performance of group testing algorithms with near constant tests per item we consider the nonadaptive group testing with n items of which k theta n theta are defective we study a test design in which each item appears in nearly the same number of tests for each item we independently pick l tests uniformly at random with replacement and place the item in those tests we analyze the performance of these designs with simple and practical decoding algorithms in a range of sparsity regimes and show that the performance is consistently improved in comparison with standard bernoulli designs we show that our new design requires roughly 23 fewer tests than a bernoulli design when paired with the simple decoding algorithms known as combinatorial orthogonal matching pursuit and definite defectives dd this gives the best known nonadaptive group testing performance for theta 0 43 and the best proven performance with a practical decoding algorithm for all theta in 0 1 we also give a converse result showing that the dd algorithm is optimal with respect to our randomized design when theta 1 2 we complement our theoretical results with simulations that show a notable improvement over bernoulli designs in both sparse and dense regimes

applications of the numerical inversion of the laplace transform to unsteady problems of the third grade fluid abstract in this article we have effectively used the numerical inversion of laplace transform to study some time dependent problems of the third grade fluid to do so we have considered three different types of unsteady flows of the third grade fluid namely a unsteady flow over a flat rigid plate with porous medium b unsteady mhd flow in a porous medium c unsteady mhd flow in a non porous space with hall currents the solution to the governing equation in each case is obtained by using the standard laplace transform however to transform the obtained solutions from laplace space back to the original space we have used the numerical inversion of the laplace transform graphical results for each case have been presented to show the effects of different parameters involved and to show how the fluid flow evolves with time

a simple converse of burnashev s reliability function in a remarkable paper published in 1976 burnashev determined the reliability function of variable length block codes over discrete memoryless channels dmcs with feedback subsequently an alternative achievability proof was obtained by yamamoto and itoh via a particularly simple and instructive scheme their idea is to alternate between a communication and a confirmation phase until the receiver detects the codeword used by the sender to acknowledge that the message is correct we provide a converse that parallels the yamamoto itoh achievability construction besides being simpler than the original the proposed converse suggests that a communication and a confirmation phase are implicit in any scheme for which the probability of error decreases with the largest possible exponent the proposed converse also makes it intuitively clear why the terms that appear in burnashev s exponent are necessary

group detection for synchronous gaussian code division multiple access channels the concept of group detection is introduced to address the design of suboptimum multiuser detectors for code division multiple access cdma channels a group detection scheme consists of a bank of p group detectors one each for detecting the information symbols of users in each group of a p group partition of the k simultaneously transmitting users in a parallel group detection scheme these group detectors operate independently whereas in a sequential scheme each group detector uses the decisions of the previous group detectors to successively cancel the interference from those users group detectors based on the generalized likelihood ratio test glrt are obtained for the synchronous gaussian cdma channel the complexity of these detectors is exponential in the group size whereas that of the optimum detector is exponential in k since the partition of users is a design parameter group sizes can be chosen to satisfy a wide range of complexity constraints a key performance result is that the glrt group detectors are optimally group near far resistant furthermore upper and lower bounds on the asymptotic efficiency of the sequential group detectors are derived these bounds reveal that the sequential group detectors can under certain conditions perform as well as glrt group detectors of much larger group sizes group detection provides a unifying approach to multiuser detection when the users are partitioned into k single user groups the glrt a modified form of glrt and the sequential group detectors reduce to previously proposed suboptimal detectors namely the decorrelator the two stage detector and the decorrelating decision feedback detector respectively for the other nontrivial partitions the group detectors are new and have a performance that is commensurate with their complexity

bayesian active learning with non persistent noise we consider the problem of noisy bayesian active learning where we are given a finite set of functions mathcal h a sample space mathcal x and a label set mathcal l one of the functions in mathcal h assigns labels to samples in mathcal x the goal is to identify the function that generates the labels even though the result of a label query on a sample is corrupted by independent noise more precisely the objective is to declare one of the functions in mathcal h as the true label generating function with high confidence using as a few label queries as possible by selecting the queries adaptively and in a strategic manner previous work in bayesian active learning considers generalized binary search and its variants for the noisy case and analyzes the number of queries required by these sampling strategies in this paper we show that these schemes are in general suboptimal instead we propose and analyze an alternative strategy for sample collection our sampling strategy is motivated by a connection between bayesian active learning and active hypothesis testing and is based on querying the label of a sample which maximizes the extrinsic jensen shannon divergence at each step we provide upper and lower bounds on the performance of this sampling strategy and show that these bounds are better than the previous bounds in the literature

solving the inverse problem of identifying an unknown source term in a parabolic equation the inverse problem of determining an unknown source term in a parabolic equation with the leading coefficient depending on time and space variables under a final overdetermination condition is considered the series pattern solution of the problem is obtained by using the weighted homotopy analysis method wham a description of the method for solving the problem and identifying the unknown source term is derived finally several examples are presented to illustrate this method

information theoretic bounds for compressed sensing in this paper we derive information theoretic performance bounds to sensing and reconstruction of sparse phenomena from noisy projections we consider two settings output noise models where the noise enters after the projection and input noise models where the noise enters before the projection we consider two types of distortion for reconstruction support errors and mean squared errors our goal is to relate the number of measurements m and snr to signal sparsity k distortion level d and signal dimension n we consider support errors in a worst case setting we employ different variations of fano s inequality to derive necessary conditions on the number of measurements and snr required for exact reconstruction to derive sufficient conditions we develop new insights on max likelihood analysis based on a novel superposition property in particular this property implies that small support errors are the dominant error events consequently our ml analysis does not suffer the conservatism of the union bound and leads to a tighter analysis of max likelihood these results provide order wise tight bounds for output noise models we show that asymptotically an snr of n together with k n k measurements is necessary and sufficient for exact support recovery furthermore if a small fraction of support errors can be tolerated a constant snr turns out to be sufficient in the linear sparsity regime in contrast for input noise models we show that support recovery fails if the number of measurements scales as o n n snr implying poor compression performance for such cases motivated by the fact that the worst case setup requires significantly high snr and substantial number of measurements for input and output noise models we consider a bayesian setup to derive necessary conditions we develop novel extensions to fano s inequality to handle continuous domains and arbitrary distortions we then develop a new max likelihood analysis over the set of rate distortion quantization points to characterize tradeoffs between mean squared distortion and the number of measurements using rate distortion theory we show that with constant snr the number of measurements scales linearly with the rate distortion function of the sparse phenomena

active sequential hypothesis testing consider a decision maker who is responsible to dynamically collect observations so as to enhance his information about an underlying phenomena of interest in a speedy manner while accounting for the penalty of wrong declaration due to the sequential nature of the problem the decision maker relies on his current information state to adaptively select the most informative sensing action among the available ones in this paper using results in dynamic programming lower bounds for the optimal total cost are established the lower bounds characterize the fundamental limits on the maximum achievable information acquisition rate and the optimal reliability moreover upper bounds are obtained via an analysis of two heuristic policies for dynamic selection of actions it is shown that the first proposed heuristic achieves asymptotic optimality where the notion of asymptotic optimality due to chernoff implies that the relative difference between the total cost achieved by the proposed policy and the optimal total cost approaches zero as the penalty of wrong declaration hence the number of collected samples increases the second heuristic is shown to achieve asymptotic optimality only in a limited setting such as the problem of a noisy dynamic search however by considering the dependency on the number of hypotheses under a technical condition this second heuristic is shown to achieve a nonzero information acquisition rate establishing a lower bound for the maximum achievable rate and error exponent in the case of a noisy dynamic search with size independent noise the obtained nonzero rate and error exponent are shown to be maximum

boolean compressed sensing and noisy group testing the fundamental task of group testing is to recover a small distinguished subset of items from a large population while efficiently reducing the total number of tests measurements the key contribution of this paper is in adopting a new information theoretic perspective on group testing problems we formulate the group testing problem as a channel coding decoding problem and derive a single letter characterization for the total number of tests used to identify the defective set although the focus of this paper is primarily on group testing our main result is generally applicable to other compressive sensing models

non adaptive group testing explicit bounds and novel algorithms we consider some computationally efficient and provably correct algorithms with near optimal sample complexity for the problem of noisy nonadaptive group testing group testing involves grouping arbitrary subsets of items into pools each pool is then tested to identify the defective items which are usually assumed to be sparse we consider nonadaptive randomly pooling measurements where pools are selected randomly and independently of the test outcomes we also consider a model where noisy measurements allow for both some false negative and some false positive test outcomes and also allow for asymmetric noise and activation noise we consider three classes of algorithms for the group testing problem we call them specifically the coupon collector algorithm the column matching algorithms and the lp decoding algorithms the last two classes of algorithms versions of some of which had been considered before in the literature were inspired by corresponding algorithms in the compressive sensing literature the second and third of these algorithms have several flavors dealing separately with the noiseless and noisy measurement scenarios our contribution is novel analysis to derive explicit sample complexity bounds with all constants expressly computed for these algorithms as a function of the desired error probability the noise parameters the number of items and the size of the defective set or an upper bound on it we also compare the bounds to information theoretic lower bounds for sample complexity based on fano s inequality and show that the upper and lower bounds are equal up to an explicitly computable universal constant factor independent of problem parameters

the influence of slip condition on thin film flow of a fourth grade fluid by the homotopy analysis method this paper provides an investigation regarding slip effects on thin film flow of a fourth grade fluid down a vertical cylinder the nonlinear problem that arises is solved for both exact and ham homotopy analysis method solutions exact and ham solutions are also compared finally we briefly describe the flow characteristics to include the effects of emerging parameters

efficient algorithms for noisy group testing group testing refers to the problem of identifying with high probability a small subset of d defectives from a large set of n items via a small number of pooled tests i e tests that have a positive outcome if at least one of the items being tested in the pool is defective else have a negative outcome for ease of presentation in this paper we focus on the regime when d mathcal o n 1 delta for some delta 0 the tests may be noiseless or noisy and the testing procedure may be adaptive the pool defining a test may depend on the outcome of a previous test or non adaptive each test is performed independent of the outcome of other tests a rich body of the literature demonstrates that theta d log n tests are information theoretically necessary and sufficient for the group testing problem and provides algorithms that achieve this performance however it is only recently that reconstruction algorithms with computational complexities that are sub linear in n have started being investigated in the scenario with adaptive tests with noisy outcomes we present the first scheme that is simultaneously order optimal up to small constant factors in both the number of tests and the decoding complexity mathcal o left d log n right in both the performance metrics the total number of stages of our adaptive algorithm is small mathcal o left log d right similarly in the scenario with non adaptive tests with noisy outcomes we present the first scheme that is simultaneously near optimal in both the number of tests and the decoding complexity via an algorithm that requires mathcal o left d log d log n right tests and has a decoding complexity of mathcal o d log n log 2 d finally we present an adaptive algorithm that only requires two stages and for which both the number of tests and the decoding complexity scale as mathcal o d log n log 2 d for all three settings the probability of error of our algorithms scales as mathcal o left 1 poly d right for each of the statements mentioned earlier about the order of the number of measurements decoding complexity and probability of error we provide explicitly computed small universal factors in our theorem statements

on the computation of analytical solutions of an unsteady magnetohydrodynamics flow of a third grade fluid with hall effects in this article a combination of lie symmetry and homotopy analysis methods ham are used to obtain solutions for the unsteady magnetohydrodynamics flow of an incompressible electrically conducting third grade fluid bounded by an infinite porous plate in the presence of hall current in particular similarity reductions are performed on the governing equations in its complex scalar and corresponding vector system forms also nontrivial conservation laws using the multiplier approach are constructed for the complex scalar equation furthermore a comparison of the results with numerical results already existing in the literature is done the analytical solutions are presented through graphs by choosing a range of the relevant physical parameters the underlying calculations were obtained via a combination of software packages in mathematica and maple in particular for the lie symmetry generators euler lagrange operators and homotopy operators the latter being towards the construction of the conserved flows

individual testing is optimal for nonadaptive group testing in the linear regime we consider nonadaptive probabilistic group testing in the linear regime where each of n items is defective independently with probability p in 0 1 and p is a constant independent of n we show that testing each item individually is optimal in the sense that with fewer than n tests the error probability is bounded away from zero

couette and poiseuille flows of an oldroyd 6 constant fluid with magnetic field abstract in this paper we develop a set of differential equations describing the steady flow of an oldroyd 6 constant magnetohydrodynamic fluid the fluid is electrically conducting in the presence of a uniform transverse magnetic field the developed non linear differential equation takes into account the effect of the material constants and the applied magnetic field we presented the solution for three types of steady flows namely i couette flow ii poiseuille flow and iii generalized couette flow homotopy analysis method ham is used to solve the non linear differential equation analytically it is found from the present analysis that for steady flow the obtained solutions are strongly dependent on the material constants non newtonian parameters which is different from the model of oldroyd 3 constant fluid numerical solutions are also given and compared with the solutions by ham

poisson group testing a probabilistic model for boolean compressed sensing we introduce a novel probabilistic group testing framework termed poisson group testing in which the number of defectives follows a right truncated poisson distribution the poisson model has a number of new applications including dynamic testing with diminishing relative rates of defectives we consider both nonadaptive and semi adaptive identification methods for nonadaptive methods we derive a lower bound on the number of tests required to identify the defectives with a probability of error that asymptotically converges to zero in addition we propose test matrix constructions for which the number of tests closely matches the lower bound for semiadaptive methods we describe a lower bound on the expected number of tests required to identify the defectives with zero error probability in addition we propose a stage wise reconstruction algorithm for which the expected number of tests is only a constant factor away from the lower bound the methods rely only on an estimate of the average number of defectives rather than on the individual probabilities of subjects being defective

group testing with random pools phase transitions and optimal strategy the problem of group testing is to identify defective items out of a set of objects by means of pool queries of the form does the pool contain at least a defective the aim is of course to perform detection with the fewest possible queries a problem which has relevant practical applications in different fields including molecular biology and computer science here we study gt in the probabilistic setting focusing on the regime of small defective probability and large number of objects p 0 and n we construct and analyze one stage algorithms for which we establish the occurrence of a non detection detection phase transition resulting in a sharp threshold overline m for the number of tests by optimizing the pool design we construct algorithms whose detection threshold follows the optimal scaling overline m propto np log p then we consider two stages algorithms and analyze their performance for different choices of the first stage pools in particular via a proper random choice of the pools we construct algorithms which attain the optimal value previously determined in mezard and toninelli arxiv 0706 3104 for the mean number of tests required for complete detection we finally discuss the optimal pool design in the case of finite p

feedback in the non asymptotic regime without feedback the backoff from capacity due to non asymptotic blocklength can be quite substantial for blocklengths and error probabilities of interest in many practical applications in this paper novel achievability bounds are used to demonstrate that in the non asymptotic regime the maximal achievable rate improves dramatically thanks to variable length coding and feedback for example for the binary symmetric channel with capacity 1 2 the blocklength required to achieve 90 of the capacity is smaller than 200 compared to at least 3100 for the best fixed blocklength code even with noiseless feedback virtually all the advantages of noiseless feedback are shown to be achievable even if the feedback link is used only to send a single signal informing the encoder to terminate the transmission stop feedback it is demonstrated that the non asymptotic behavior of the fundamental limit depends crucially on the particular model chosen for the end of packet control signal fixed blocklength codes and related questions concerning communicating with a guaranteed delay are discussed in which situation feedback is demonstrated to be almost useless even non asymptotically

nonadaptive group testing based on sparse pooling graphs an information theoretical analysis of nonadaptive group testing schemes based on sparse pooling graphs is presented a pooling graph is a bipartite graph for which the adjacency matrix is a pooling matrix the binary status of the objects to be tested are modeled by independent identically distributed bernoulli random variables with probability p an l r n regular pooling graph is a bipartite graph in which the left nodes have degree l the right nodes have degree r and n is the number of left nodes the main contribution of this paper is a direct coding theorem that gives the conditions for the existence of an estimator that can achieve an arbitrarily small probability of error an estimator is a function that uses observations to infer the state of an object the direct coding theorem is proved by averaging the upper bound on the probability of the estimation error of the typical set estimator over an l r n regular pooling graph ensemble numerical results indicate sharp threshold behaviors in the asymptotic regime these results can provide a concrete benchmark for nonadaptive group testing of existing and emerging detection algorithms over a noiseless system

the geometry of generalized binary search this paper investigates the problem of determining a binary valued function through a sequence of strategically selected queries the focus is an algorithm called generalized binary search gbs gbs is a well known greedy algorithm for determining a binary valued function through a sequence of strategically selected queries at each step a query is selected that most evenly splits the hypotheses under consideration into two disjoint subsets a natural generalization of the idea underlying classic binary search this paper develops novel incoherence and geometric conditions under which gbs achieves the information theoretically optimal query complexity i e given a collection of n hypotheses gbs terminates with the correct function after no more than a constant times logn queries furthermore a noise tolerant version of gbs is developed that also achieves the optimal query complexity these results are applied to learning halfspaces a problem arising routinely in image processing and machine learning

how many queries are needed to learn we investigate the query complexity of exact learning in the membership and proper equivalence query model we give a complete characterization of concept classes that are learnable with a polynomial number of polynomial sized queries in this model we give applications of this characterization including results on learning a natural subclass of dnf formulas and on learning with membership queries alone query complexity has previously been used to prove lower bounds on the time complexity of exact learning we show a new relationship between query complexity and time complexity in exact learning if any honest class is exactly and properly learnable with polynomial query complexity but not learnable in polynomial time then p np in particular we show that an honest class is exactly polynomial query learnable if and only if it is learnable using an oracle for p 4

on the homotopy analysis method for nonlinear problems a powerful easy to use analytic tool for nonlinear problems in general namely the homotopy analysis method is further improved and systematically described through a typical nonlinear problem i e the algebraically decaying viscous boundary layer flow due to a moving sheet two rules the rule of solution expression and the rule of coefficient ergodicity are proposed which play important roles in the frame of the homotopy analysis method and simplify its applications in science and engineering an explicit analytic solution is given for the first time with recursive formulas for coefficients this analytic solution agrees well with numerical results and can be regarded as a definition of the solution of the considered nonlinear problem

computation of conservation laws for nonlinear lattices an algorithm to compute polynomial conserved densities of polynomial nonlinear lattices is presented the algorithm is implemented in mathematica and can be used as an automated integrability test with the code diffdens m conserved densities are obtained for several well known lattice equations for systems with parameters the code allows one to determine the conditions on these parameters so that a sequence of conservation laws exist

peristaltically induced motion of a mhd third grade fluid in a deformable tube abstract we consider incompressible magnetohydrodynamic mhd third grade fluid confined in a circular cylindrical tube the tube surfaces are electrically non conducting travelling sinusoidal wave is imposed on the tube which induces peristaltic motion in the fluid the stream function axial velocity and pressure gradient are determined analytically for small deborah number the analysis consists of a perturbation expansion in terms of the deborah number up to o solutions for pressure rise and frictional force per wavelength are given through the use of numerical integration the general solution of highly non linear equation for the hydrodynamic third grade fluid is developed the results are finally compared with the third grade hydrodynamic and newtonian fluid cases it is found that the axial velocity increases by increasing deborah number but decreases for large values of hartmann number

extrinsic jensen shannon divergence applications to variable length coding this paper considers the problem of variable length coding over a discrete memoryless channel with noiseless feedback this paper provides a stochastic control view of the problem whose solution is analyzed via a newly proposed symmetrized divergence termed extrinsic jensen shannon ejs divergence it is shown that strictly positive lower bounds on ejs divergence provide nonasymptotic upper bounds on the expected code length this paper presents strictly positive lower bounds on ejs divergence and hence nonasymptotic upper bounds on the expected code length for the following two coding schemes 1 variable length posterior matching and 2 maxejs coding scheme that is based on a greedy maximization of the ejs divergence as an asymptotic corollary of the main results this paper also provides a rate reliability test variable length coding schemes that satisfy the condition s of the test for parameters r and e are guaranteed to achieve a rate r and an error exponent e the results are specialized for posterior matching and maxejs to obtain deterministic one phase coding schemes achieving capacity and optimal error exponent for the special case of symmetric binary input channels simpler deterministic schemes of optimal performance are proposed and analyzed