integer $n$ into factors larger than $1$. We show that the number of distinct values of $f(n)$, less than or equal to $x$, is at most $\exp \left( C \sqrt{\frac{\log x}{\log \log x}} \left( 1 + o(1) \right) \right)$, where $C=2π\sqrt{2/3}$ and $x$ is sufficiently large. This improves upon a previous result of the first author and F. Luca.

math.NT↗
arXiv2016

Sum-free subsets of finite abelian groups of type III

A finite abelian group $G$ of cardinality $n$ is said to be of type III if every prime divisor of $n$ is congruent to 1 modulo 3. We obtain a classification theorem for sum-free subsets of largest possible cardinality in a finite abelian group $G$ of type III. This theorem, when taken together with known results, gives a complete characterisation of sum-free subsets of the largest cardinality in any finite abelian group $G$. We supplement this result with a theorem on the structure of sum-free subsets of cardinality "close" to the largest possible in a type III abelian group $G$. We then give two applications of these results. Our first application allows us to write down a formula for the number of orbits under the natural action of ${\rm Aut}(G)$ on the set of sum-free subsets of $G$ of the largest cardinality when $G$ is of the form $({\mathbf{Z}}/m{\mathbf{Z}})^r$, with all prime divisors of $m$ congruent to 1 modulo 3, thereby extending a result of Rhemtulla and Street. Our second application provides an upper bound for the number of sum-free subsets of $G$. For finite abelian groups $G$ of type III and with {\em a given exponent} this bound is substantially better than that implied by the bound for the number of sum-free subsets in an arbitrary finite abelian group, due to Green and Ruzsa.

math.NT↗
arXiv2015

Catalan's Conjecture over Number Fields

Catalan conjecture/Mihailescu theorem is a theorem in number theory that was conjectured by Mathematician Eugene Charles Catalan in 1844 and was proved completely by Preda Mihailescu in 2005. Some form of problem dates back atleast to Gersonides who seems to have proved a special case of the conjecture in 1343. The note stating the problem was not given the due imprtance at the begining and appeared among errata to papers which had appeared in the earlier volume of Crelle journal, however the problem got its due considration after work of Cassles and Ko Chao in 1960s. The Catalan problem asks that the equation $x^m-y^n=1$ has no solution for x,y,m,n in +ve integers other than the trivial solution $ 3^2-2^3=1 $. An important and first ingredient for the proof is Cassles criteria which says that whenever we have a solution of $x^p-y^q=1$ with p,q primes then $q|x$ and $p|y$ . Here we look a generalization of the problem, namely we will consider the equation $x^p-y^q=1$ where x,y takes value in ring of integers ${O}_K$ of a number field K and p,q are rational primes. In this article we supply a possible formulation of Cassles criterion and a proof for that in some particular cases of number fields. After this work one can expect to follow Mihailescu and Characterize solutions of Catalan over number fields.

math.NT↗
arXiv2015

On Selberg's approximation to the twin prime problem

In his Classical approximation to the Twin prime problem, Selberg proved that for $x$ sufficiently large, there is an $n \in (x,2x)$ such that $2^{Ω(n)}+2^{Ω(n+2)} \leq λ$ with $λ=14$, where $Ω(n)$ is the number of prime factors of $n$ counted with multiplicity. This enabled him to show that for infinitely many $n$, $n(n+2)$ has atmost $5$ prime factors, with one having atmost $2$ and the other having atmost $3$ prime factors. By adopting Selberg's approach and using a refinement suggested by Selberg, we improve this value of $λ$ to about $λ=12.59$.

math.NT↗
arXiv2014

Mean-Value of Product of Shifted Multiplicative Functions and Average Number of Points on Elliptic Curves

In this paper, we consider the mean value of the product of two real valued multiplicative functions with shifted arguments. The functions $F$ and $G$ under consideration are close to two nicely behaved functions $A$ and $B$, such that the average value of $A(n-h)B(n)$ over any arithmetic progression is only dependent on the common difference of the progression. We use this method on the problem of finding mean value of $K(N)$, where $K(N)/\log N$ is the expected number of primes such that a random elliptic curve over rationals has $N$ points when reduced over those primes.

math.NT↗
arXiv2014

On Additive Representation Functions

Let $\A=\{a_1<a_2<a_3.....<a_n<...\}$ be an infinite sequence of integers and let $R_2(n)=|\{(i,j):\ \ a_i+a_j=n;\ \ a_i,a_j\in \A;\ \ i\le j\}|$. We define $S_k=\s_{l=1}^k(R_2(2l)-R_2(2l+1))$. We prove that, if $L^{\infty}$ norm of $S_k^+(=\max\{S_k,0\})$ is small then $L^1$ norm of $\frac{S_k^+}{k}$ is large.

math.NT↗