This paper describes the basis for techniques such as stochastic subdivision in the theory of random processes and estimation theory. The popular stochastic subdivision construction is then generalized to provide control of the autocorrelation and spectral properties of the synthesized random functions. The generalized construction is suitable for generating a variety of perceptually distinct high-quality random functions, including those with non-fractal spectra and directional or oscillatory characteristics. It is argued that a spectral modeling approach provides a more powerful and somewhat more intuitive perceptual characterization of random processes than does the fractal model. Synthetic textures and terrains are presented as a means of visually evaluating the generalized subdivision technique. Categories and Subject Descriptors: I.3.3 [Computer Graphics]: Picture/Image Generation; I.3.7 [Computer Graphics]: Three Dimensional Graphics and Realism -<F11.

Abstract — The second-order statistical properties of complex signals are usually characterized by the covariance function. However, this is not sufficient for a complete second-order description, and it is necessary to introduce another moment called the relation function. Its properties, and especially the conditions that it must satisfy, are analyzed both for stationary and nonstationary signals. This leads to a new perspective concerning the concept of complex white noise as well as the modeling of any signal as the output of a linear system driven by a white noise. Finally, this is applied to complex autoregressive signals, and it is shown that the classical prediction problem must be reformulated when the relation function is taken into consideration. I.

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Abstract. We examine the problem of predicting the evolution of solutions of the Kuramoto-Sivashinsky equation when initial data are missing. We use the optimal prediction method to construct equations for the reduced system. The resulting equations for the resolved components of the solution are random integrodifferential equations. The accuracy of the predictions depends on the type of projection used in the integral term of the optimal prediction equations and on the choice of resolved components. The novel features of our work include the first application of the optimal prediction formalism to a nonlinear, non-Hamiltonian equation and the use of a non-invariant measure constructed through inference from empirical data.

A brief account of my life and work This account is not as short as I wanted to make it due to my long life while my life and my work cannot be separated. I belong to a generation which was too young for active duty in the First World War and too old for the Second. To give the story some structure I divide it into significant periods, 1. Romania 1903–1922. I was born on April 21, 1903, in Galatz, Romania, as the youngest of four children. My father, trained as an accountant, inherited the lumber business of my grandfather Isaac Segal, on my mother’s side, but my father was not a good business man. The fortunes of the family improved in 1910 when we moved to Jassy, Romania, where my father eventually became Sub-Director (Vice President) of the newly founded “Banca Moldova”. The building of the Banca Moldova, at present a main Post Office of Jassy, was new in 1910, and our family occupied one-half of its second floor. We four children enjoyed the cultural interests of our parents, especially of mother’s who wrote and spoke French fluently. My father was an Austro-Hungarian subject and Consul in Jassy of the Dual Monarchy. This caused difficulties during the First World War after

laboratory in which faculty members and graduate students from numerous academic departments conduct research. The research reported in this document was made possible in

A sequence which converges is a Cauchy sequence, although the converse is not necessarily true. If the converse is true for all Cauchy sequences in a given inner product space, then the space is called complete. We shall assume that all of the spaces we work with from now on are complete. A few more de nitions from real analysis: De nition 3. An open ball centered at x with radius r > 0 is the set B r (x) = fu : ku xk < rg De nition 4. A set S is closed if x 2 S whenever there exists a sequence fu n g 2 S such that fu n g ! x. De nition 5. A set S is open if, for all x 2 S, there exists an open ball B r (x) such that B r (x) S. An example of a closed set is the closed interval [0; 1] R. An example of an open set is the open interval (0; 1) R. The complement of an open set is closed, and the complement of a closed set is open. Given a set S and some point b outside of S, we want to determine under what conditions there is a point b 2 S closest to b, i.e., such that

Let {Z(s)} be a Gaussian process on Rn, i.e., a collection of jointly normal random variables Z(s) associated with n-dimensional locations s ∈ Rn. The joint distribution of {Z(s)} depends only on the means µ(s) = EZ(s) and the covariances C(s,t) = E ( Z(s) − µ(s)) ( Z(t) − µ(t) ). The process is called stationary or translation invariant if the distribution wouldn’t change under a rigid translation of the entire collection of locations, i.e., if µ(s) = µ(s + h) and C(s + h,t + h) = C(s,t) for all h; in this case µ(s) ≡ µ is constant and C(s,t) = C(s − t,0) can only depend on the difference h = (s − t) between the two locations, so must be of the form C(s,t) = C0(s − t) for some function C0(h) = C(h,0) on Rn. Not just any function C0(h) can be a covariance function; let’s see what the choices are. It’s easy to see that the function C0 must be even, i.e., must satisfy C0(h) = C0(−h), since C(s − t) = E ( Z(s) − µ(s)) ( Z(t) − µ(t) ) = C(t − s). But more is true: if {sj} any collection of locations, then complex linear combinations aT (Z − µ) = ∑ aj(Zj − µj) of the centered random variables Zj = Z(sj) (with means µj = µ(sj)) must have nonnegative squared modulus E ∣ ∑ aj(Zj − µj) ∣ 2 = ∑ ajC(sj − sk)āk ≥ 0 for every set of complex numbers {aj} ⊂ C. A function C0(h) is called positive semi-definite if it always satisfies the inequality ∑ jk ajC(sj −sk)āk ≥ 0 for any locations sj and complex numbers aj; this is equivalent to asking that C(h) = C(−h) for every h ∈ Rn and that ∑ ajC(sj −sk)ak ≥ 0 for all real numbers aj ∈ R. One way to get a symmetric positive semi-definite function C0(h) is by taking the Fourier transform C0(h) = e ih·ω G(ω)d n ω R n of any positive function G(ω) on R n or, more generally, of any finite positive measure G(dω), because then ajC(sj − sk)āk = jk

Let {Z(s)} be a Gaussian process on Rn, i.e., a collection of jointly normal random variables Z(s) associated with n-dimensional locations s ∈ Rn. The joint distribution of {Z(s)} depends only on the means µ(s) = EZ(s) and the covariances C(s, t) = E ( Z(s) − µ(s)) ( Z(t) − µ(t) ). The process is called stationary or translation invariant if the distribution wouldn’t change under a rigid translation of the entire collection of locations, i.e., if µ(s) = µ(s + h) and C(s + h, t + h) = C(s, t) for all h; in this case µ(s) ≡ µ is constant and C(s, t) = C(s − t, 0) can only depend on the difference h = (s − t) between the two locations, so must be of the form C(s, t) = C0(s − t) for some function C0(h) = C(h, 0) on Rn. Not just any function C0(h) can be a covariance function; let’s see what the choices are. It’s easy to see that the function C0 must be even, i.e., must satisfy C0(h) = C0(−h), since C(s − t) = E ( Z(s) − µ(s)) ( Z(t) − µ(t) ) = C(t − s). But more is true: if {sj} any collection of locations, then complex linear combinations aT (Z − µ) = ∑ aj(Zj − µj) of the centered random variables Zj = Z(sj) (with means µj = µ(sj)) must have nonnegative squared modulus E ∣ ∑ aj(Zj − µj) ∣ 2 = ∑ ajC(sj − sk)āk ≥ 0 for every set of complex numbers {aj} ⊂ C. A function C0(h) is called positive semi-definite if it always satisfies the inequality ∑ jk ajC(sj −sk)āk ≥ 0 for any locations sj and complex numbers aj; this is equivalent to asking that C(h) = C(−h) for every h ∈ Rn and that ∑ ajC(sj − sk)ak ≥ 0 for all real numbers aj ∈ R. One way to get a symmetric positive semi-definite function C0(h) is by taking the Fourier transform C0(h) = e ih·ω G(ω) d n ω R n of any positive function G(ω) on R n or, more generally, of any finite positive measure G(dω), because then ajC(sj − sk)āk = jk

Let {Z(s)} be a Gaussian process on Rn, i.e., a collection of jointly normal random variables Z(s) associated with n-dimensional locations s ∈ Rn. The joint distribution of {Z(s)} depends only on the means µ(s) = EZ(s) and the covariances C(s, t) = E ( Z(s) − µ(s)) ( Z(t) − µ(t) ). The process is called stationary or translation invariant if the distribution wouldn’t change under a rigid translation of the entire collection of locations, i.e., if µ(s) = µ(s + h) and C(s + h, t + h) = C(s, t) for all h; in this case µ(s) ≡ µ is constant and C(s, t) = C(s − t, 0) can only depend on the difference h = (s − t) between the two locations, so must be of the form C(s, t) = C0(s − t) for some function C0(h) = C(h, 0) on Rn. Not just any function C0(h) can be a covariance function; let’s see what the choices are. It’s easy to see that the function C0 must be even, i.e., must satisfy C0(h) = C0(−h), since C(s − t) = E ( Z(s) − µ(s)) ( Z(t) − µ(t) ) = C(t − s). But more is true: if {sj} any collection of locations, then complex linear combinations aT (Z − µ) = ∑ aj(Zj − µj) of the centered random variables Zj = Z(sj) (with means µj = µ(sj)) must have nonnegative squared modulus E ∣ ∑ aj(Zj − µj) ∣ 2 = ∑ ajC(sj − sk)āk ≥ 0 for every set of complex numbers {aj} ⊂ C. A function C0(h) is called positive semi-definite if it always satisfies the inequality ∑ jk ajC(sj −sk)āk ≥ 0 for any locations sj and complex numbers aj; this is equivalent to asking that C(h) = C(−h) for every h ∈ Rn and that ∑ ajC(sj − sk)ak ≥ 0 for all real numbers aj ∈ R. One way to get a symmetric positive semi-definite function C0(h) is by taking the Fourier transform C0(h) = Rn e ih·ω G(ω) d n ω of any positive function G(ω) on R n or, more generally, of any finite positive measure G(dω), because then ajC(sj − sk)āk = jk