Many Sparse Cuts via Higher Eigenvalues

Cheeger's fundamental inequality states that any edge-weighted graph has a vertex subset $S$ such that its expansion (a.k.a. conductance) is bounded as follows: \phi(S) \defeq \frac{w(S,\bar{S})}{\min \set{w(S), w(\bar{S})}} \leq 2\sqrt{\lambda_2} where $w$ is the total edge weight of a subset or a cut and $\lambda_2$ is the second smallest eigenvalue of the normalized Laplacian of the graph. Here we prove the following natural generalization: for any integer $k \in [n]$, there exist $ck$ disjoint subsets $S_1, ..., S_{ck}$, such that $$\max_i \phi(S_i) \leq C \sqrt{\lambda_{k} \log k}$$ where $\lambda_i$ is the $i^{th}$ smallest eigenvalue of the normalized Laplacian and $c0$ are suitable absolute constants. Our proof is via a polynomial-time algorithm to find such subsets, consisting of a spectral projection and a randomized rounding. As a consequence, we get the same upper bound for the small set expansion problem, namely for any $k$, there is a subset $S$ whose weight is at most a \bigO(1/k) fraction of the total weight and $\phi(S) \le C \sqrt{\lambda_k \log k}$. Both results are the best possible up to constant factors. The underlying algorithmic problem, namely finding $k$ subsets such that the maximum expansion is minimized, besides extending sparse cuts to more than one subset, appears to be a natural clustering problem in its own right

Comparative study between Direction of arrival for wide band & narrow band Signal using Music Algorithm

Direction of arrival is a key parameter in array signal processing. It is one of the important problem in field such as sonar, radar and wireless communication. Traditional DOA estimation algorithm consists of large no of snapshot and are not reliable in application such as underwater array processing. There are many sources such as seismic wave ,acoustic signals, speech and signal processing which is wide band signal and estimation parameters such as snapshot ,side lobes, resolution is an important task. In the recent advancement of technology wide band signal are more favoured over narrow band signals. Wide band signal are able to estimate  DoAs efficiently with less side lobes and snapshots. In this paper a comparative analysis of direction of arrival for wide band and narrow band by analysing angular spectrum of MUSIC algorithm. We will   estimate the position of spectral with different scanning grid size. We will search the spectral peak position and estimates final DOA Therefore it become important to study and analyzed wide band signal specially application such as 5G m-MIMO systems

Comparative study between Direction of arrival for wide band & narrow band Signal using Music Algorithm

Direction of arrival is a key parameter in array signal processing. It is one of the important problem in field such as sonar, radar and wireless communication. Traditional DOA estimation algorithm consists of large no of snapshot and are not reliable in application such as underwater array processing. There are many sources such as seismic wave ,acoustic signals, speech and signal processing which is wide band signal and estimation parameters such as snapshot ,side lobes, resolution is an important task. In the recent advancement of technology wide band signal are more favoured over narrow band signals. Wide band signal are able to estimate  DoAs efficiently with less side lobes and snapshots. In this paper a comparative analysis of direction of arrival for wide band and narrow band by analysing angular spectrum of MUSIC algorithm. We will   estimate the position of spectral with different scanning grid size. We will search the spectral peak position and estimates final DOA Therefore it become important to study and analyzed wide band signal specially application such as 5G m-MIMO systems

Dynamics and Control of Ball and Beam System

In this paper modeling and dynamics of Ball and Beam system have been studied and presented. Ball beam system is very useful system is widely used in control system laboratory as an experimental arrangement. Its basic principles of control is similar to control principles used in many industrial applications. So understanding control of ball and beam system makes one to understand and design control strategy for industrial application. Due to such resemblances and simplicity this system has been widely used and studied and controlled using different techniques. Here in this paper system dynamics and control of the system using PD controlled is presented

The Complexity of Approximating Vertex Expansion

We study the complexity of approximating the vertex expansion of graphs $G = (V,E)$, defined as $$\Phi^V := \min_{S \subset V} n \cdot \frac{|N(S)|}{|S| |V \backslash S|}.$$ We give a simple polynomial-time algorithm for finding a subset with vertex expansion $O(\sqrt{OPT \log d})$ where $d$ is the maximum degree of the graph. Our main result is an asymptotically matching lower bound: under the Small Set Expansion (SSE) hypothesis, it is hard to find a subset with expansion less than $C\sqrt{OPT \log d}$ for an absolute constant $C$. In particular, this implies for all constant $\epsilon > 0$, it is SSE-hard to distinguish whether the vertex expansion $< \epsilon$ or at least an absolute constant. The analogous threshold for edge expansion is $\sqrt{OPT}$ with no dependence on the degree; thus our results suggest that vertex expansion is harder to approximate than edge expansion. In particular, while Cheeger's algorithm can certify constant edge expansion, it is SSE-hard to certify constant vertex expansion in graphs. Our proof is via a reduction from the {\it Unique Games} instance obtained from the \SSE hypothesis to the vertex expansion problem. It involves the definition of a smoother intermediate problem we call {\sf Analytic Vertex Expansion} which is representative of both the vertex expansion and the conductance of the graph. Both reductions (from the UGC instance to this problem and from this problem to vertex expansion) use novel proof ideas

Optical Coherence Tomography and its application in prognosis of disease through ayurveda.

A review of OCT as diagnostic tool and its application in ayurved. Optical Coherence Tomography (OCT) is one of the elite diagnostic measures in modern science which not only helps in diagnosing diseases but also in establishing relation between modern science and ancient science. OCT is usually used in diagnosing and managing diseases like Diabetic Macular Edema (DME), Myopia, Diabetic Retinopathy (DR), Central Serous Retinopathy (CSR), Glaucoma, etc. Depending on the disease condition; result of OCT can be co-related with modern aspect as well as ancient aspect. The primary objective of this literary review is concerned with gunas of vataj, pittaj, kaphaj dosha with clinical findings as seen in OCT. Ayurvedic classics state that guna of vata, guna of pitta, guna of pitta, guna of kapha are evident in shrotas and shrotojanya vyadhi. In the present era changes in lifestyle, food habits, uninhibited use of steroids had led to disorders of retina which is visible as changes in normative findings of OCT. Therefore, a proper understanding of doshas and its guna will help in decoding the findings of OCT with ayurvedic perspective

Viddha karma in timira roga - A single case study

Conditions with gradual loss of vision leading to blindness, is considered as Timira (Myopic Astigmatism). The clinical features of timira are dominated by the type of dosha vitiated where as the severity of the disease is dependant upon the number of patalas involved. As per Acharya Vagbhatta Drishti Mandal of Netra is developed from Kapha and Rakta[1]. Drishti Indriya is also developed from Teja Mahabhuta[2]. A case study of Timira Roga had been taken for understanding the effect of Vidhha Karma in Timira roga presented by a 21-year old female who came to Shalakya Netra roga OPD in  Dr. D.Y. Patil College of Ayurved &amp; Research Centre, Pimpri, Pune – 18 of Dr. D.Y. Patil Vidyapeeth, Pimpri, Pune (Deemed to be University), Maharashtra, India. Alochaka pitta is situated in netra. Rakta dhatu is the ashray sthana of pitta dosha as per ashrayaashrayi bhaav. In Timira roga, vitiated dosha which is located at the patala comes out with the rakta by the help of sira-vedhana. Sira vedhana or viddha karma causes samprapti- bhang of timira roga and gives clear vision to the patient. In vidhha karma, avyakta rakta srava is always attained, therefore vidhha karma is useful in timira roga