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LIPICS - Leibniz International Proceedings in Informatics, Oct 2017
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D I S C Neuro-RAM Unit with Applications to Similarity Testing and Compression in Spiking Neural Networks? Nancy Lynch 0 Cameron Musco 0 Merav Parter 0 0 CSAIL, MIT , Cambridge, USA CSAIL, MIT, Cambridge , USA Weizmann Institute , Rehovot , Israel We study distributed algorithms implemented in a simplified biologically inspired model for stochastic spiking neural networks. We focus on tradeoffs between computation time and network complexity, along with the role of noise and randomness in efficient neural computation. It is widely accepted that neural spike responses, and neural computation in general, is inherently stochastic. In recent work, we explored how this stochasticity could be leveraged to solve the 'winner-take-all' leader election task. Here, we focus on using randomness in neural algorithms for similarity testing and compression. In the most basic setting, given two n-length patterns of firing neurons, we wish to distinguish if the patterns are equal or -far from equal. Randomization allows us to solve this task with a very compact network, using O ?n log n auxiliary neurons, which is sublinear in the input size. At the heart of our solution is the design of a t-round neural random access memory, or indexing network, which we call a neuro-RAM. This module can be implemented with O(n/t) auxiliary neurons and is useful in many applications beyond similarity testing - e.g., we discuss its application to compression via random projection. Using a VC dimension-based argument, we show that the tradeoff between runtime and network size in our neuro-RAM is near optimal. To the best of our knowledge, we are the first to apply these techniques to stochastic spiking networks. Our result has several implications - since our neuro-RAM can be implemented with deterministic threshold gates, it shows that, in contrast to similarity testing, randomness does not provide significant computational advantages for this problem. It also establishes a separation between feedforward networks whose gates spike with sigmoidal probabilities, and well-studied deterministic sigmoidal networks, whose gates output real number sigmoidal values, and which can implement a neuro-RAM much more efficiently. 1998 ACM Subject Classification F.1.1 Models of Computation - Self-modifying machines (e.g., neural networks), F.2.2 Nonnumerical Algorithms and Problems and phrases spiking neural networks; biological distributed algorithms; circuit design - Biological neural networks are arguably the most fascinating distributed computing systems in our world. However, while studied extensively in the fields of computational neuroscience ? Full version available at <URL> and artificial intelligence, they have received little attention from a distributed computing perspective. Our goal is to study biological neural networks through the lens of distributed computing theory. We focus on understanding tradeoffs between computation time, network complexity, and the use of randomness in implementing basic algorithmic primitives, which can serve as building blocks for high level pattern recognition, learning, and processing tasks. Spiking Neural Network (SNN) Model. We work with biologically inspired spiking neural networks (SNNs) [18, 19, 12, 15], in which neurons fire in discrete pulses in synchronous rounds, in response to a sufficiently high membrane potential. This potential is induced by spikes from neighboring neurons, which can have either an excitatory or inhibitory effect (increasing or decreasing the potential). As observed in biological networks, neurons are either strictly inhibitory (all outgoing edge weights are negative) or excitatory. As we will see, this restriction can significantly affect the power of these networks. A key feature of our model is stochasticity ? each neuron is a probabilistic threshold unit, spiking with probability given by applying a sigmoid function to its potential. While a rich literature focuses on deterministic circuits [21, 13] we employ a stochastic model as it is widely accepted that neural computation is stochastic [1, 24, 9]. Computational Problems in SNNs. We consider an n-bit binary input vector X, which represents the firing status of a set of input neurons. Given a (possibly multi-valued) function f : {0, 1}n ? {0, 1}m, we seek to design a network of spiking neurons that converges to an output vector Z = f (X) (or any Z ? f (X) if f is multi-valued) as quickly as possible using few auxiliary (non-input or output) neurons. The number of auxiliary neurons used corresponds to the ?node complexity? of the network [14]. Designing circuits with small node complexity has received a lot of attention ? e.g., the work of [10] on PARITY and [3] on AC0. Much less is known, however, on what is achievable in spiking neural networks. For most of the problems we study, there is a trivial solution that uses ?(n) auxiliary neurons for inputs of size n. Hence, we primarily focus on designing sublinear size networks ? with n1?c auxiliary neurons for some c. Past Work: WTA. Recently, we studied the ?winner-take-all? (WTA) leader election task in SNNs [17]. Given a set of firing input neurons, the network is required to converge to a single firing output ? corresponding to the ?winning? input. In that work, we critically leveraged the noisy behavior of our spiking neuron model: randomness is key in breaking the symmetry between initially identical firing inputs. This Paper: Similarity Testing and Compression. In this paper, we study the role of randomness in a different setting: for similarity testing and compression. Consider the basic similarity testing problem: given X1, X2 ? {0, 1}n, we wish to distinguish the case when X1 = X2 from the case when the Hamming distance between the vectors is large ? i.e., dH(X1, X2) ? n for some parameter . This problem can be solved very efficiently using randomness ? it suffices to sample O(log n/ ) indices and compare X1 and X2 at these positions to distinguish the two cases with high probability. Beyond similarity testing, similar compression approaches using random input subsampling or hashing can lead to very efficient routines for a number of data processing tasks. 1.1 A Neuro-RAM Unit To implement the randomized similarity testing approach described above, and to serve as a foundation for other random compression methods in spiking networks, we design a basic indexing module, or random access memory, which we call a neuro-RAM. This module solves: I Definition 1 (Indexing). Given X ? {0, 1}n and Y ? {0, 1}log n which is interpreted as an integer in {0, ..., n ? 1}, the indexing problem is to output the value of the Y th bit of X1. Our neuro-RAM uses a sublinear number of auxiliary neurons and solves indexing with high probability on any input. We focus on characterizing the trade-off between the convergence time and network size of the neuro-RAM, giving nearly matching upper and lower bounds. Generally, our results show that a compressed representation (e.g., the index Y ) can be used to access a much larger datastore (e.g., X), using a very compact neural network. While binary indexing is not very ?neural? we can imagine similar ideas extending to more natural coding schemes used, for example, for memory retrieval, scent recognition, or other tasks. Relation to Prior Work. Significant work has employed random synaptic connections between neurons ? e.g., the Johnson-Lindenstrauss compression results of [2] and the work of Valiant [26]. While it is reasonable to assume that the initial synapses are random, biological mechanisms for changing connectivity (functional plasticity) act over relatively large time frames and cannot provide a new random sample of the network for each new input. In contrast, stochastic spiking neurons do provide fresh randomness to each computation. In general, transforming of a network with m possible random edges to a network with fixed edges and stochastic neurons requires ?(m) auxiliary neurons and thus fails to fulfill our sublinearity goal, as there is typically at least one possible outgoing edge from each input. Our neuro-RAM can be thought of as improving the naive simulation ? by reading a random entry of an input, we simulate a random edge from the specified neuron. Beyond similarity testing, we outline how our result can be used to implement Johnson-Lindenstrauss compression similar to [2] without assuming random connectivity. 1.2 1.2.1 Our Contributions Efficient Neuro-RAM Unit Our primary upper bound result is the following: I Theorem 2 (t-round Neuro-RAM). For every integer t ? ?n, there is a ( recurrent) SNN with O(n/t) auxiliary neurons that solves the indexing problem in t rounds with high probability. In particular, there exists a neuro-RAM unit that contains O(?n) auxiliary neurons and solves the indexing problem in O(?n) rounds. Above, and throughout the paper ?with high probability? or w.h.p. to denotes with probability at least 1 ? 1/nc for some constant c. Theorem 2 is proven in Section 3. Neuro-RAM Construction. The main idea is to first ?encode? the firing pattern of the input neurons X into the potentials of t neurons. These encoding neurons will spike with some probability dependent on their potential. However, simply recording the firing rates of the neurons to estimate this probability is too inefficient. Instead, we use a ?successive decoding 1 Here, and throughout, for simplicity we assume n is a power of 2 so log n is an interger. strategy?, in which the firing rates of the encoding neurons are estimated at finer and finer levels of approximation, and adjusted through recurrent excitation or inhibition as decoding progresses. The strategy converges in O(n/t) rounds ? the smaller t is the more information is contained in the potential of a single neuron, and the longer decoding takes. Theorem 2 shows a significant separation between our networks and traditional feedforward circuits where significantly sublinear sized indexing units are not possible. I Fact 3 (See Lower Bounds in [16]). A circuit solving the indexing problem that consists of AND/OR gates connected in a feedforward manner requires ?(n) gates. A feedforward circuit using linear threshold gates requires ?(n/ log n) gates. We note, however, that our indexing mechanism does not exploit the randomness of the spiking neurons, and in fact can also be implemented with deterministic linear threshold gates. Thus, the separation between Theorem 2 and Fact 3 is entirely due to the recurrent (non-feedforward) layout of our network. Since any recurrent network using O(m) neurons and converging in t rounds can be ?unrolled? into a feedforward circuit using O(mt) neurons, Fact 3 shows that the tradeoff between network size and runtime in Theorem 2 is optimal up to a log n factor, if we use our spiking neurons in this restricted way. However, it does not rule out improvements using more sophisticated randomized strategies. 1.2.2 Lower Bound for Neuro-RAM in Spiking Networks Surprisingly, we are able to show that despite the restricted way in which we use our spiking neuron model, significant improvements are not possible: I Theorem 4 (Lower Bound for Neuro-RAM in SNNs). Any SNN that solves indexing in t rounds with high probability in our model must use at least ? t long2 n auxiliary neurons. Theorem 4, whose proof is in Section 4, shows that the tradeoff in Theorem 2 is within a log2 n factor of optimal. It matches the lower bound of Fact 3 for deterministic threshold gates up to a log n factor, showing that there is not a significant difference in the power of stochastic neurons and deterministic gates in solving indexing. Reduction from SNNs to Deterministic Circuits. We first argue that the output distribution of any SNN is identical to the output distribution of an algorithm that first chooses a deterministic threshold circuit from some distribution and then applies it to the input. This is a powerful observation as it lets us apply Yao?s principle: an SNN lower bound can be shown via a lower bound for deterministic circuits on any input distribution [27]. Deterministic Circuit Lower Bound via VC Dimension. We next show that any deterministic circuit that succeeds with high probability on uniform random inputs cannot be too small. The bound is via a VC dimension-based argument, which extends the work of [16]. As far as we are aware, we are the first to give a VC dimension-based lower bound for probabilistic and biologically plausible networks and we hope our work significantly expands the toolkit for proving lower bounds in this area. In contrast to our lower bounds on the WTA problem [17], which rely on indistinguishability arguments based on network structure, our new techniques allow us to give more general bounds for any network architecture. Separation of Network Models. Aside from showing that randomness does not give significant advantages in constructing a neuro-RAM (contrasting with its importance in WTA and similarity testing), our proof of Theorem 4 establishes a separation between feedforward spiking networks and deterministic sigmoidal circuits. Our neurons spike with probability computed as a sigmoid of their membrane potential. In sigmoidal circuits, neurons output real numbers, equivalent to our spiking probabilities. A neuro-RAM can be implemented very efficiently in these networks: I Fact 5 (See [16], along with [19] for similar bounds). There is a feedforward sigmoidal circuit solving the indexing problem using O(?n) gates.2 In contrast, via an unrolling argument, the proof of Theorem 4 shows that any feedforward spiking network requires ? logn2 n gates to solve indexing with high probability. It has been shown that feedforward sigmoidal circuits can significantly outperform standard feedforward linear threshold circuits [20, 16]. However, previously it was not known that restricting gates to spike with a sigmoid probability function rather than output the real value of this function significantly affected their power. Our lower bound, along with Fact 5, shows that in some cases it does. This separation highlights the importance of modeling spiking neuron behavior in understanding complexity tradeoffs in neural computation. 1.2.3 Applications to Randomized Similarity Testing and Compression As discussed, our neuro-RAM is widely applicable to algorithms that require random sampling of inputs. In Section 5 we discuss our main application, to similarity testing ? i.e., testing if X1 = X2 or if dH(X1, X2) ? n. It is easy to implement an exact equality tester using ?(n) auxiliary neurons. Alternatively, one can solve exact equality with three auxiliary neurons using mixed positive and negative edge weights for the outgoing edges of inputs. However this is not biologically plausible ? neurons typically have either all positive (excitatory) or all negative (inhibitory) outgoing edges, a restriction included in our model. Designing sublinear sized exact equality testers under this restriction seems difficult ? simulating the three neuron solution requires at least ?(n) auxiliary neurons ? ?(1) for each input. By relaxing to similarity testing and applying our neuro-RAM, we can achieve sublinear sized networks. We can use ?(log n/ ) neuro-RAMs, each with O(?n) auxiliary neurons to check equality at ?(log n/ ) random positions of X1 and X2 distinguishing if X1 = X2 or if dH(X1, X2) ? n with high probability. This is the first sublinear solution for this problem in the spiking neural networks. In Section 5, we discuss possible additional applications of our neuro-RAM to Johnson-Lindenstrauss random compression, which amounts to multiplying the input by a sparse random matrix ? a generalization of input sampling. 2 2.1 Computational Model and Preliminaries Network Structure We now give a formal definition of our computational model. A Spiking Neural Network (SNN) N = hX, Z, A, w, bi consists of n input neurons X = {x1, . . . , xn}, m output neurons Z = {z1, . . . , zm}, and ` auxiliary neurons A = {a1, ..., a`}. The directed, weighted synaptic 2 Note that [20] shows that general deterministic sigmoidal circuits can be simulated by our spiking model. However, the simulation blows up the size of the circuit size by ?n, giving ?(n) auxiliary neurons. connections between X, Z, and A are described by the weight function w : [X ? Z ? A] ? [X ? Z ? A] ? R. A weight w(u, v) = 0 indicates that a connection is not present between neurons u and v. Finally, for any neuron v, b(v) ? R?0 is the activation bias ? as we will see, roughly, v?s membrane potential must reach b(v) for a spike to occur with good probability. The weight function defining the synapses in our networks is restricted in a few notable ways. The in-degree of every input neuron xi is zero. That is, w(u, x) = 0 for all u ? [X?Z?A] and x ? X. This restriction bears in mind that the input layer might in fact be the output layer of another network and so incoming connections are avoided to allow for the composition of networks in higher level modular designs. Additionally, each neuron is either inhibitory or excitatory: if v is inhibitory, then w(v, u) ? 0 for every u, and if v is excitatory, then w(v, u) ? 0 for every u. All input and output neurons are excitatory. 2.2 Network Dynamics An SNN evolves in discrete, synchronous rounds as a Markov chain. The firing probability of every neuron at time t depends on the firing status of its neighbors at time t ? 1, via a standard sigmoid function, with details given below. For each neuron u, and each time t ? 0, let ut = 1 if u fires (i.e., generates a spike) at time t. Let u0 denote the initial firing state of the neuron. Our results will specify the initial input firing states xj0 = 1 and assume that u0 = 0 for all u ? [Z ? A]. For each non-input neuron u and every t ? 1, let pot(u, t) denote the membrane potential at round t and p(u, t) denote the corresponding firing probability (Pr[ut = 1]). These values are calculated as: pot(u, t) = X v?X?Z?A 1 wv,u ? vt?1 ? b(u) and p(u, t) = 1 + e?pot(u,t)/? (1) where ? > 0 is a temperature parameter, which determines the steepness of the sigmoid. It is easy to see that ? does not affect the computational power of the network. A network can be made to work with any ? simply by scaling the synapse weights and biases appropriately. 1 For simplicity we assume that ? = ?(log n) . Thus by (1), if pot(u, t) ? 1, then ut = 1 w.h.p. and if pot(u, t) ? ?1, ut = 0 w.h.p. (recall that w.h.p. denotes with probability at least 1 ? 1/nc for some constant c). Aside from this fact, the only other consequence of (1) we use in our constructions is that pot(u, t) = 0 =? p(u, t) = 1/2. That is, we use our spiking neurons entirely as random threshold gates, which fire w.h.p. when the incoming potential from their neighbors? spikes exceeds b(u), don?t fire w.h.p. when the potential is below b(u), and fire randomly when the input potential equals the bias. It is an open question if there are any problems which require using the full power of the sigmoidal probability function. 2.3 Additional Notation For any vector x we let xi denote the value at its ith position, starting from x0. Given binary x ? {0, 1}n, we use dec(x) to indicate the integer encoded by x. That is, dec(x) = Pin=?01 xi ?2i. Given an integer x we use bin(x) to denote its binary encoding, where the number of digits used in the encoding will be clear from context. We will often think of the firing pattern of a set of neurons as a binary string. If B = {y1, ..., ym} is a set of m neurons then Bt ? {0, 1}m is the binary string corresponding to their firing pattern at time t. Since the input is typically fixed for some number of rounds, we often just write X to refer to the n-bit string corresponding to the input firing pattern. Boolean Circuits. We mention that SNNs are similar to boolean circuits, which have received enormous attention in theoretical computer science. A circuit consists of gates (e.g., threshold gates, probabilistic threshold gates) connected in a directed acyclic graph. This restriction means that a circuit does not have feedback connections or self-loops, which we do use in our SNNs. While we do not work with circuits directly, for our lower bound, we show a transformation from an SNN to a linear threshold circuit. We sometimes refer to circuits as feedforward networks, indicating that their connections are cycle-free. 3 Neuro-RAM Network In this section we prove our main upper bound: I Theorem 6 (Efficient Neuro-RAM Network). There exists an SSN with O(?n) auxiliary neurons that solves indexing in 5?n rounds. Specifically, given inputs X ? {0, 1}n, and Y ? {0, 1}log n, which are fixed for all rounds t ? {0, ..., 5?n}, the output neuron z satisfies: if Xdec(Y ) = 1 then z5?n = 1 w.h.p. Otherwise, if Xdec(Y ) = 0, z5?n = 0 w.h.p. Theorem 6 easily generalizes to other network sizes, giving Theorem 2, which states the full size-time tradeoff. Here we discuss the intuition behind the basic construction. The full details and proof are given in Appendices A.1 and A.2 of our full paper. We divide the n input neurons X into ?n buckets each containing ?n neurons3: X0 = {x0, ..., x?n?1}, ..., X?n?1 = {x(?n?1)?n, ..., xn?1}. Throughout, all our indices start from 0. We encode the firing pattern of each bucket Xi via the potential of a single neuron ei. Set w(xi?n+j , ei) = 2?n?j for all i, j ? 0. Thus, for every round t, the total potential contributed to ei by the firing of the inputs in bucket Xi is: winhethree X?pireilsi mthienarervieesr.saWloef Xseit abn(deid)e=c(?)2?ginv+e2s +the2?dnecimal value of a binary string, as defined ? 1. We will see later why this is an appropriate value. We defer detailed discussion of the remaining connections to ei for now, first giving a general description of the network construction. In addition to the encoding neurons e0, ..., e?n?1, we have decoding neurons d0,k, ..., d?n?1,k for k = 1, 2, 3 (3?n neurons total). The idea is to select a bucket Xi (via ei) using the first log ?n = log n bits in the index Y . Let Y1d=ef{y0, ..., y log n ?1} and Y2d=ef{y log n , ..., ylog n?1} 2 2 2 be the higher and lower order bits of Y respectively. It is not hard to see that using O(?n) neurons we can construct a network that processes Y1 and uses it to select ei with i = dec(Y1). When a bucket is selected, the potential of any ej with j 6= dec(Y1) is significantly depressed compared to that of ei and so after this selection stage, only ei fires. We then use the decoding neurons to ?read? each bit of the potential encoded in ei. The final output is selected from each of these bits using the lower order bits Y2, which can again be done efficiently with O(?n) neurons. We call this phase the decoding phase since ei encodes the value (in decimal) of its bucket Xi, and we need to decode from that value the bit of the appropriate neuron inside that bucket. 3 Throughout we assume for simplicity that n = 22m for some integer m. This ensures that ?n, log n, and log ?n are integers. It will be clear that if this is not the case, we can simply pad the input, which only affects our time and network size bounds by constant factors. ?n?1 X xi?n+j ? 2?n?j = 2 ? dec(X?i). j=0 (2) is onT.heNodteecotdhiantgtphreowceesisghwtofrrkosmasthf oisllobwits:toineiitiiasll2y?,eni awnidll tfihruesomnloyreif tthhaenfidrostubbilte othfebuwcekiegthti from any other input bit. Thus, by appropriately setting b(ei), we can ensure that the setting of this single bit determines if ei fires initially. If the first bit is the correct bit to output (i.e., if the last log n bits of the index Y2 encode 2 position 0), this will trigger the output z to fire. Otherwise, we iterate. If ei in fact fired, this triggers inhibition that cancels out the potential due to the first bit of bucket i. So ei will now only fire if the second bit of Xi is on. If ei did not fire, the opposite will happen. Further excitation will be given to ei again ensuring that it can fire as long as the second bit of Xi is on. The network iterates in this way, successively reading each bit, until we reach the one encoded by Y2 and the output fires. The first decoding neuron for position j, dj,1, is responsible to triggering the output to fire if j is the correct bit encoded by Y2. The second decoding neuron dj,2 is responsible for providing excitation when ei does not fire. Finally, the third decoding neuron dj,3 provides inhibition when ei does fire. In Appendix A.1 of our full paper, we describe the first stage in which we use the first log n/2 index bits to select the bucket to which the desired index belongs to. In Appendix A.2, we discuss the second phase where we use the last log n/2 bits of Y , to select the desired index inside the bucket i. Our successive decoding process is synchronized by a clock mechanism. This clock mechanism consists of chain of ?(?n) neurons that govern the timing of the ?(?n) steps of our decoding scheme. Roughly, traversing the ?n bits of the chosen ith bucket from left to right, we spend O(1) rounds checking if the current index is the one encoded by Y2. If yes, we output the value at that index and if not, the clock ?ticks? and we move to the next candidate. Note that our model and the proof of Theorem 6 assume that no auxiliary neurons or the output neuron fire in round 0. However, in applications it will often be desirable to run the neuro-RAM for multiple inputs, with execution not necessarily starting at round 0. We can easily add a mechanism that ?clears? the network once it outputs, giving: I Observation 7 (Running Neuro-RAM for Multiple Inputs). The neuro-RAM of Theorem 6 can be made to run correctly given a sequence of multiple inputs. 4 Lower Bound for Neuro-RAM in Spiking Networks In this section, we show that our neuro-RAM construction is nearly optimal. Specifically: I Theorem 8. Any SNN solving indexing with probability ? 1 ? 21n in t rounds must use ` = ? t long2 n auxiliary neurons. This result matches the lower bound for deterministic threshold gates of Fact 3 up to a log n factor, demonstrating that the use of randomness cannot give significant runtime advantages for the indexing problem. Even if one just desires a constant (e.g., 2/3) probability of success, a lower bound of ? t long3 n applies: by replicating any network with success probability 2/3, ?(log n) times and taking the majority output (which can be computed with just a single additional auxiliary neuron), we obtain a network that solves the problem w.h.p. 4.1 High Level Approach and Intuition The proof of Theorem 8 proceeds in a number of steps, which we overview here. Reduction to Deterministic Indexing Circuit. We first observe that a network with ` auxiliary neurons solving the indexing problem in t rounds can be unrolled into a feedforward circuit with t layers and ` neurons per layer. We then show that the output distribution of a feedforward stochastic spiking circuit is identical to the output distribution if we first draw a deterministic linear threshold circuit (still with t layers and ` neurons per layer) from a certain distribution, and evaluate our input using this random circuit. This equivalence is powerful since it allows us to apply Yao?s principle [27]: assuming the existence of a feedforward SNN solving indexing with probability ? 1 ? 21n , given any distribution of the inputs X, Y , there must be some deterministic linear threshold circuit ND which solves indexing with probability ? 1 ? 21n over this distribution. If we consider the uniform distribution over X, Y , this success probablity ensures via an averaging argument that for at least 1/2 of the 2n possible values of X, ND succeeds for at least a 1 ? 21n fraction of the possible Y inputs. Note, however, that the Y can only take on n possible values ? thus this ensures that for 1/2 the possible values of X, ND succeeds for all possible values of the index Y . Let X be the set of ?good inputs? for which ND succeeds. Lower Bound for Deterministic Indexing on a Subset of Inputs. We have now reduced our problem to giving a lower bound on the size of a deterministic linear threshold circuit which solves indexing on an arbitrary subset X of 12 ? 2n = 2n?1 inputs. We do this using VC dimension techniques inspired by the indexing lower bound of [16]. The key idea is to observe that if we fix some input X ? X , then given Y , ND evaluates the function fX : {0, 1}log n ? {0, 1}, whose truth table is given by X. Thus ND can be viewed as a circuit for evaluating any function fX (Y ) for X ? X , where the X inputs are ?programmable parameters?, which effectively change the thresholds of some gates. It can be shown that the VC dimension of the class of functions computable by a fixed a linear threshold circuit with m gates and variable thresholds is O(m log m). Thus for a circuit with t layers and ` gates per layer, the VC dimension is O(`t log(`t)) [5]. Further, as a consequence of Sauer?s Lemma [23, 25, 4], defining the class of functions F = {fX for any X ? X }, since |F | = |X | = 2n?1, we have V C(F ) = ?(n/ log n). These two VC dimension bounds, in combination with the fact that we know ND can compute any function in F if its input bits are fixed appropriately, imply that `t ? log(`t) = ?(n/ log n). n Rearranging gives ` = ? t log2 n , completing Theorem 8. 4.2 Reduction to Deterministic Indexing Circuit We now give the argument explained above in detail, first describing how any SNN that solves indexing w.h.p. implies the existence of a deterministic feedforward linear threshold circuit which solves indexing for a large fraction of possible inputs X. I Lemma 9 (Conversion to Feedforward Network). Consider any SNN N with ` auxiliary neurons, which given input X ? {0, 1}n that is fixed for rounds {0, ..., t}, has output z satisfying Pr[zt = 1] = p. Then there is a feedforward SNN NF (an SNN whose directed edges form an acyclic graph) with (t ? 1) ? (` + 1) auxiliary neurons also satisfying Pr[zt = 1] = p when given X which is fixed for rounds {0, ..., t}. Proof. Let B = A ? z ? all non-input neurons. We produce t ? 1 duplicates of each auxiliary neuron a ? A : {a1, ..., at?1} and of z : {z1, ..., zt?1}, which are split into layers B1, ..., Bt?1. For each incoming edge from a neuron u to v and each i ? 2 we add an identical edge from ui?1 to vi. Any incoming edges from input neurons to u are added to each ui for all i ? 1. Finally connect z to the appropriate neurons in Bt?1 (including zt?1 if there is a self-loop). In round 1, the joint distribution of the spikes B11 in NF is identical to the distribution of B1 in N since these neurons have identical incoming connections from the inputs, and since any incoming connections from other auxiliary neurons are not triggered in N since none of these neurons fire at time 0. Assuming via induction that Bii is identically distributed to Bi, since Bi+1 only has incoming connections from Bi and the inputs which are fixed, then the distribution of Bii++11 identical to that of Bi+1. Thus Btt??11 is identically distributed to Bt?1, and since the output in NF is only connected to Bt?1 its distribution is the same in round t as in N . J I Lemma 10 (Conversion to Distribution over Deterministic Threshold Circuits). Consider any spiking sigmoidal network N with ` auxiliary neurons, which given input X ? {0, 1}n that is fixed for rounds {0, ..., t}, has output neuron z satisfying Pr[zt = 1] = p. Then there is a distribution D over feedforward deterministic threshold circuits with (t ? 1) ? (` + 1) auxiliary gates that, for ND ? D with output z, PrD[zt = 1] = p when presented input X. Proof. We start with NF obtained from Lemma 9. This circuit has t?1 layers of `+1 neurons B1, ..., Bt?1. Given X ? {0, 1}n that is fixed for rounds {0, ..., t}, NF has Pr[zt = 1] = p, which matches the firing probability of the output z in N in round t. Let D be a distribution on deterministic threshold circuits that have identical edge weights to NF . Additionally, for any (non-input) neuron u ? NF , letting u? be the corresponding neuron in the deterministic circuit, set the bias b(u?) = ?, where ? is distributed according to a logistic distribution with mean ? = b(u) and scale s = ?. The random bias is chosen independently for each u. It is well known that the cumulative density function of this distribution is equal to the sigmoid function. That is: 1 Pr[? ? x] = 1 + e? x?b(u) . (3) ? Consider ND ? D and any neuron u in the first layer B1 of NF . u only has incoming edges from the input neurons X. Thus, its corresponding neuron u? in ND also only has incoming edges from the input neurons. Let W = Px?X w(x, u) ? x0. Then we have: Pr[u?1 = 1] = Pr[W ? ? ? 0] = Pr[? ? W ] D 1 = 1 + e? W??b(u) = Pr[u1 = 1]. (Deterministic threshold) (Logistic distribution CDF (3)) (Spiking sigmoid dynamics (1)) Let B?i denote the neurons in ND corresponding to those in Bi. Since in round 1, all neurons in B1 fire independently and since all neurons in B?1 fire independently as their random biases are chosen independently, the joint firing distribution of B11 is identical to that of B?11. By induction assume that B?ii is identically distributed (over the random choice of deterministic network ND ? D) to Bii. Then for any u ? Bi+1 we have by the same argument as above, conditioning on some fixed firing pattern V of Bi in round i: Pr[u?i+1 = 1|B?ii = V ] = Pr[ui+1 = 1|Bii = V ]. D Conditioned on Bii = V , the neurons in Bi+1 fire independently in round i + 1. So do the neurons of B?i+1 due to their independent choices of random biases. Thus, the above implies that the distribution of B?ii++11 conditioned on B?ii = V is identical to the distribution of Bii++11. This holds for all V , so, the full joint distribution of B?ii++11 is identical to that of Bii++11. We conclude by noting that the same argument applies for the outputs of NF and ND since B?tt??11 is identically distributed to Btt??11. J Lemma 10 is simple but powerful ? it demonstrates that the output distribution of a spiking sigmoid network is identical to the output distribution of a deterministic feedforward threshold circuit drawn from some distribution D. Thus, the performance of any SNN is equivalent to the performance of a randomized algorithm which first selects a linear threshold circuit using D and then applies this circuit to the input. This lets us show: I Lemma 11 (Application of Yao?s Principle). Assume there exists an SNN N with ` auxiliary neurons, which given any inputs X ? {0, 1}n and Y ? {0, 1}log n which are fixed for rounds {0, ..., t}, solves indexing with probability ? 1 ? ? in t rounds. Then there exists a feedforward deterministic linear threshold circuit ND with (t ? 1) ? (` + 1) auxiliary gates which solves indexing with probability ? 1 ? ? given X, Y drawn uniformly at random. Proof. We use the idea of Yao?s principle, employing an averaging argument to show that the existence of a randomized circuit succeeding with high probability implies the existence of a deterministic circuit succeeding with high probability on uniform random inputs. Specifically, given X, Y drawn uniformly at random, N solves indexing with probability ? 1 ? ? (since by assumption, it succeeds with this probability for any X, Y ). By Lemma 10, N performs identically to an algorithm which selects a deterministic circuit from some distribution D and then applies it to the input. So at least one circuit in the support of D must succeed with probability ? 1 ? ? on X, Y drawn uniformly at random, since the success probability of N on the uniform distribution is just an average over the deterministic success probabilities. J From Lemma 11 we have a corollary which concludes our reduction from our spiking sigmoid lower bound to a lower bound on deterministic indexing circuits. I Corollary 12 (Reduction to Deterministic Indexing on a Subset of Inputs). Assume there exists an SNN N with ` auxiliary neurons, which, given inputs X ? {0, 1}n and Y ? {0, 1}log n which are fixed for rounds {0, ..., t}, solves indexing with probability ? 1 ? 21n in t rounds. Then there exists some subset of inputs X ? {0, 1}n with |X | ? 2n?1 and a feedforward deterministic linear threshold circuit ND with (t ? 1) ? (` + 1) auxiliary gates which solves indexing given any X ? X and any index Y ? {0, 1}log n. Proof. Applying Lemma 11 yields ND which solves indexing on uniformly random X, Y with probability 1 ? 21n . Let I(X, Y ) = 1 if ND solves indexing correctly on X, Y and 0 otherwise. Then: If n1 PY ?{0,1}log n (1 ? I(X, Y )) 6= 0 then n1 PY ?{0,1}log n (1 ? I(X, Y )) ? n1 just by the fact that the sum is an integer. Thus, for (4) to hold, we must have n1 PY ?{0,1}log n (1?I(X, Y )) = 0 for at least 12 of the inputs X ? {0, 1}n. That is, ND solves indexing for every input index on some subset X with |X | ? 12 |{0, 1}n| ? 2n?1. J Lower Bound for Deterministic Indexing on a Subset of Inputs With Corollary 12 in place, we now turn to lower bounding the size of a deterministic linear threshold circuit ND which solves the indexing problem on some subset of inputs X with |X | ? 2n?1. To do this, we employ VC dimension techniques first introduced for bounding the size of linear threshold circuits computing indexing on all inputs [16]. Consider fixing some input X ? X , such that the output of ND is just a function of the index Y . Specifically, with X fixed, ND computes the function fX : {0, 1}log n ? {0, 1} whose truth table is given by X. Note that the output of ND with X fixed is equivalent to the output of a feedforward linear threshold circuit NDX where each gate with an incoming edge from xi ? X has its threshold adjusting to reflect the weight of this edge if xi = 1. We define two sets of functions. Let F = {fX |X ? X } be all functions computable using some NDX as defined above. Further, let G be the set of all functions computabled by any circuit N D0 which is generated by removing the input gates of ND and adjusting the threshold on each remaining gate to reflect the effects of any inputs with xi = 1. We have F ? G and hence, letting V C(?) denote the VC dimension of a set of functions have: V C(F ) ? V C(G). We can now apply two results. The first gives a lower bound V C(F ): I Lemma 13 (Corollary 3.8 of [4] ? Consequence of Sauer?s Lemma [23, 25]). For any set of boolean functions H = {h} with h : {0, 1}log n ? {0, 1}: V C(H) ? loglong+|Hlo|g e . We next upper bound V C(G). We prove in Appendix B of our full paper: I Lemma 14 (Linear Threshold Circuit VC Bound). Let H be the set of all functions computed by a fixed feedforward linear threshold circuit with m ? 2 gates (i.e., fixed edges and weights), where each gate has a variable threshold. Then: V C(H) ? 3m log m. Applying the bounds of Lemmas 13 and 14 along with V C(F ) ? V C(G) gives: I Lemma 15 (Deterministic Circuit Lower Bound). For any set X ? {0, 1}n with |X | ? 2n?1, any feedforward deterministic linear threshold circuit ND with m non-input gates which solves indexing given any X ? X and any index Y ? {0, 1}log n must have m = ? log2 n . n Proof. Let F and G be as defined in the beginning of the section. We have V C(F ) ? V C(G). log |F| log |X | At the same time, by Lemma 13 we have V C(F ) ? log n+log e = log n+log e ? locgnn for some fixed constant c. By Lemma 14 we have V C(G) ? 3m log m. We thus can conclude that locgnn ? 3m log m, and so m = ? logn2 n . J We conclude by proving our main lower bound: Proof of Theorem 8. The existence of a spiking sigmoidal network with ` auxiliary neurons, solving indexing with probability ? 1 ? 21n in t rounds implies via Corollary 12 the existence of a feedforward deterministic linear threshold circuit with (t ? 1)` + 1 non-input gates solving indexing on some subset of inputs X with |X | ? 2n?1. So by Lemma 15, `?t = ? log2 n . J n 5 5.1 Applications to Similarity Testing and Compression Similarity Testing I Theorem 16 (Similarity Testing). There exists an SNN with O auxilary neurons that solves the approximate equality testing problem in O(?n) rounds. Specifically, given inputs X1, X2 ? {0, 1}n which are fixed for all rounds t ? {0, .., 5?n + 2}, the output z satisfies w.h.p. z5?n+2 = 1 if dH(X1, X2) ? n. Further if X1 = X2 then z5?n+2 = 0 w.h.p. ?n log n Our similarity testing network uses K = ? log n copies of our neuro-RAM network from Theorem 6, labeled S1,k and S2,k for all k ? {1, ..., K}. The idea is to employ log n auxiliary neurons Yk = y1,k, ..., ylog n,k whose values encode a random index i ? {0, ..., n ? 1}. By feeding the inputs (X1, Yk) and (X2, Yk) into S1 and S2, we can check whether X1 and X2 match at position i. Checking ? log n different random indices suffices to identify if dH(X1, X0) ? n w.h.p. Further, if X1 = X0, they will never differ at any of the checks, and so the output will never be triggered. We use: I Observation 17. Consider X1, X2 ? {0, 1}n with dH(X1, X0) ? n. Let i1, ..., iT be chosen independently and uniformly at random in {0, ..., n ? 1}. Then for T = c ln n , 1 Pr[(X1)it = (X2)it for all t ? 1, ..., T ] ? nc . Proof. For any fixed t, Pr[(X1)it = (X2)it ] = 1 ? nn = 1 ? as we select indices at random. Additionally, each of these events is independent since i1, ...iT are chosen independently so: Pr[(X1)it = (X2)it for all t ? 1, ..., T ] ? (1 ? )T = 1 ? )1/ c ln n ? ec 1ln n ? n1c . J 5.1.1 Implementation Sketch It is clear that the above strategy can be implemented in the spiking sigmoidal network model ? we sketch the construction here. By Theorem 6, we require O ?n log n auxiliary neurons for the 2K = ? log n neuro-RAMs employed, which dominates all other costs. It suffices to present a random index to each pair of neuro-RAMs S1, k an S2, k for 5?n rounds (the number of rounds required for the network of Theorem 6 to process an n-bit input). To implement this strategy, we need two simple mechanisms, described below. Random Index Generation: For each of the log n index neurons in Yk we set b(yi) = 0 and add a self-loop w(yi, yi) = 2. In round 1, since they have no-inputs, each neuron has potential 0 and fires with probability 1/2. Thus, Yk1 represents a random index in {0, ..., n ? 1}. To propagate this index we can use a single auxiliary inhibitory neuron g, which has bias b(g) = 1 and w(x, g) = 2 for every input neuron x. Thus, g fires w.h.p. in round 1 and continues firing in all later rounds, as long as at least one input fires. We add an inhibitory edge from g to yi for all i with weight w(g, yi) = ?1. The inhibitory edges from g will keep the random index ?locked? in place. The inhibitory weight of ?1 prevents any yi without an active self-loop from firing w.h.p. but allows any yi with an active self-loop to fire w.h.p. since it will still have potential b(yi) + w(yi, yi) ? 1 = 1. If both inputs are 0, g will not fire w.h.p. However, here our network can just output 0 since X1 = X2 so it does not matter if the random indices stay fixed. Comparing Outputs: We next handle comparing the outputs of S1,k and S2,k. We use two neurons ? f1,k and f2,k. f1,k is excitatory and fires w.h.p. if at least one of S1,k or S2,k has an active output. f2,k is an inhibitor that fires only if both S1,k and S2,k have active outputs. We then connect f1,k to z with weight w(f1,k, z) = 2 and connect f2,k with weight w(f2, z) = ?2 for all k. We set b(z) = 1. Thus, z fires in round 5?n + 2 w.h.p. if for some k, exactly one of S1,k or S2,k has an active output in round 5?n and hence an inequality is detected. Otherwise, z does not fire w.h.p. This gives the output condition of Theorem 16. 5.2 Randomized Compression We conclude by discussing informally how our neuro-RAM can be applied beyond similarity testing to other randomized compression schemes. Consider the setting where we are given n input vectors Xi ? {0, 1}d. Let X ? {0, 1}n?d denote the matrix of all inputs. Think of d as being a large ambient dimension, which we would like to reduce before further processing. One popular technique is Johnson-Lindenstrauss (JL) random projection, where X is multiplied by a random matrix ? ? Rd?d0 with d0 << d to give the compressed dataset X? = X?. Regardless of the initial dimension d, if d0 is set large enough, X? preserves significant information about X. d0 = O?(log n) is enough to preserve the distances between all points, d0 = O?(k) is enough to use X? for approximate k-means clustering or k-rank approximation [6, 8], and d0 = O?(n) preserves the full covariance matrix of the input and so X? can be used for approximate regression and many other problems [7, 22]. JL projection has been suggested as a method for neural dimensionality reduction [2, 11], where ? is viewed as a matrix of random synapse weights, which connect the input neurons representing X to the output neurons representing X? . While this view is quite natural, we often want to draw ? with fresh randomness for each input X. This is not possible using changing synapse weights, which evolve over a relatively long time scale. Fortunately, it is possible to simulate these random connections using our neuro-RAM module. Typically, ? is sparse so can be multiplied by efficiently. In the most efficient construction [7], it has just a single nonzero entry in each row which is a randomly chosen ?1 placed in a uniform random position. Thus, computing a single bit of X? = X? requires selecting on average d/d0 random columns of X, multiplying their entries by a random sign and summing them together. This can be done with a set of neuro-RAMS, each using O(?d) auxiliary neurons which select the random columns of X. In total, we need O?(d/d0) networks ? the maximum column sparsity of ? with high probability, yielding O(d3/2/d0) auxiliary neurons total. In contrast, a naive simulation of random edges using spiking neurons requires ?(d) auxiliary neurons, which is less efficient whenever d0 > d1/2. Additionally, our neuro-RAMs can be reused to compute multiple entries of X? , which is not the case for the naive simulation. Traditionally, the value of an entry of X? is a real number, which cannot be directly represented in a spiking neural network. In our construction, the value of the entry is encoded in its potential, and we leave as an interesting open question how this potential should be decoded or otherwise used in downstream applications of the compression. Acknowledgments. We thank Mohsen Ghaffari ? the initial ideas regarding the importanace of the indexing module came up while Merav Parter was visiting him at ETH Zurich. We also thank Sergio Rajsbaum, Ron Rothblum, and Nir Shavit for helpful discussions. <DIGITS> Christina Allen and Charles F Stevens . An evaluation of causes for unreliability of synaptic transmission . PNAS , 1994 . Zeyuan Allen-Zhu , Rati Gelashvili, Silvio Micali, and Nir Shavit . Sparse sign-consistent Johnson-Lindenstrauss matrices: Compression with neuroscience-based constraints . PNAS , 2014 . Eric Allender . A note on the power of threshold circuits . In Proceedings of the 30th Annual IEEE Symposium on Foundations of Computer Science (FOCS) , 1989 . Cambridge University Press, 2009 . Eric B Baum and David Haussler . What size net gives valid generalization? In Advances in Neural Information Processing Systems 2 (NIPS) , 1989 . Christos Boutsidis , Anastasios Zouzias, and Petros Drineas . Random projections for kmeans clustering . In Advances in Neural Information Processing Systems 23 (NIPS) , 2010 . Kenneth L Clarkson and David P Woodruff . Low rank approximation and regression in input sparsity time . In Proceedings of the 45th Annual ACM Symposium on Theory of Computing (STOC) , 2013 . Dimensionality reduction for k-means clustering and low rank approximation . In Proceedings of the 47th Annual ACM Symposium on Theory of Computing (STOC) , 2015 . A Aldo Faisal , Luc PJ Selen, and Daniel M Wolpert. Noise in the nervous system . Nature Reviews Neuroscience , 9 ( 4 ): 292 - 303 , 2008 . Merrick Furst , James B Saxe, and Michael Sipser . Parity, circuits, and the polynomial-time hierarchy . Theory of Computing Systems , 17 ( 1 ): 13 - 27 , 1984 . Surya Ganguli and Haim Sompolinsky . Compressed sensing, sparsity, and dimensionality in neuronal information processing and data analysis . Annual Review of Neuroscience , 2012 . Wulfram Gerstner and Werner M Kistler. Spiking neuron models: Single neurons, populations, plasticity . Cambridge University Press, 2002 . John J Hopfield , David W Tank, et al. Computing with neural circuits- a model . Science , 233 ( 4764 ): 625 - 633 , 1986 . Bill G Horne and Don R Hush. On the node complexity of neural networks . Neural Networks , 7 ( 9 ): 1413 - 1426 , 1994 . Eugene M Izhikevich. Which model to use for cortical spiking neurons ? IEEE Transactions on Neural Networks , 15 ( 5 ): 1063 - 1070 , 2004 . Pascal Koiran . VC dimension in circuit complexity . In Proceedings of the 11th Annual IEEE Conference on Computational Complexity , 1996 . Nancy Lynch , Cameron Musco, and Merav Parter . Computational tradeoffs in biological neural networks: Self-stabilizing winner-take-all networks . In Proceedings of the 8th Conference on Innovations in Theoretical Computer Science (ITCS) , 2017 . Wolfgang Maass . On the computational power of noisy spiking neurons . In Advances in Neural Information Processing Systems 9 (NIPS) , 1996 . Wolfgang Maass . Networks of spiking neurons: the third generation of neural network models . Neural Networks , 10 ( 9 ): 1659 - 1671 , 1997 . Wolfgang Maass , Georg Schnitger, and Eduardo D Sontag . On the computational power of sigmoid versus boolean threshold circuits . In Proceedings of the 32nd Annual IEEE Symposium on Foundations of Computer Science (FOCS) , 1991 . Marvin Minsky and Seymour Papert . Perceptrons. 1969 . Tamas Sarlos . Improved approximation algorithms for large matrices via random projections . In Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS) , 2006 . Norbert Sauer . On the density of families of sets . Journal of Combinatorial Theory , Series A , 13 ( 1 ): 145 - 147 , 1972 . Current Opinion in Neurobiology, 4 ( 4 ): 569 - 579 , 1994 . Saharon Shelah . A combinatorial problem; stability and order for models and theories in infinitary languages . Pacific Journal of Mathematics , 41 ( 1 ): 247 - 261 , 1972 . Leslie G Valiant . Circuits of the Mind . Oxford University Press on Demand, 2000 . In Proceedings of the 18th Annual IEEE Symposium on Foundations of Computer Science (FOCS) , 1977 .


I have known for a couple of weeks, but was not allowed to tell folks - but now I can. I am going to be a grandmother!My daughter and her boyfriend are having a baby - she is just under 11 weeks along. How absolutely amazing this is for me and Elaine. We both feel so blessed, it's difficult to put into words. And.....
I spent most of the day yesterday posting earrings to my website. I finally got through all the ones that match necklaces, but none of the ones on my 4 earring racks (each of which hold 70 pairs). I don't think those will ever make it to my website because of the amount of work it takes. Plus I am not thrilled with how they look. I seem to love everything posted on the website having consistency - which doesn't work with the earrings - whaa!I never thought of myself as being so obsessive but I sure am when it comes to my business. Some of it's a good thing, like wanting my pieces to look just right and to know that the jewelry will hold up well for years to come. But there are other aspects to my business that aren't so great, like obsessing about how my earrings look on the website, or how to learn html code to have a spot for people to sign up for a newsletter, and of course how to make my business profitable enough so that I can stop worrying about money. All these huge issues can keep me awake at night. As if I didn't have enough problems already!


2.0 out of 5 starsbut I felt like it was largely a brush with poems drilled in ...
This book was purchased for my daughter for school. She passed this course, so other than that, I cannot really review this book. I personally enjoy Garrison Keillor, and would've liked to see this book come back home!


If you make a mistake at the Oscars, chances are, Twitter will find you.
That’s what happened to People editor-in-chief Jess Cagle on Sunday when he confused the Oscar nominated films Hidden Figures and Fences during a pre-ceremony red carpet show.
This isn’t the first time the "Hidden Fences" flub has occurred. Jenna Bush Hager and Michael Keaton made the same mix-up during the Golden Globes, leaving Twitter hyper-vigilant to make sure the two films with majority-black casts were given the respect they deserved. But much to viewers' dismay, it happened 10 minutes into the ABC broadcast.
So even though Cagle corrected his mistake, that did not stop people from tweeting their frustration and disappointment about the red carpet host.


Addressing a ceremony marking the Ninth National Export Day in Tehran on Tuesday, Mohammad Reza Nematzadeh also said, "To reach an 8% economic growth within the next five years [a], we need to spur exports."
Foreign Minister Mohammad Javad Zarif, who also attended the event, said: "The anti-Iran atmosphere prior to the July 14 nuclear deal denied Iranians the execution of the policies of Resistance Economy on the international front and advancing the objectives of national development. It is our hope to enter into negotiations with major world powers and cement ties with their economies. The repeat of sanctions regime would be a remote possibility in this climate, since other countries' economies would intertwine with our economy."
Also in attendance at the Tuesday event, First Vice President Es'haq Jahangiri said: "The only way to achieve development is by aiming for an outward-looking economy, which is within reach through boosting exports."


Tresource represents an attempt to catalohe IRIS Center offers a searchable webpage to locate portrayals of people with disabilities in movies and film.
While no footage exists on this page itself, it informs you of where to locate it. While some of the footage you locate may be disrespectful or incorrect (the site is careful to point this out and not to endorse any of the portrayals), it still offers fodder for great discussion. Use leads you receive from it to engage your training audiences in discussion, activities and assignments. Youth, in particular, respond to using current day media as a training tool.


Ondo state Governor, Oluwarotimi Akeredolu says the late legal icon and Emeritus Professor, David Ijalaye, SAN, lived a fulfilled life, impacting both the academic world and legal profession extensively.
The Governor while on a condolence visit to the Ijalayes in Ile-Ife charged the children to emulate their father’s legacies.
Orange FM’s Olumide Adeluwoye who was at Ile Ife with the governor reports that Akeredolu said the deceased and his late father were close friends describing the legal luminary as a rare gem and valuable icon teaching and practice of law.
Akeredolu emphasised that his academic pursuit at Obafemi Awolowo University was with the support of the deceased.
He commended late Ijalaye’s dedication to the service of the country, having mentored great men who are doing well in their chosen careers, especially legal practitioners.
Adebayo commended Akeredolu for doing the university proud by ensuring good governance in Ondo State.
Speaking on behalf of the children of the deceased, Mr Niyi Ijalaye congratulated the Governor on his second term inauguration and thanked him for honouring their father.
Present at the condolence visit includeDeputy Vice Chancellor Administration of Obafemi Awolowo University, Professor Ajila and members of the faculty of law as well as family and friends.


Ann Street Gallery, NY / “Mythology” Exhibition / Call for Artists
All works must arrive in reusable packaging and include a prepaid return shipping label.
Ars


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An on-demand bond is also referred to as an Unconditional Bond. Contractor’s performance bonds come in two basic varieties to meet the needs of different project owners. An on-demand bond results in an automatic cash payith a conditional performance bond the burden of proof lies with the owner or developer. This briefing paper concentrates upon the on-demand bond.
In the Vossloh Case [2010] EWHC 2443 (Ch) the judge explained the “performance bond” in the following way: This brings me to the so-called “performance bond”, sometimes known as a “performance guarantee”, often as a “demand bond” or “demand guarantee” or even as a “first demand guarantee”. In the context of the present dispute I prefer the expression “demand bond”. In essence it is a particularly stringent contract of indemnity. It is a contractual undertaking by a person, usually a bank, to pay a specified amount of money to a third party on the occurrence of a stated event, usually the non-fulfilment of a contractual obligation by the principal to that third party. Sometimes the wording of the contract has the result that the liability of the person who has given the bond arises on mere demand by the creditor notwithstanding that it may be evident that the principal is not in any way in default or even that the creditor himself is in default under his contract with the principal. It all depends on the wording of the instrument. It is often a difficult question to determine whether, on its true construction, a particular contract which provides for payment on demand is a performance or demand bond (where the obligation to pay is triggered by a demand alone or by a demand accompanied by the provision of specified documents) or whether it is a guarantee (strictly so called) where the obligation to pay is of the “see to it” kind, i.e. conditional on proof by the creditor of default of the principal.”
The courts will usually only block payment under an on-demand bond in the case of established or obvious fraud. However, in the case of Simon Carves Ltd v Ensus UK Ltd [2011] EWHC 657, the Court added another ground on which to restrain performance. The written contract incorporated the General Conditions of Contract for Lump Sum Contractsin 2001, known as the Red Book. The contract said that as soon as the plant has passed all the performance tests, the Purchaser’s project manager shall issue the Acceptance Certificate. Some of the Special Conditions related to the Bond, namely:
– The Performance Bond shall be provided as security for any and all of the Contractor’s obligations and liabilities under the Contract
– The Performance Bond shall initially be for an amount equivalent to 12% of the Contract Price
– Upon the issue of the Acceptance Certificate the Performance Bond shall become null and void (save in respect of any pending or previously notified claims)
It can be noted that Simon Carves procured a Performance Bond from Standard Chartered Bank in the due amount of £18,480,000. The on-demand wording said “The Bank’s obligation to make payments under this Bond shall arise on receipt of a demand made in accordance with the provisions of this Bond without any further proof or condition and without any right of set-off or counterclaim and the Bank shall not be required or permitted to make any other investigation or enquiry.”
As with all cases in the High Court there are complex issues in this instance, however, an Acceptance Certificate was issued on 19 August 2010. It included the words, “the Plant is accepted …”
On the 25 February 2011 Simon Carves sought an injunction to restrain Ensus from making any demand under the Bond. The injunction was granted on the 25 February 2011. In this case the judge found that, “The balance of convenience favours the continuing of the injunction in my view.”
There are occasions when Contractors cannot obtain a Performance Bond ad buyers may note that in the Simon Carves case judgment at para 40 it says, “The banks providing the bonds will usually have security and counter-indemnities so that they are secured when and if they have to pay out on the bond to the beneficiaries. It is often the case that banks will not provide more than a certain number of bonds or bonds beyond a certain value to any one contractor (our emphasis).
Our readers should now be convinced that the specialised area of bonds should be actively researched prior to including them in a contract. This briefing paper does not offer legal advice. You should seek formal legal advice from your organisation’s in-house legal services team or, when used, your external legal advisers. When negotiations become necessary on the wording of the document, legal advice and presence is strongly advised.
Next we will cover Conditional Bonds.
Empty vessels


1st October 2015 30th September 2015 by Innocent Charms Chats,	in category Review
Thomas is well-loved in our home, I have spoken of the Take n Play toys and the kids fondness of them before, to be honest any Thomas Toy that comes in the house is usually met with excited screams, even more so since we went to watch the premiere of Thomas Sodor's Legend of the Lost Treasure in London.
Shining his Thomas Torch on Thomas
But once built it brings serious fun and excitement to little (and if honest big) faces. Daredevil Thomas climbs the incline (all on his own as his is motorized) to the ship's mast and then quicker than you can blink he turns upside down into a 360° spiral and great speed.
In Addison's words "Wow Thomas look at you go, it's just like in the film Mummy".
It has brought some serious fun on rainy days these past weeks, they love it when we set it up and they watch it go round and round. I will say I think it is definitely more suited to the guided 3+ as Deacon wanted to keep playing with the motorized Thomas as he does his little Take N Play ones, where as Addison at 4 loves to watch it over and over as I did Hornby Trains as a child.
Te is now available in shops, it is on Addison and Deacon's list to Santa already as they loved it first time round. As will all Thomas fans.
Thomas The Tank Engine / Sodor's Legend Of The Lost Treasure Premiere
Review // Fisher-Price Skyway
Review // Mamas & Papas Armadillo Dolls Pushchair
Review // My Little Pony Rainbow Dash Style And Groom
Tags: blog review, motorized rail set, review, shipwrecked rails set, Thomas & Friends, thomas & friends legend of lost treasure, thomas the tank engine, , toy, toy review
Addison's Rocking #9
Kids // Learning About Money


Inter.

A
Albany parade from Electric railway journal.
Ale
An Arab Sheikh from Travels between the years 1765 and 1773 through para.
An illustration of Axel Cronstedt's recommended pocket mineralogist's toolkit from An essay towards a system of mineralogy.
Artillery with Russia's Great Army fromBarren Island from Volcanoes and earthquakes.
Bison from Animate creation : popular edition of "Our living world" : a natural history.
Braided edge Mexican hammock from The American Garden.
BBureau centrale telephonique from Les nouvelles conquetes de la science.


the 15th AnnuIUW). The one-day student-organized workshop dedicated to human factors engineering is being hosted by the University of Buffalo on November 15, 2014.


“I suppose the argument would be that RGDP is also understating actual growth, as lots of goods and services are not being measured by the government”
If RGDP is being understated than I assume NGDP is too. How big of a problem is it for NGDPTthe thing that is being targeted ?


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HomeBlogJune 2015Renting in the Washington, DC Area Archives
Renting in the Washington, DC Area 6/15/2015
Selected by George Washington to be this nation's capital in 1790, Washington, DC is part federal government and part bustling urban American city with its treasures of art, culture, music, and historWashington, DC area, either in Northern Virginia or Maryes to consider. Washington, DC Leesburg, Reston, Springfield, and Sterling. In Maryland, our locations include Frederick, Olney, and Silver SpringWashington, DC – the best of both worlds! Washington, DCWashington, DCWashington, DC Area to lease your new apartment home.
Washington, DC Area Features and Attractions
The Capitol Building iscity. The geographic quadrants of the city were marked from the building's location. The city is also home to many world famous museums—including the ewseum – a museum dedicated to the news. Washington DC also plays host to some famous theaters, including Ford’s illmore in neighboring Silver Spring, Maryland. Furthermore, DC plays host to many professional sports teams, including football’s Washington Redskins, basketball’s Washington Wizards, and baseball’s Washington Nationals, who make their home in DC’s first green-certified ballpark. Washing's clean and efficient, and travels into Virginia and Maryland.
Like any diverse American city, Washington and its surrounding areas have eclectic neighborhoods. Adams Morgan Restaurants feature cuisine from around the world. Georgetown is a popular place to walk or linger in casual cafés and trendy shops. Lafayette Park is home to fascinating museums. The Shaw/U Street Corridor is the place for cool jazz. Old Town Alexandria, Virginia’s waterfront, with shops and restaurants, shouldn’t be missed, and historic Mount Vernon is a short drive away.
Washington, DC Area Economy
Washington is only a short drive from Baltimore, Richmond, Philadelphia, the Virginia Peninsula/Tidewater Area, and the Chesapeake Bay beaches.


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Bring your team, appetite, and business card (we're not giving away empanadas to just anyone!). And get them while they're hot– they're sure to go quick!


The Department of Art and Design Faculty Art Exhibit Reception was held on September 8 from 4:00 p.m. to 6:00 p.m., in the Blue Gallery at the ZT Johnson Cafeteria.
The title of the exhibition is Artist as Truthteller, which is derived from an essay delivered by Asbury’s own Dr. Linda Stratford. “Art is a mirror of our soul and our society,” explained gallery coordinator Professor Elissa Morley at Asbury Collegian. “Sometimes the truth hurts. Sometimes it’s stabilizing or comforting. I hope that every member of the community will find something valuable in their experience of the exhibition, that they will come into contact with the Truth as a Nobody, namely Jesus Christ.
The exhibition started on August 22 and will continue until October 12. The exhibition includes 2D and 3D visual works by Professor Emeritus Rudy Medlock, photographs by Keith Barker, paintings by Chris Segre-Lewis and works by Professor Morley. that deal with landscape and space. Dovetail Practice, a collaborative project of Josh and Margaret Smith, also features various print articles and a video piece, focusing on unconventional approaches to art objects.
Together, the works on display seek to portray the truth of Jesus Christ and the gospel.


Having determined the absolute rotation matrix for every single source image, and given the angle of view (which can be derived from the focal length, which is assumed to be known), we basically have a pixel mapping onto a sphere around the viewpoint. We can then use an arbitrary projection to visualize the sphere. We implemented three different projections, a planar one, a spherical one (limited to 180 degrees views), and an angular one (good for 360 degree panoramas, but distorts for large vertical span). All of the images we show here are in angular projection.
In the ideal case, once the source images are registered, they can be overlaid to get the resulting panorama. In practice though there are small errors in the registration, perspective distortions, and nonlinear barrel distortions. Therefore the images will not fit exactly at their respective edges.
Hence there is a need to smoothly blend over between images. When the source images are "faded" in towards their center, then in practice the seams that are visible in simpler overlay solutions are virtually gone. Sometimes this blending also reduces ghosts resulting from moving objects in the scene.
Figure 1 shows a possible arrangement of source images. It can clearly be seen that multiple (more than two) images can overlap at the same time. The blending algorithm has to weigh each image's pixels in the overlap region with a factor such that the edge pixel's weight is zero. As soon as there is no overlap any more, the weight has to approach one. In between a smooth transition has to occur. Adding the irregular geometry of the arrangement, it is obvious that an exact solution to this problem will be rather complicated [Burt83].
where M is the number of overlapping images and d the distance to a source image's closest edge. In practice, a value for n between 3 and 4 has proven to yield the best results. The higher this parameter n is, the sharper the transition between the images, and the less blurred the transition regions appear. However, in the presence of large registration errors a smaller n can at least partly disguise the misalignments. Approaches similar to this are widely used in image mosaicing and have been described e.g. by Bao et al. [Bao98].
Figures 2 and 3 show an example of a case of gross misregistration. In addition, the leaves of the tree are moved by the wind between the shots, so the tree looks extremely blurred in the un-blended image. This can be compensated to a certain extent by the blending. Some artifacts and shadows still remain in the enhanced version, but it looks a lot better then the original version.
One problem that arised as we were rendering our first panoramas was the presence of contrast discontinuities in the final image, that, beside making the panorama pretty unrealistic, made seams visible unless a very heavy averaging of the images was done. It actually comes from the fact that digital cameras have automatic gain control, which leads to different exposure levels in adjacent images. To compensate for this, we proceed approximately as we do for spatial registration, except that we now work in the pixel value space instead of pixel coordonnates space. Similarly we first need to model the pixel value transformations between images. We should ideally account for gamma correction, exposure level differences, and any non-linear operation achieved by the camera.
However we realised that this method was yielding incorrect results. Indeed a slight spatial misregistration of the images introduces some non-correlated noise in the linear regression, which has the effect to flatten the correlation slope and increase accordingly the constant factor of the linear mapping. Which lead to situations where the derived transform from A to B was not the inverse from the the transform from B to A.
Consequently, we considered the constant factor as irrelevant and restricted to a relative gain between images. We also simplified the exposure registration to make it rougthly independent from the quality of the spatial registration. We actually computed the gain as the ratio of the mean luminances for each images in the overlap area.
Global registration consists in running all the relative gains through the least-squares global optimizer and derive an unique absolute gain for each image. That gain is used in the blending operation to adjust the image brightness according to the mapping .
The following images show the kind of artefacts that appear due to different exposure levels in adjacent images and the compensated result : the images of the back of the room are taken against the light, so they seem very dark compared to the wall on the right. In the right image, the gain discontinuity is corrected by darkening the wall.


their votes, here are the top travel nursing companies for 2013 (with the best ratio thumbs up to thumbs down):
Top 5 rated companies to date on Travel Nursing Blogs:
*companies with less than 10 thumbs up are not figured into this list. </s> Vestvågøy
- Registrations from 1970, supplement registrations by K?re Ringstad. </s> Tips on All-Inclusive Resorts in Mexico
The legal age to purchase or consume alcohol in Mexico is 18, which is different from the United States and important for parents traveling with older teenagers to know. Your 18-year-old has the ability to buy and drink alcohol in the country and order alcoholic drinks from the resort's bars or restaurants. Many resorts also have bars directly on the beach that sell alcohol to teenagers. Those of legal drinking age receive a wristband, designating themselves as an adult.
All-inclusive doesn’t necessarily mean that tipping is included in the total cost of the resort. As of 2010, the general tipping policy is $100 for a week-long stay. As some resorts include tips in your cost, check your agreement beforehand. Keep in mind that those who do an exceptionally good job or do something special for you deserve an additional tip. If the concierge got you into a closed activity or a server brought you a special dish, leave a small tip as a thank you.
Mexican all-inclusive resortstennis. You’re typically expected to pay for this rental equipment if you don’t bring your own. Most resorts also offer activities away from the resort that cost extra and aren’t included in your total costs. These may include tours of local missions and historical landmarks, hot air balloon rides, swimming with dolphins and helicopter tours. Your resort may have a deal with one of the local companies for a discounted rate on those trips or activities, so check with the concierge before booking.Destinations > Canada > Kamloops
...is just one of the wonderful parks, which are situated in this region. The BC Wildlife Park Kamloops exhibits 65 different wildlife species, native to this untouched land, for example grizzly bears, birds of prey, snakes, mountain goats, moose, birds of prey and wolves. The park also features an Indoor Discovery Centre for those, who want to escape from rainy days. Entertaining festivals and events are held in the Wildlife Park as well. Check out the Easter Eggscitement, Mother's Day Event, Father's Day Event and the Animal Encounters Summer Festival for Kids.
...exhibits the stunning works by art stuExplore the world of art by taking a look at ce.
in Kamloops represents the stunning history of the Shuswap people through tales and other legends in order to provide people with a great and new insight on former Kamloops roots.
...Time travel by climbing aboard the heritage train 2141 “Spirit of Kamloops”Not only exciting for “historians” and “engine drivers” among us. Also in the city is the so-called Spirit of Christmas Train. Check out the train again during Christmas time to experience a “Silent night, holy night!”
Kamloops’ Cemetery Tour Looking for some extraordinary entertainment? This tour is sponsored by the city’s Museum. Just explore the Pleasant Street Cemetery. Various historical and popular locals of our past will take centre-stage through story-telling of your tourist guide.
Rockhounding If you want to experience archaeology at its greatest, there’s one event which offers you a special family day by exploring the Kamloops fossils.It might be an educational field trip for your children. Check it out!
Wendy’s If you’re looking for some traditional and well-seasoned hamburgers and


His assertion puts at rest speculation that moves were underway to resume dialogue with Pakistan in the near future. Referring to the release of Jamaat-ud-Dawa chief Hafiz Mohammad Saeed, Krishna said, "That
On US Krishna said, "W"


using a Magnolia ‘Little Gem’ or Magnolia ‘Teddy Bear’ as these are dwarf varieties of the….
Wood … 7 Innovative & Beautiful Ways to Store Firewood Read More » </s> Suspected Militants hurled Grenade at his residence followed by firing which was retaliated by security by security forces. However no loss of life or injury reporte. </s>  Ram, I'm a Senior Level Embedded Software / Firmware Engineer with 8 years of experience. I have keen interest and wide exposure in Product Development, Validation, Customer Support, Innovation and Process Improvement.
In my current role as a Senior Firmware Engineer I have been Instrumental in successfully bringing up the firmware related security features in their 1st Generation Dual Actuator Drives ( 2 processors in 1 hard disk )
During my stint as an Embedded Software Engineer , have worked on POC's for major OEMs and involved in the pilot integration of embedded secure element into Samsung Smart Phones and was in Korea for 1 year providing on-site customer support for the same
Is there a particular soft skill that you take pride in?
Versatile in working and coordinatand adapting to changes quickly without impacting project milestones.
Started looking out recently only , so looking forward to it !!!
Always on the lookout for new challenges that would push my boundaries and help both the company and myself to reach greater heights . </s> Security News and Events
January 28th is International Data Privacy Day. It’s a reminder to take stock of what you do to ensure your personal data is protected.
Daily Information Security Tips </s> Importance of Copyrights for Children
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- Copyright also includes for wide range of creative work like intellectual, scientific, artistic form of works which includes poems, composition of songs, music, videos, dances, sculptures, software, and many more. They specifically vary by concerned. </s> by Clear Peaceful Mind
Is every woman on this race a mother?
We exist because of these mothers but how many of us still have our mothers alive and how many of us here as women have good relationships with our mothers. Just because someone gave birth to you, it doesn’t mean that there is a love shared between the two of you, a motherly bond. How many women were actually forced into motherhood in the hopes of keeping a failing relationship alive, sexual abuse – being raped as a kid only to realize that you are having a baby when your tummy is well rounded and are due for labour. One of your close family members died, leaving you with no choice but to raise her kids.
Mothers’ day is a joyous occasion for those who are mothers but how those who are not mothers feel. People never think about it on Mother’s day when they walk up to a woman and say HAPPY MOTHER’S DAY.
What people fail to realize is that some of us women may not have children. There are women who may have lost their child or can’t have children etc. or just are childless because of financial situations or are unable to find a suitable partner to have that child with.
So, Mother’s Day for some women is pure torture and it’s sad and humiliating.
I wonder if anyone out there ever thought about this before wishing a woman a happy mothers day?
We call ourselves Mom’s and we don’t care about what we gave up and instead LOVE what we get in return!
That’s what being a Mom is all about!! Send this to all the brilliant Mom’s you know…
2013 April 13:


Soudan, June 27, 2018.
Leaders of South Sudan’s warring parties have reached a peace agreement that will see rebel leader Riek Machar reinstated to the vice presidency and a possible end to the 5-year-old civil war.


Pan Am Flight 103 Wreckage Book
The Bombing of Pan Am Flight 103
ARCHIVE: Lockerbie Bombing 1988 - BBC Newsnight
On 21 December , N16 crewth a total of people killed, it iKingdom. In , Abdelbaset al-Megrahi ,counts of murder in connection with the bombing.
Libya said it had to accept responsibility due to Megrahi's status as a government employee. During the Libyan Civil War in , ed, [2] .
Some relatives of the dead, including the Lockerbie campaigner Dr Jim Swire, believe the bomb was planted at Heathrow airport and not sent via feeder flights from Malta, as the US and UK claim. PA was bookable as .
At London Heathrow Airport , passengers and their luggage as well as an unaccompanied interline , which originated from Los Angeles and arrived from San Francisco though repeated, with reference, below , , left Heathrow airport on time. After the bombing, the flight number was changed, in accordance with standard practice among airlines after disasters.
At n On scope, the aircraft showed transponder code , or "squawk", and flight level At s ' CVR at Fe. The explosion punched a cm inch hole on the left side of the fuselage. There was no evidence of a distress signal; a The aircraft's elevator and rudder control cables had beenleft. AAIB  was effectively blown off, and was . The nose cone was briefly held on by a band of metal but facing aft, like the lid of a can. It then sheared off, up and backwards to starboard, striking off the No.
Af "tto station es 2. All
Of the total fatalities, were American citizens and 43 British citizens. Nineteen other nationalities were represented, with four or fewer passengers per country. Flight e 55 , an experienced pilot with almost 11, flight hours, of which more than 4, had been accrued in aircraft. First Officer Ray Wagner 52 had approximately 5, flight hours in the and a total of almost 12, hours. Flight Engineer Jerry Don Avritt 46 had more than 8, hours of flying experience, with nearly hours in the He had come to Pan Am through the merger with National Airlines.
The cockpit crew was based at JFK.
Six of the 13 cabin crew members becaAll were originally hired by Pan Am and ranged from 28 years to nine months in seniorityseveral First C Tundergarth.

Some passengers may have remained alive briefly after impact; a pathologist's report concluded that at least two of these passengers might have survived if they had been found soon enough.
Prominent among the passenger victims was the year-old UN Commissioner for Namibia , Bernt Carlsson the following day.
There was a party of US intelligence specialists on board the flight. Their presence gave rise to speculations and conspiracy theories , in which one or more of them were said to have been targeted.
A family of four were kill. Their sonbeen repairing his sister's bicycle. Father Patrick Keegans, Lockerbie's Roman Catholic priest, was preparing to visit the Henrys at approximately 7 pm that evening with his mother, having recently been appointed a parish priest of the town.
The same source claims that, following this, "the shaking stopped and to his surprise he was uninjured".
Mrs Keegans was also unharmed, having been shielded from debris by a fridge-freezer. Dee dead
Ts.
The BBC 's Scotland cay. On 5 December 16 days prior to the attack , the Federal Aviation Administration FAAi , Finland ion ; he said a Finnish woman would carry the bomb on board as an unwitting couriert On 13 December, the warning was posted on bulletin boards in the US Embassy in Moscow and eventually distributed to the entire American .
Just days before the sabotage of the aircraft, security forces in European countries, including the UK, were put on alert after a warning from the Palestine Liberation Organization PLO that extremists might launch terrorist attacks to undermine the thenday of the bombingwas informed by their British counterpart that the English suspected the Libyans to be behind the bombing.
According to a CIA analysis dated 22 December  The list's author noted, "," but the analysis concluded, "We cannot assign responsibility for this tragedy to any terrorist group at this time.in ,attack. A flood of warnings immediately preceding the disaster had included one that read: 'team of Palestinians not associated with PLO intends to attack US targets in Europe.
Time frame is present. On Dalkamoni's account bombs made by Khreesat were at large somewhere.  satellite imaging, and a search of the area by police and soldiers. The wreckage of the crash was scattered over 2, square kilometres and AAIB investigators were confronted by a massive jigsaw puzzle in trying to piece the plane back together.
In total, 4 million pieces of wreckage were collected and registered on computer files. More than 10, p The perpetrators had apparentlycontrary to statement, with reference, above , trail of evidence.
Exe. i ce.
Gauci was interviewed 23 times, giving contradictory evidence about who had bought the clothes, that person's age and appearance, and the date of purchase, but later identified Abdelbaset al-Megrahi.
As Megrahi had only been in Malta on 7 December, that date was assumed to be the purchase date.
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This date is in doubt as Gauci had testified that Malta's Christmas lights had not been on when the clothes had been purchased; it was subsequently determined that the lights had been switched on on 6 December. Scottish police had also failed to inform the defence that another witness had testified seeing Libyan men making a similar purchase on a different day. Gauci then identified al-Megrahi for the court stating — "He is the man on this side.
He resembles him a lot".  onely for carrying materials for a Semtex bomb.
The timer was allegedly traced through its Swiss manufacturer, Mebo ,r , testified at the trial that the Scottish police had originally shown him a fragment of a brown 8-ply circuit board from a prototype timer which had never been supplied to Libya.
Yet the sample he was asked to identify at the trial was a green 9-ply circuit board that Mebo had indeed supplied to Libya. Bollier wanted to pursue this discrepancy, but was told by trial judge Lord Sutherland that he could not do so.n, Dumfries and Galloway Constabulary.
UN to Scottish police at Camp Zeist, Netherlands , which was selected as a neutral venue for their trial. Both the accused,life imprisonment, but Fhimah was acquitted. , andMegrahi served just over 10 years of his sentence beginning 5 April , [63] first in Barlinnie prison, Glasgow, and latterly in Greenock prison, Renfrewshire, throughout which tim20 August In October , Scottish prosecutors announced that they wanted to interview two Libyan nationals, whom they had identified as new suspects, over the bombing.
Megrahi was found guilty of counts of murder on 31 January , and was sentenced to life imprisonment in Scotland; his co-defendant, Fhimah, was found not guilty. The Lockerbie judgment stated: " at Heathrow.
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Start ups are exciting but they can be pretty daunting too. So, if you are still trying to find the courage to launch that business idea that's been bubbling away in the forefront of your mind for far too long already, get set…go!
As Steve Jobs said, "it i't give up."
Be meticulous in your financial planning. Factor in every business expense that you can think of including rent, rates, supplies, marketing, phone bills, equipment, even petrol. Then whatever the amount is, double it!


scale for measuring change in these patients. LHS in stroke patients correlates with other outcome measures, yet they cannot be used interchangeably. A significant relationship between education and level of handicap exists.
DOI: 10.1191/0269215504cr731oa
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SIXTYTHREE BED AND BREAKFAST“ Eternal City”s very easy to reach both for who arrives in Rome by train and for who reaches both metro lines and of the bus stop will make easier to get to any other part of the city.Many of the main monuments and historycal places such a Domus Aurea, are within walkin distance; our location is also perfect for reaching La Sapienza University.T “Palazzo Umbertino” with elevator.The apartment was completely restructured on December 2005. Each room has private bathroom, air conditioning, sat-tv with LCD monitor and hairdryer. Free wifi broadband internet access is also available for our guests.
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1 Container Each </s> Harvey Korman
Harvey Herschel Kormen, professionally known as Harvey Korman (b. February 15, 1927) is an actor born in Chicago, Illinois. He has performed in television and movie productions in the U.S. since 1960. His first (1963-1967),(1967-1978) (1974).
February 15 - 1927 - Actor - Chicago, Illinois - Television - Movie - U.S. - 1960 - The Danny Kaye Show - 1963 - 1967 - The Carol Burnett Show - 1978 - Mel Brooks - Blazing Saddles - 1974The Flintstones (1965-1966). Recently he has done voice work for the live-action version of The Flintstones (1994) as well as the animated ' (1998).
Voice-over - Tom and Jerry - 1965 - The Flintstones - 1966 - The Flintstones - 1994 - 1998
Emmy Awards - 1969 - 1971 - 1972 - 1974 - Golden Globe - 1975


The Cuillin (An Cuilthionn or An Cuiltheann) is the draw for many people to visit Skye, it is a perfect place to spend time with friends in high places. The Black Cuillin should be distinguished from the Red Cuillin (na Beanntan Dearga, Red Hills) which are lower less rocky and have fewer scrambles or climbs. The most famous individual Red Cuillin is Glamaig, the focus for an annual hill run, regarded by many as the toughest hill race in the UK. The two ranges are separated by Glen Sligachan, a beautiful scenic place to walk in its own right The highest point of the Cuillin, is Sgurr Alasdair at 992 m (3,255 ft).
Making recommendations is always difficult but if it is your first visit to Skye for the Black Cuillin we would suggest Bruach na Frithe (Slope of the deer) or Sgurr na Banchdich (the disputed gaelic translation being small pox) while on the Red Cuillin – Marsco (seagull hill – norse translation) will always provides an exhilarating expedition. The mountain also featured in a Runrig (www.runrig.co.uk) song “nightfall on Marsco in the 1981 album “Recovery”.


Isabel is a slave. But this is not a plantation in the Civil War South; it is 1776, and New York is bristling with the news of an impending British invasion. Isabel and her younger sister Ruth’s former master died, and instead of being set free, like they thought they would be, they are sold to a couple of Tories: Mr. and Madame Lockton. Then they are taken to New York, where they are caught in the middle of the revolution. It’s just a matter of time, and circumstances, before Isabel decides which side she will be on. And what price that will cost her.
First: this book is beautiful. I’m not usually a tactile reader, but in this instance, I kept looking at the more-than-perfect cover, stroking the pages, and loving the font. Especially the blurb on the back cover. I could tell much care was taken with the design of the book. And I, for one, appreciated it.
In the interview at the back of the book, Anderson says that the whole slave issue cannot be broken down into “good guys” and “bad guys”. Which is an understatement. There are sympathetic characters on both sides of the revolution — while the politics of the revolution play a role in this book, it is not an indicator of character. (Nor should it ever be.) Chains is thoroughly complex and unflinching, presenting the issues at hand — freedom, slavery, revolution — with honesty. Anderson doesn’t write down to the reader; the book is quite brutal at times. That’s not to say the book is harsh. Rather, interspersed with all the brutality aremoments of absolute poignancy. The book just about ripped my heart in two at parts. Isabel as a character is not just compelling; she’s strong and captivating, and honest. I felt for her, I adored her, I cheered for her.
In short, it’s historical fiction at its finest. But then, it’s Laurie Halse Anderson.
Rated: Mild for mild swearing and some intense situations.
One Response to Chains
Forge | Rated Reads says:
[…] front, so you know, this book, the sequel to Chains, isn’t lighthearted fluff. It picks up several months after Isabel and Curzon escape the […]


Challenge: How long can you hold a Wall Squat for?
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Start your seek for jobs with different locationsg totally different elements tCenter from problems associated with most cancers surgery. Phlebotomy is a wonderful career alternativeis pupil didn’t must take any formal career evaluation, because her face stated it all. In case you’v. will have the ability to objectively observe which things make you light up and which of them do not as a part of the general career exploration and assessment process.