Volume 8 | Issue 3 |

 Abstract

In today�s era, there is a large amount of increase in the crime rate due to the research gap between technologies and the optimal usage of Investigation. Identifying and analysing the patterns for crime prevention is one of the big challenge. Also, due to some technological limitations, having large amount of data it is difficult to analyse crimes. The goal of this paper is to propose employment of algorithms that works efficiently on large amount of data. This paper is concentrated on crime prevention by concerning various incidents occurred in various states.

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Creative Commons Attribution 4.0 and The Open Definition

 Keywords

Crime Analysis, Location Based Systems, Data Mining, Database, K Mean.

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Creative Commons Attribution 4.0 and The Open Definition

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Supply chain management in pharmaceutical industry is a complex process which comprises of managing tasks like manufacturing, storage, and sales of medical drugs. Though supply chain management is a challenge on every industry, in health care, compromised supply chain adds risk to consumers� safety. Increased adoption of technology, globalization and industry populated with multiple stakeholders in various jurisdictions has given rise to a complicated health supply chain. In this operation process, due to imbalance and asymmetry of information among the stakeholders arises a fraud problem such as compromising the consumer drug information. In such cases, in order to improve the security and reliability of drug information we suggest blockchain based data storage technology. In this paper we discuss about hoe fraudulent drug information problem can be resolved using blockchain and by the way presenting reliable information to the concerning consumers. We discuss about smart contract based on Consensus algorithm. Blockchain in pharmaceutical supply chain not only reduces the risk of counterfeiting and theft, but also allows efficient inventory management

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Creative Commons Attribution 4.0 and The Open Definition

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Supply chain, blockchain, pharmaceutical industry, proof of work, medical drug information, drug counterfeiting.

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Creative Commons Attribution 4.0 and The Open Definition

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ANPAD RAMPUR ME JAL SANSADHANO KA BHAUGOLIC VISHLESHAN

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Creative Commons Attribution 4.0 and The Open Definition

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ANPAD RAMPUR ME JAL SANSADHANO KA BHAUGOLIC VISHLESHAN

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Creative Commons Attribution 4.0 and The Open Definition

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Emotion recognition is that a part of speech recognition which is gaining more popularity and wish for it increases enormously. Although there are methods to know emotion using machine learning techniques, this project attempts to use deep learning and image classification method to acknowledge emotion and classify the emotion consistent with the speech signals. Our proposed model outperforms previous state-of-the-art methods in assigning data to at least one of 4 emotion categories (i.e., angry, happy, sad and neutral) when the model is applied to the RAVDESS dataset, as re?ected by accuracies starting from 68.8% to 71.8%. Various datasets are investigated and explored for training emotion recognition model are explained during this paper.

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Creative Commons Attribution 4.0 and The Open Definition

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Emotion recognition

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Creative Commons Attribution 4.0 and The Open Definition

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documentclass[11pt]{article} usepackage{fullpage} usepackage{amsfonts} usepackage{amsmath} usepackage{amssymb} usepackage{amsthm} ewtheorem{theorem}{Theorem}[section] egin{document} itle{Hardy Hilbert spaces and its applications} author{Praveen Sharma Department of Mathematics, University of Delhi} date{} maketitle section*{Abstract} In this paper , we discuss the Hardy Hilbert Space on the open disk with center origin and radius unity. We have proved that $H^2$ Space is isomorphic to proper subspace of $L^2$ Space which has various applications in Quantumm Mechanics. section{Preliminaries} subsubsection{Definition(Inner Product Space)} An extit{inner product space} is an vector space $W$ (over field $K=mathbb{R}ormathbb{C}$) with an inner product defined on it. Here, an inner product is an function $<,> : W imes W o K$ which satisfies the following properties:-- egin{enumerate} item $<alpha u+v,w>hspace{1cm} =hspace{1cm} alpha<u,w> + <v,w>$ item $ overline{<u,v>} hspace{2cm} =hspace{1cm} <v,u> $ item $<u,u> hspace{2cm} geq 0$ item $ <u,u> = 0 hspace{1cm} Leftrightarrow u = 0$ hspace{1cm} ( extbf{ for all scalers $alphain K$ and for all vectors $u,v,win W$}) paragraph{Note1} extbf{ Every inner product space is an normed spaces with the norm induced by the inner product is given by} egin{center} $||u|| = sqrt{<u,u>}$ end{center} paragraph{Note2} extbf{An normed space $(W,||.||)$ is said to be complete if each cauchy sequence coverges in $W$.} end{enumerate} subsection{Hilbert Space} An Hilbert Space is defined as the complete inner product space.smallskip extbf{Example:-} smallskip egin{center} egin{large} extbf{$l^2 = {(x_0,x_1,ldots ) : x_n in mathbb{C}, sum_{n=0}^infty |x_n|^2 <infty }$}end{large} end{center} i.e. all the elements of $l^2$ are the sequence of all the complex numbers that are square-summable. Inner product on $l^2$ is given by :- egin{center} extbf{$ <(x_n)_{n=0}^infty , (y_n)_{n=0}^infty> = sum_{n=0}^infty{x_noverline{y_n}}$} hspace{1cm} ( extsc{it is an hilbert sequence space)} end{center} subsection{Definition(Orthonormal sets and sequences)} An subset $X$ of an inner product space is said to be orthonormal if for all $u, v in X $ we have , $$ <u,v> = left{egin{array}{ll} 0 & mathrm{if} u eq v ||u||^2 & mathrm{if} u=v end{array} ight. .$$ paragraph{Note} extbf{If norm of each element of an orthogonal set $X$ is 1 then the set is said to be orthogonal. i.e for all $u,v in X$ we have, } $$ <u,v> = left{ egin{array}{ll} 0 & mathrm{if} u eq v 1 & mathrm{if} u =v end{array} ight. $$ subsection{Definition(Orthonormal basis)} An orthonormal subset $X$ of Hilbert space $W$ is said to be an orthonormal basis if span of $X$ is dense in $W$.i.e. $$overline{spanX} = W $$ paragraph{Note} Every Hilbert space $W eq{0}$ has an orthonormal basis. subsection{Definition(Separable Hilbert Space)} An Hilbert-Space $W$ is said to be extit{separable} if there exist an countable set which is dense in $W$.smallskip extbf{Example:} $l^2$ is an separable Hilbert space paragraph{Note} Each orthonormal basis of an separable Hilbert space are countable.Therefore orthonormal basis of $l^2$ are countable subsection*{Recall} egin{enumerate} item An orthonormal sequence $(e_n)_{n=0}^infty$ is an orthonormal basis of an Hilbert - Space $W$ egin{LARGE}$$Leftrightarrow$$end{LARGE} for all $uin W$ we havehspace{2cm}$ sum_{n=0}^infty|<u,e_n>|^2 = ||u||^2$ cite{kreyszig1978introductory} hspace{0.5cm} extbf{Parseval identity} item Let $(e_n)$ be an orthonormal sequence in an Hilbert-space then $$ sum_{n=0}^inftyalpha_ne_n$$ converges in $W$ $$iff$$ the series $$sum_{n=0}^infty|alpha_n|^2$$ converges in $mathbb{R}$ end{enumerate} section{THE HARDY-HILBERT SPACE} subsection{DEFINITION} { It is defined as the space of all the analytic functions which have a power series representation about origin with square-summable complex coefficients. It is denoted by extbf{ $H^2$}. $$ H^2 = { f : f(z) = sum_{n=0}^infty alpha_nz^n : sum_{n=0}^infty|alpha_n|^2 <infty}$$ Inner Product on $H^2$ is given by $$<f,g> =sum_{n=0}^infty a_noverline{b_n}$$ for $ f(z) = sum_{n=0}^infty a_nz^n hspace{0.5cm} and hspace{0.5cm} g(z) = sum_{n=0}^infty b_nz^nhspace{0.3cm} in hspace{0.2cm} H^2$ egin{theorem} extbf{The Hardy-Hilbert space is an separable Hilbert Space.} end{theorem} egin{proof} Define an function;-- $$ phi:l^2 o H^2$$ given by $$ (a_n)_{n=0}^infty o sum_{n=0}^infty a_nz^n$$ egin{itemize} item extbf{underline{ $phi$ is well defined}} since hspace{1cm} $ (a_n)_{n=0}^infty in l^2 Rightarrow sum_{n=0}^infty |a_n|^2 <infty Rightarrow sum_{n=0}^infty a_nz^n $ hspace{1cm} which being an power series is an analytic function whose coefficients are square summable hence is in $H^2$ $ herefore phi$ is well defined item extbf{underline{Clearly $phi$ is linear}} item extbf{underline{ $phi$ is isometric}} Fix $(a_n)_{n=0}^infty in l^2 $ then we have $$phi((a_n)_{n=0}^infty) = ||sum_{n=0}^infty a_nz^n||_{H^2} = sqrt{sum_{n=0}^infty |a_n|^2} =||(a_n)_{n=0}^infty||_{l^2} $$ $ herefore phi$ is an isometric $ herefore phi$ preserves the norm so that the inner product item extbf{since isometry property implies one one property$ herefore phi$ is one one } cite{bhatia2009notes} item extbf{underline{$phi$ is onto}}smallskip Let $f in H^2$ then $f(z) = sum_{n=0}^infty a_nz^n$ where $sum_{n=0}^infty|a_n|^2<infty$medskip extbf{ define} $x=(a_0,a_1,ldots)$ extbf{Since} $$||x||^2 =sum_{n=0}^infty |a_n|^2 <infty$$ extbf{$$ herefore xin l^2$$} extbf{and}$$phi(x) = f $$ extbf{$$ hereforephi hspace{0.5cm} is hspace{0.5cm} onto$$} end{itemize} extbf{Therefore $phi$ is an vector space isomorphism which also preserves the inner product. Since $l^2$ is an separable Hilbert space hence $H^2$ is also an separable Hilbert Space } end{proof} paragraph{Notations} $mathbb{D}= {z:|z|<1}$ denotes the open unit disk about origin in $mathbb{C}$ $mathbb{S^1} ={z:|z|=1}$ denotes the unit circle about origin in $mathbb{C}$ egin{theorem} extbf{Radius of convergence of each function in $H^2$ is atleast $1$} (i.e. each function in $H^2$ is analytic in the open unit disk $mathbb{D} $) end{theorem} egin{proof} Let $z_0in mathbb{D}$ is fixed $Rightarrowhspace{0.5cm} |z_0|<1$ $ herefore$ the geometric series $sum_{n=0}^infty |z_0|^n$ converges. Let $fin H^2$ is arbitrary.Then $$f(z) = sum_{n=0}^infty a_nz^n hspace{2cm} where hspace{1cm} sum_{n=0}^infty|a_n|^2 <infty$$ Since the series $sum_{n=0}^infty|a_n|^2$ converges $Rightarrow$ $|a_n|^2 longrightarrow 0 Rightarrow|a_n|longrightarrow 0$ $ herefore (|a_n|)_{n=0}^infty$ is an convergent sequence hence bounded. $ herefore exists M>0 $ such that $$ |a_n|leq M hspace{2cm} forall hspace{1cm} ngeq0$$ Now $$sum_{n=0}^infty|a_nz_0^n| leq Msum_{n=0}^infty|z_0|^n $$ where being an geometric series right hand side converges. $ herefore$ By Comparison test the series $ sum_{n=0}^infty a_nz_0^n$ converges absolutely . Since in Hilbert space absolute convergence implies convergence. $ herefore$ the series $sum_{n=0}^infty a_nz_0^n$ converges in $H^2$ since $z_0in H^2$ is arbitrary $ herefore$ each function in $H^2$ is analytic in the unit disk $mathbb{D}$ end{proof} subsection{Definition ($L^2(mathbb{S^1})$ space)} It is defined as the space of all the equivalence classes of functions cite{royden2010real} that are Lebesgue measurable on $S^mathbb{1}$ and square integrable on $S^mathbb{1}$ with respect to Lebesgue measure normalized such that measure of $S^mathbb{1}$ is $1$. $$ L^2(S^mathbb{1}) = {f: f hspace{0.2cm} ishspace{0.2cm} Lesbesgue hspace{0.2cm} measurable hspace{0.2cm} onhspace{0.2cm} mathbb{S^1} hspace{0.2cm} andhspace{0.3cm} frac{1}{2pi}int_0^{2pi}|f(e^{iota heta)}|^2d heta<infty } $$ Inner product on $L^2(mathbb{S^1})$ is given by - $$<f,g> = frac{1}{2pi}int_0^{2pi} f(e^{iota heta})overline{g(e^{iota heta})}d heta$$ paragraph{Note} $L^2(mathbb{S^1})$ is an Hilbert-space with the orthonormal basis given by ${e_n: nin mathbb{Z}}$ where $e_n(e^{iota heta})= e^{iota n heta}.$ extbf{Therefore} $$L^2(mathbb{S^1}) =left {f:f=sum_{n=-infty}^{n=infty}<f,e_n>e_n ight}.$$.cite{martinez2007introduction} subsubsection{Definition ($widehat{H^2}$ space)} $$widehat{H^2} = {f in L^2(mathbb{S^1}): <f,e_n> =0hspace{0.2cm} forhspace{0.2cm} negativehspace{0.2cm} valuehspace{0.2cm} ofhspace{0.2cm} n }$$ $$widehat{H^2} = left{f in L^2(mathbb{S^1}) : f= sum_{n=0}^infty <f,e_n>e_n ight }.$$ extbf{$widehat{H^2}$ is an subspace of $L^2(mathbb{S^1})$ whose negative Fourier coefficients are 0 $ herefore $ ${e_n : n=0,1,ldots }$ are orthonormal basis of $widehat{H^2}$} egin{theorem} egin{LARGE} extbf{$widehat{H^2}$ is an Hilbert-space}end{LARGE} end{theorem} egin{proof} Let $f in overline{widehat{H^2}}$ then there exist an sequence $(f_n)_{n=0}^infty$medskip such that hspace{2cm} $f_n longrightarrow f$ as $nlongrightarrowinfty$medskip Since hspace{2cm} $f_nin widehat{H^2}hspace{1cm} forall hspace{1cm} ngeq0$medskip $ herefore hspace{3cm} <f_n,e_k>=0 hspace{1cm} forall ngeq0 hspace{1cm} and hspace{1cm} forall k<0$medskip extbf{Now for each $k<0$ we have}medskip $|<f_n,e_k> - <f,e_k>| leq |<(f_n-f,e_k>| leq||f_n - f||longrightarrow 0 hspace{0.3cm} ashspace{0.1cm} nlongrightarrowinfty$(Schwarz Inequality cite{kreyszig1978introductory})medskip since $ hspace{3cm} <f_n,e_k>=0 hspace{1cm} forall ngeq0 hspace{1cm} Rightarrow hspace{1cm} <f,e_k>=0 $medskip Since $k<0$ is arbitrary $ herefore hspace{1cm}<f,e_k>=0 hspace{1cm}forallhspace{1cm} k<0$medskip $$ hereforehspace{1cm} fin widehat{H^2}$$ extbf{Therefore $widehat{H^2}$ is an closed subspace of $L^2(mathbb{S^1})$ Hence an Hilbert-Space} end{proof} egin{theorem} egin{large} extbf{The Hardy-Hilbert space can be identified as a subspace of $L^2(mathbb{S^1})$} end{large} end{theorem} egin{proof} Define an function extbf{$$psi:H^2 owidehat{H^2}$$} $$f o ilde{f}$$ where $f(z)=sum_{n=0}^infty a_nz^n$hspace{1cm} and hspace{1cm} $ ilde{f}=sum_{n=0}^infty a_ne_n$ egin{itemize} item extbf{underline{$psi$ is well defined}}medskip Let $f in H^2$ hspace{0.5cm} Then hspace{0.5cm} $f(z)=sum_{n=0}^infty a_nz^n$ hspace{0.5cm} where hspace{0.5cm} $sum_{n=0}^infty |a_n|^2 <infty$medskip Then by extbf{(recall 2)} the series $ ilde{f} =sum_{n=0}^infty a_ne_n $ converges in $widehat{H^2}$medskip $ hereforepsi$ hspace{0.5cm} is hspace{0.5cm} wellhspace{0.5cm} defined item extbf{underline{Clearly $psi$ is linear}}medskip item extbf{underline{$psi$ is an isometry}}medskip For any arbitrary $fin H^2$ where $f(z)=sum_{n=0}^infty a_nz^n$ we have:- $$ || psi(f)|| = || ilde{f}|| = frac{1}{2pi}int_0^{2pi}| ilde{ f}(e^{iota heta}|^2d heta $$medskip extbf{Now} $$frac{1}{2pi}int_0^{2pi} | ilde{f}(e^{iota heta})|^2 d heta = frac{1}{2pi}int_0^{2pi}(sum_{n=0}^infty a_ne^{iota n heta})(overline{sum_{m=0}^infty a_me^{iota m heta}}) $$ hspace{7cm} = $frac{1}{2pi}int_0^{2pi}sum_{n=0}^infty sum_{m=0}^infty a_noverline{a_m}e^{iota(n-m) heta} d heta$ hspace{7cm} = $sum_{n=0}^infty |a_n|^2$ hspace{1cm} extbf{(since $frac{1}{2pi} int_0^{2pi} e^{iota(n-m) heta} = delta_{nm}$)} hspace{7cm} = $||f||^2$ Since $fin H^2$ is arbitrary extbf{$$ herefore ||psi(f)||hspace{1cm} = hspace{1cm}||f||hspace{1cm} forall hspace{0.5cm} fin H^2$$} egin{large} extbf{Therefore $psi$ is an isometry. Hence it preserves the inner product Isometry $Rightarrow$ one one property. $ herefore psi$ is one one.} end{large} item extbf{underline{$psi$ is Onto}} Let $ ilde{f}in widehat{H^2}$. Then $ ilde{f}=sum_{n=0}^infty<f,e_n>e_n$ where $<f,e_1>,<f,e_2>,ldots$ are Fourier coefficients of f with respect to the orthonormal basis ${e_n: nin mathbb{N}}.$ extbf{Then by Parseval relation we have} $$sum_{n=0}^infty |<f,e_n>|^2 = ||f||^2 < infty $$ extbf{ Define} $$ f =sum_{n=0}^infty a_nz^n hspace{1cm} where hspace{1cm} a_n = <f,e_n>hspace{1cm} forallhspace{1cm} ngeq0$$ extbf{ Since} $$ sum_{n=0}^infty |a_n|^2 = sum_{n=0}^infty |<f,e_n>|^2 = ||f||^2 < infty $$ extbf{Therefore} $$fin H^2$$ extbf{That is for each $ ilde{f}in widehat{H^2}$ there exist $fin H^2$ such that $psi(f) = ilde{f}$ Therefore $psi$ is onto}medskip extbf{ That is $psi$ is an vector space isomorphism which also preserves the norm. Therefore $H^2$ can br identified as a subspace of the $L^2(mathbb{S^1})$ space} end{itemize} end{proof} section{ extbf{ Applications} } egin{enumerate} item In the mathematical rigrous formulation of Quantum Mechanics, developed by extbf{Joh Von Neumann}' the position and momentum states for a single non relavistic spin 0 Particle is the space of all the square integrable functions($L^2$). But $L^2$ have some undesirable properties and $H^2$ is much well behaved space so we work with $H^2$ instead of $L^2$. end{enumerate} ibliographystyle{plain} ibliography{my} end{document} documentclass[11pt]{article} usepackage{fullpage} usepackage{amsfonts} usepackage{amsmath} usepackage{amssymb} usepackage{amsthm} ewtheorem{theorem}{Theorem}[section] egin{document} itle{Hardy Hilbert spaces and its applications} author{Praveen Sharma Department of Mathematics, University of Delhi} date{} maketitle section*{Abstract} In this paper , we discuss the Hardy Hilbert Space on the open disk with center origin and radius unity. We have proved that $H^2$ Space is isomorphic to proper subspace of $L^2$ Space which has various applications in Quantumm Mechanics. section{Preliminaries} subsubsection{Definition(Inner Product Space)} An extit{inner product space} is an vector space $W$ (over field $K=mathbb{R}ormathbb{C}$) with an inner product defined on it. Here, an inner product is an function $<,> : W imes W o K$ which satisfies the following properties:-- egin{enumerate} item $<alpha u+v,w>hspace{1cm} =hspace{1cm} alpha<u,w> + <v,w>$ item $ overline{<u,v>} hspace{2cm} =hspace{1cm} <v,u> $ item $<u,u> hspace{2cm} geq 0$ item $ <u,u> = 0 hspace{1cm} Leftrightarrow u = 0$ hspace{1cm} ( extbf{ for all scalers $alphain K$ and for all vectors $u,v,win W$}) paragraph{Note1} extbf{ Every inner product space is an normed spaces with the norm induced by the inner product is given by} egin{center} $||u|| = sqrt{<u,u>}$ end{center} paragraph{Note2} extbf{An normed space $(W,||.||)$ is said to be complete if each cauchy sequence coverges in $W$.} end{enumerate} subsection{Hilbert Space} An Hilbert Space is defined as the complete inner product space.smallskip extbf{Example:-} smallskip egin{center} egin{large} extbf{$l^2 = {(x_0,x_1,ldots ) : x_n in mathbb{C}, sum_{n=0}^infty |x_n|^2 <infty }$}end{large} end{center} i.e. all the elements of $l^2$ are the sequence of all the complex numbers that are square-summable. Inner product on $l^2$ is given by :- egin{center} extbf{$ <(x_n)_{n=0}^infty , (y_n)_{n=0}^infty> = sum_{n=0}^infty{x_noverline{y_n}}$} hspace{1cm} ( extsc{it is an hilbert sequence space)} end{center} subsection{Definition(Orthonormal sets and sequences)} An subset $X$ of an inner product space is said to be orthonormal if for all $u, v in X $ we have , $$ <u,v> = left{egin{array}{ll} 0 & mathrm{if} u eq v ||u||^2 & mathrm{if} u=v end{array} ight. .$$ paragraph{Note} extbf{If norm of each element of an orthogonal set $X$ is 1 then the set is said to be orthogonal. i.e for all $u,v in X$ we have, } $$ <u,v> = left{ egin{array}{ll} 0 & mathrm{if} u eq v 1 & mathrm{if} u =v end{array} ight. $$ subsection{Definition(Orthonormal basis)} An orthonormal subset $X$ of Hilbert space $W$ is said to be an orthonormal basis if span of $X$ is dense in $W$.i.e. $$overline{spanX} = W $$ paragraph{Note} Every Hilbert space $W eq{0}$ has an orthonormal basis. subsection{Definition(Separable Hilbert Space)} An Hilbert-Space $W$ is said to be extit{separable} if there exist an countable set which is dense in $W$.smallskip extbf{Example:} $l^2$ is an separable Hilbert space paragraph{Note} Each orthonormal basis of an separable Hilbert space are countable.Therefore orthonormal basis of $l^2$ are countable subsection*{Recall} egin{enumerate} item An orthonormal sequence $(e_n)_{n=0}^infty$ is an orthonormal basis of an Hilbert - Space $W$ egin{LARGE}$$Leftrightarrow$$end{LARGE} for all $uin W$ we havehspace{2cm}$ sum_{n=0}^infty|<u,e_n>|^2 = ||u||^2$ cite{kreyszig1978introductory} hspace{0.5cm} extbf{Parseval identity} item Let $(e_n)$ be an orthonormal sequence in an Hilbert-space then $$ sum_{n=0}^inftyalpha_ne_n$$ converges in $W$ $$iff$$ the series $$sum_{n=0}^infty|alpha_n|^2$$ converges in $mathbb{R}$ end{enumerate} section{THE HARDY-HILBERT SPACE} subsection{DEFINITION} { It is defined as the space of all the analytic functions which have a power series representation about origin with square-summable complex coefficients. It is denoted by extbf{ $H^2$}. $$ H^2 = { f : f(z) = sum_{n=0}^infty alpha_nz^n : sum_{n=0}^infty|alpha_n|^2 <infty}$$ Inner Product on $H^2$ is given by $$<f,g> =sum_{n=0}^infty a_noverline{b_n}$$ for $ f(z) = sum_{n=0}^infty a_nz^n hspace{0.5cm} and hspace{0.5cm} g(z) = sum_{n=0}^infty b_nz^nhspace{0.3cm} in hspace{0.2cm} H^2$ egin{theorem} extbf{The Hardy-Hilbert space is an separable Hilbert Space.} end{theorem} egin{proof} Define an function;-- $$ phi:l^2 o H^2$$ given by $$ (a_n)_{n=0}^infty o sum_{n=0}^infty a_nz^n$$ egin{itemize} item extbf{underline{ $phi$ is well defined}} since hspace{1cm} $ (a_n)_{n=0}^infty in l^2 Rightarrow sum_{n=0}^infty |a_n|^2 <infty Rightarrow sum_{n=0}^infty a_nz^n $ hspace{1cm} which being an power series is an analytic function whose coefficients are square summable hence is in $H^2$ $ herefore phi$ is well defined item extbf{underline{Clearly $phi$ is linear}} item extbf{underline{ $phi$ is isometric}} Fix $(a_n)_{n=0}^infty in l^2 $ then we have $$phi((a_n)_{n=0}^infty) = ||sum_{n=0}^infty a_nz^n||_{H^2} = sqrt{sum_{n=0}^infty |a_n|^2} =||(a_n)_{n=0}^infty||_{l^2} $$ $ herefore phi$ is an isometric $ herefore phi$ preserves the norm so that the inner product item extbf{since isometry property implies one one property$ herefore phi$ is one one } cite{bhatia2009notes} item extbf{underline{$phi$ is onto}}smallskip Let $f in H^2$ then $f(z) = sum_{n=0}^infty a_nz^n$ where $sum_{n=0}^infty|a_n|^2<infty$medskip extbf{ define} $x=(a_0,a_1,ldots)$ extbf{Since} $$||x||^2 =sum_{n=0}^infty |a_n|^2 <infty$$ extbf{$$ herefore xin l^2$$} extbf{and}$$phi(x) = f $$ extbf{$$ hereforephi hspace{0.5cm} is hspace{0.5cm} onto$$} end{itemize} extbf{Therefore $phi$ is an vector space isomorphism which also preserves the inner product. Since $l^2$ is an separable Hilbert space hence $H^2$ is also an separable Hilbert Space } end{proof} paragraph{Notations} $mathbb{D}= {z:|z|<1}$ denotes the open unit disk about origin in $mathbb{C}$ $mathbb{S^1} ={z:|z|=1}$ denotes the unit circle about origin in $mathbb{C}$ egin{theorem} extbf{Radius of convergence of each function in $H^2$ is atleast $1$} (i.e. each function in $H^2$ is analytic in the open unit disk $mathbb{D} $) end{theorem} egin{proof} Let $z_0in mathbb{D}$ is fixed $Rightarrowhspace{0.5cm} |z_0|<1$ $ herefore$ the geometric series $sum_{n=0}^infty |z_0|^n$ converges. Let $fin H^2$ is arbitrary.Then $$f(z) = sum_{n=0}^infty a_nz^n hspace{2cm} where hspace{1cm} sum_{n=0}^infty|a_n|^2 <infty$$ Since the series $sum_{n=0}^infty|a_n|^2$ converges $Rightarrow$ $|a_n|^2 longrightarrow 0 Rightarrow|a_n|longrightarrow 0$ $ herefore (|a_n|)_{n=0}^infty$ is an convergent sequence hence bounded. $ herefore exists M>0 $ such that $$ |a_n|leq M hspace{2cm} forall hspace{1cm} ngeq0$$ Now $$sum_{n=0}^infty|a_nz_0^n| leq Msum_{n=0}^infty|z_0|^n $$ where being an geometric series right hand side converges. $ herefore$ By Comparison test the series $ sum_{n=0}^infty a_nz_0^n$ converges absolutely . Since in Hilbert space absolute convergence implies convergence. $ herefore$ the series $sum_{n=0}^infty a_nz_0^n$ converges in $H^2$ since $z_0in H^2$ is arbitrary $ herefore$ each function in $H^2$ is analytic in the unit disk $mathbb{D}$ end{proof} subsection{Definition ($L^2(mathbb{S^1})$ space)} It is defined as the space of all the equivalence classes of functions cite{royden2010real} that are Lebesgue measurable on $S^mathbb{1}$ and square integrable on $S^mathbb{1}$ with respect to Lebesgue measure normalized such that measure of $S^mathbb{1}$ is $1$. $$ L^2(S^mathbb{1}) = {f: f hspace{0.2cm} ishspace{0.2cm} Lesbesgue hspace{0.2cm} measurable hspace{0.2cm} onhspace{0.2cm} mathbb{S^1} hspace{0.2cm} andhspace{0.3cm} frac{1}{2pi}int_0^{2pi}|f(e^{iota heta)}|^2d heta<infty } $$ Inner product on $L^2(mathbb{S^1})$ is given by - $$<f,g> = frac{1}{2pi}int_0^{2pi} f(e^{iota heta})overline{g(e^{iota heta})}d heta$$ paragraph{Note} $L^2(mathbb{S^1})$ is an Hilbert-space with the orthonormal basis given by ${e_n: nin mathbb{Z}}$ where $e_n(e^{iota heta})= e^{iota n heta}.$ extbf{Therefore} $$L^2(mathbb{S^1}) =left {f:f=sum_{n=-infty}^{n=infty}<f,e_n>e_n ight}.$$.cite{martinez2007introduction} subsubsection{Definition ($widehat{H^2}$ space)} $$widehat{H^2} = {f in L^2(mathbb{S^1}): <f,e_n> =0hspace{0.2cm} forhspace{0.2cm} negativehspace{0.2cm} valuehspace{0.2cm} ofhspace{0.2cm} n }$$ $$widehat{H^2} = left{f in L^2(mathbb{S^1}) : f= sum_{n=0}^infty <f,e_n>e_n ight }.$$ extbf{$widehat{H^2}$ is an subspace of $L^2(mathbb{S^1})$ whose negative Fourier coefficients are 0 $ herefore $ ${e_n : n=0,1,ldots }$ are orthonormal basis of $widehat{H^2}$} egin{theorem} egin{LARGE} extbf{$widehat{H^2}$ is an Hilbert-space}end{LARGE} end{theorem} egin{proof} Let $f in overline{widehat{H^2}}$ then there exist an sequence $(f_n)_{n=0}^infty$medskip such that hspace{2cm} $f_n longrightarrow f$ as $nlongrightarrowinfty$medskip Since hspace{2cm} $f_nin widehat{H^2}hspace{1cm} forall hspace{1cm} ngeq0$medskip $ herefore hspace{3cm} <f_n,e_k>=0 hspace{1cm} forall ngeq0 hspace{1cm} and hspace{1cm} forall k<0$medskip extbf{Now for each $k<0$ we have}medskip $|<f_n,e_k> - <f,e_k>| leq |<(f_n-f,e_k>| leq||f_n - f||longrightarrow 0 hspace{0.3cm} ashspace{0.1cm} nlongrightarrowinfty$(Schwarz Inequality cite{kreyszig1978introductory})medskip since $ hspace{3cm} <f_n,e_k>=0 hspace{1cm} forall ngeq0 hspace{1cm} Rightarrow hspace{1cm} <f,e_k>=0 $medskip Since $k<0$ is arbitrary $ herefore hspace{1cm}<f,e_k>=0 hspace{1cm}forallhspace{1cm} k<0$medskip $$ hereforehspace{1cm} fin widehat{H^2}$$ extbf{Therefore $widehat{H^2}$ is an closed subspace of $L^2(mathbb{S^1})$ Hence an Hilbert-Space} end{proof} egin{theorem} egin{large} extbf{The Hardy-Hilbert space can be identified as a subspace of $L^2(mathbb{S^1})$} end{large} end{theorem} egin{proof} Define an function extbf{$$psi:H^2 owidehat{H^2}$$} $$f o ilde{f}$$ where $f(z)=sum_{n=0}^infty a_nz^n$hspace{1cm} and hspace{1cm} $ ilde{f}=sum_{n=0}^infty a_ne_n$ egin{itemize} item extbf{underline{$psi$ is well defined}}medskip Let $f in H^2$ hspace{0.5cm} Then hspace{0.5cm} $f(z)=sum_{n=0}^infty a_nz^n$ hspace{0.5cm} where hspace{0.5cm} $sum_{n=0}^infty |a_n|^2 <infty$medskip Then by extbf{(recall 2)} the series $ ilde{f} =sum_{n=0}^infty a_ne_n $ converges in $widehat{H^2}$medskip $ hereforepsi$ hspace{0.5cm} is hspace{0.5cm} wellhspace{0.5cm} defined item extbf{underline{Clearly $psi$ is linear}}medskip item extbf{underline{$psi$ is an isometry}}medskip For any arbitrary $fin H^2$ where $f(z)=sum_{n=0}^infty a_nz^n$ we have:- $$ || psi(f)|| = || ilde{f}|| = frac{1}{2pi}int_0^{2pi}| ilde{ f}(e^{iota heta}|^2d heta $$medskip extbf{Now} $$frac{1}{2pi}int_0^{2pi} | ilde{f}(e^{iota heta})|^2 d heta = frac{1}{2pi}int_0^{2pi}(sum_{n=0}^infty a_ne^{iota n heta})(overline{sum_{m=0}^infty a_me^{iota m heta}}) $$ hspace{7cm} = $frac{1}{2pi}int_0^{2pi}sum_{n=0}^infty sum_{m=0}^infty a_noverline{a_m}e^{iota(n-m) heta} d heta$ hspace{7cm} = $sum_{n=0}^infty |a_n|^2$ hspace{1cm} extbf{(since $frac{1}{2pi} int_0^{2pi} e^{iota(n-m) heta} = delta_{nm}$)} hspace{7cm} = $||f||^2$ Since $fin H^2$ is arbitrary extbf{$$ herefore ||psi(f)||hspace{1cm} = hspace{1cm}||f||hspace{1cm} forall hspace{0.5cm} fin H^2$$} egin{large} extbf{Therefore $psi$ is an isometry. Hence it preserves the inner product Isometry $Rightarrow$ one one property. $ herefore psi$ is one one.} end{large} item extbf{underline{$psi$ is Onto}} Let $ ilde{f}in widehat{H^2}$. Then $ ilde{f}=sum_{n=0}^infty<f,e_n>e_n$ where $<f,e_1>,<f,e_2>,ldots$ are Fourier coefficients of f with respect to the orthonormal basis ${e_n: nin mathbb{N}}.$ extbf{Then by Parseval relation we have} $$sum_{n=0}^infty |<f,e_n>|^2 = ||f||^2 < infty $$ extbf{ Define} $$ f =sum_{n=0}^infty a_nz^n hspace{1cm} where hspace{1cm} a_n = <f,e_n>hspace{1cm} forallhspace{1cm} ngeq0$$ extbf{ Since} $$ sum_{n=0}^infty |a_n|^2 = sum_{n=0}^infty |<f,e_n>|^2 = ||f||^2 < infty $$ extbf{Therefore} $$fin H^2$$ extbf{That is for each $ ilde{f}in widehat{H^2}$ there exist $fin H^2$ such that $psi(f) = ilde{f}$ Therefore $psi$ is onto}medskip extbf{ That is $psi$ is an vector space isomorphism which also preserves the norm. Therefore $H^2$ can br identified as a subspace of the $L^2(mathbb{S^1})$ space} end{itemize} end{proof} section{ extbf{ Applications} } egin{enumerate} item In the mathematical rigrous formulation of Quantum Mechanics, developed by extbf{Joh Von Neumann}' the position and momentum states for a single non relavistic spin 0 Particle is the space of all the square integrable functions($L^2$). But $L^2$ have some undesirable properties and $H^2$ is much well behaved space so we work with $H^2$ instead of $L^2$. end{enumerate} ibliographystyle{plain} ibliography{my} end{document} In this paper , we discuss the Hardy Hilbert Space on the open disk with center origin and radius unity. We have proved that H2 Space is isomorphic to proper subspace of L2 Space which has various applications in Quantumm Mechanics.

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Lebesgue , Parseval Identity , inner product space , separable

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Creative Commons Attribution 4.0 and The Open Definition

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Chetan Bhagat has been writing since 2004 while Aravind Adiga from 2008. Both writers detail the prevailing culture to suit the expectations and current trends of the society. The average Indian�s lifestyle and the language are well established. English language has become a beautiful bridge across the various local Indian languages and hence Kashmir to Kanyakumari and Kutch to Kibithu amass realize that all Indians live with similar values and lifestyle. Actually a middle class house may have region, religion, caste or any other identity but economically all families have the same level of advantages and disadvantages. Their open style aligning with the realities in society has bought them readership and international awards. Indians at heart feel they have mostly projected the weaknesses inherent in our society and culture as that has more scoop value. Hence they are at times accused of having washed the Indian dirty linen in public

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Indian English Writers, Chetan Bhagat, Aravind Adiga, Language writing style, Indian Culture, Indian Economy and Indian Society

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Creative Commons Attribution 4.0 and The Open Definition

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Creative Commons Attribution 4.0 and The Open Definition

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Creative Commons Attribution 4.0 and The Open Definition

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Finger vein pattern has been an effective biometric for human identification in recent years as each and every individual have an unique pattern of veins in the finger concealed under the skin surface. Finger vein biometric is a new technology, it has received much attention in recent years by providing the high level of security in data transfer. Finger vein identification method based on the self-learned features is adopted in this article. The Convolutional Neural Network (CNN) algorithm relies on the weights and biases of the biometric features which can offer an accurate matching. The deep learning approach based on the CNN algorithm is used in this system, achieved very good results. we use layers of CNN which include convolution layers.. It performs better than traditional algorithm.

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Creative Commons Attribution 4.0 and The Open Definition

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biometrics, finger vein recognition, deep learning, convolutional neural network, GSM module, Zigbee transmitter, Zigbee receiver , Arduino , LED , Buzzer

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Creative Commons Attribution 4.0 and The Open Definition

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The most common problem faced by students these days is related to Attendance when they miss to submit an application of leave. They get short of attendance because in the case of emergency, they are unable to send their application. Sometimes due to lack of co-operation they lose their attendance of workshops they actually attended. Sometimes they do not even get the information about the upcoming events which would be beneficial for them. Our technique coming up with ideas, known as brainstorming. Keeping these problems in our mind over all the possible solutions, we generate an idea of creating an application which would solve all above stated problems. In the discussion we create such a system which manage the all type of application of student. This idea overcome all the traditional file system. This application focuses on all type of application as well as attendance of student. It provides all the related details of working days, holidays, events & workshops taking place in the college. AMS is internet base web application that can access by anyone, anytime, anywhere throughout the department or the organization. This system can be automating the workflow and their approval. AMS is reduce the work of paper and pen.

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Creative Commons Attribution 4.0 and The Open Definition

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AMS (Application management system), Attendance, Application, Students, Approval, Report generator.

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Creative Commons Attribution 4.0 and The Open Definition

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A wideband H-shaped DRA Fed by Offset Microstripline is proposed in this paper ,micro stripline feed impedance is 50?.Antenna is designed to operating frequency 3.09GHz to 4.25GHz with 32.62% fractional Bandwidth.Wideband Achieved using H-shape of DRA .Resonant Frequency is 3.52GHz.when Broadband Characteristics using H-shaped DRA getting directional pattern at gain of 1.55dB.TE10 and TM01 mode generate at 3.52GHz .S-parameter get -27.82dB at 3.52GHz.bandwidth enhancement Using H-shaped DRA.Shifting frequency Using Width of Substrate. The DR operate at Dominant mode in Rectangular waveguide .Its radiation Pattern is Broadside. The differential feeding method for improvement of DRA .Rectangular DRA is easy to fabricated .Time Domain Analysis is used for simulation.It is work high Frequency Problem .Q factor Also depend on Bandwidth.If loaded Q factor is low Bandwidth is high.Bandwidth is inversely propotional to Q-factor.

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Creative Commons Attribution 4.0 and The Open Definition

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H-shaped DRA; offset Micro strip line Feed,5G Application, impedance matching, High Impedance Bandwidth, VSWR.

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Creative Commons Attribution 4.0 and The Open Definition