1 Parameter estimation Since we cannot observe the hidden variablesH1:T , we cannot solve the parameter estimation problem by simply counting events from a dataset. Instead we must apply some method that can give us an approximate solution. The canonical way of parameter estimation in HMM is to use EM, and therefore we shall also adopt this technique for our GBN-HMMs. As before, let o1:T repres...

Non-local transformation, which connects the 3-dimensional Burgers-type equation with a linear heat equation, is constructed. Via this transformation, nonlinear superposition formulae for solutions are obtained and the conditional non-local symmetry of this equation is studied. The multidimensional generalization of the Burgers equation L1(u) = u0 − u|∇u| − u = 0, (1) is called further the Burg...

Let Θ be an arbitrary variety of algebras and H an algebra in Θ. Along with algebraic geometry in Θ over the distinguished algebra H, a logical geometry in Θ over H is considered. This insight leads to a system of notions and stimulates a number of new problems. Some logical invariants of algebras H ∈ Θ are introduced and logical relations between different H1 and H2 in Θ are analyzed. The pape...

R2 SH1(τ, ν)S ? H2(τ, ν)d(τ, ν), H1, H2 ∈ H with the induced norm ‖H‖H = √ 〈H,H〉H. We will consider the set of operatorsQ ⊂ H, defined as Q = {H ∈ H : SH(τ, ν) = 0, (τ, ν) 6⊆ Θ} where Θ is the rectangle Θ = [0, T )× [0, 1/T ), T ∈ R. Let us next formally define the notion of identifiability. The set Q is said to be identifiable from an input-output measurement, if there exists a probing signal ...

Here, for convenience, the mutually mutated gene is specified by a vector of binary random variables H = (H1, . . . ,Hn), only one of which can be assigned value 1 at a time: P (Hg = 1) = 1 n , and Hg = 1 implies that Hg′ = 0 for all g ′ 6= g. To obtain P (y|θ), the observed likelihood for observation y (equation 1 in the main text), we need to marginalize the hidden variables out. This likelih...

We prove a relation between several easy-to-define numerical invariants of generic knots in H × S, and discuss its implication to contact geometry. 1. Definitions of Invariants The 2-dimensional hyperbolic space H2 is defined to be the upper half plane {(x, y) ∈ R2| y > 0} equipped with the metric ds2 = dx +dy y2 . STH2 is the unit tangent bundle of H2, i.e., the S1-bundle over H2 formed by tan...

Let H1,H2, . . . ,Hk+1 be a sequence of k + 1 finite, undirected, simple graphs. The (multicolored) Ramsey number r(H1,H2, . . . ,Hk+1) is the minimum integer r such that in every edge-coloring of the complete graph on r vertices by k + 1 colors, there is a monochromatic copy of Hi in color i for some 1 ≤ i ≤ k + 1. We describe a general technique that supplies tight lower bounds for several nu...

Let X1,X2, . . . be a discrete-time stochastic process with a distribution Pθ, θ ∈ Θ, where Θ is an open subset of the real line. We consider the problem of testing a simple hypothesis H0 : θ = θ0 versus a composite alternative H1 : θ > θ0, where θ0 ∈ Θ is some fixed point. The main goal of this article is to characterize the structure of locally most powerful sequential tests in this problem. ...

i.e., there exists a constant C such that SN(θ, u) ≤ CN . We define δ(θ, u) to be the infimum of the δ satisfying H1 for θ and u. About H1, in the case where θ takes its values in U (the set of complex numbers of modulus 1), it is clear that for all sequences θ and u, δ(θ, u) is smaller than or equal to 1 and it is well-known (see [Ka] for example) that it is greater than or equal to 1/2. Few e...

▸ For any class F of total functions in N, we define what it means for a real function to be conditionally F-computable. This notion extends the notion of uniform F-computability of real functions introduced in the paper [SkWeGe10]. ▸ If F consists of recursive functions then the conditionally F-computable real functions are computable in the sense of [Gr 55] extended by allowing the used compu...