scholarly journals Almost periodic generalized functions

1983 ◽  
Vol 94 (1) ◽  
pp. 149-166
Author(s):  
H. Burkill ◽  
B. C. Rennie
Keyword(s):  
Generalized Functions ◽  
Inner Product ◽  
Almost Periodic ◽  
Test Functions ◽  
Complex Functions ◽  
Entire Complex ◽  
Order Of Magnitude ◽  
The Individual ◽  
Image Position ◽  
Derivatives Of

In (4) a space C of generalized functions was defined which is rather larger than the simple space used to such effect by Lighthill in (3). At the core of C is the space C0 = T of test functions. These are entire (complex) functions f such that all derivatives of f and its Fourier transform F have order of magnitude not exceeding as x → ± ∞, where c is a positive number depending on the individual derivative concerned. If f, g∈ T, the inner product 〈f | g〉 is defined to be

scholarly journals On the Gauss-Green theorem

1968 ◽  
Vol 8 (3) ◽  
pp. 385-396 ◽  
Author(s):  
B. D. Craven
Keyword(s):  
Lebesgue Measure ◽  
Inner Product ◽  
Open Set ◽  
The Individual ◽  
Image Position ◽  
Vector Valued Function ◽  
Vector Valued ◽  
Dimensional Lebesgue Measure ◽  
Lebesgue Integration ◽  
Riemann Integration

In a previous paper [1], Green's theorem for line integrals in the plane was proved, for Riemann integration, assuming the integrability of Qx−Py, where P(x, y) and Q(x, y) are the functions involved, but not the integrability of the individual partial derivatives Qx and Py. In the present paper, this result is extended to a proof of the Gauss-Green theorem for p-space (p ≥ 2), for Lebesgue integration, under analogous hypotheses. The theorem is proved in the form where Ω is a bounded open set in Rp (p-space), with boundary Ω; g(x) =(g(x1)…, g(xp)) is a p-vector valued function of x = (x1,…,xp), continuous in the closure of Ω; μv,(x) is p-dimensional Lebesgue measure; v(x) = (v1(x),…, vp(x)) and Φ(x) are suitably defined unit exterior normal and surface area on the ‘surface’ ∂Ω and g(x) · v(x) denotes inner product of p-vectors.

Summability of Hermite polynomial expansions of generalized functions

1970 ◽  
Vol 68 (1) ◽  
pp. 129-139
Author(s):  
Z. Ditzian
Keyword(s):  
Hermite Polynomial ◽  
Generalized Functions ◽  
Hermite Functions ◽  
Test Functions ◽  
Expansion Theorem ◽  
Tempered Distribution ◽  
Image Position ◽  
Polynomial Expansions

1. Introduction: The well-known result that every function f(x) ∈ L2(–∞, ∞) can be expanded in L2(– ∞,∞) by where and was recently followed by an expansion theorem for generalized functions. In papers by Wildlund(8), Giertz(4) and Zemanian(9) it was shown that a tempered distribution f∈S′ can be expanded by Hermite functions that isfor all ψ∈S (the space of rapidly decreasing test functions).

A Multivariable form of the Fundamental Theorem of Algebra

1983 ◽  
Vol 26 (3) ◽  
pp. 271-272
Author(s):  
Pablo M. Salzberg
Keyword(s):  
Fundamental Theorem ◽  
Homogeneous Polynomial ◽  
Inner Product ◽  
Closed Field ◽  
Sufficient Condition ◽  
Algebraically Closed Field ◽  
Fundamental Theorem Of Algebra ◽  
Image Position ◽  
Derivatives Of

AbstractLet H(x) be a homogeneous polynomial in n indeterminates over an algebraically closed field K. A necesssary and sufficient condition is given for H(x) to admit a factorization of the forma, b∈ Kn, and “∘” is the usual inner product. This condition involves the linear derivatives of H(x).

The Coarse Scale Velocity

2017 ◽  
Author(s):  
Philip Isett
Keyword(s):  
Velocity Field ◽  
Building Blocks ◽  
Coarse Scale ◽  
Divergence Equation ◽  
Higher Derivatives ◽  
Order Of Magnitude ◽  
Derivatives Of

This chapter deals with the coarse scale velocity. It begins the proof of Lemma (10.1) by choosing a double mollification for the velocity field. Here ∈ᵥ is taken to be as large as possible so that higher derivatives of velement are less costly, and each vsubscript Element has frequency smaller than λ‎ so elementv⁻¹ must be smaller than λ‎ in order of magnitude. Each derivative of vsubscript Element up to order L costs a factor of Ξ‎. The chapter proceeds by describing the basic building blocks of the construction, the choice of elementv and the parametrix expansion for the divergence equation.

The role of secondary alcoholic groups of D-galacturonan in its degradation with exo-D-galacturonanase

1980 ◽  
Vol 45 (2) ◽  
pp. 427-434 ◽  
Author(s):  
Kveta Heinrichová ◽  
Rudolf Kohn
Keyword(s):  
Acetyl Derivative ◽  
Competitive Inhibitor ◽  
Hydroxyl Groups ◽  
Pectic Acid ◽  
Substrate Complex ◽  
Enzyme Substrate ◽  
The Individual ◽  
Degradation Degree ◽  
Derivatives Of

The effect of exo-D-galacturonanase from carrot on O-acetyl derivatives of pectic acid of variousacetylation degree was studied. Substitution of hydroxyl groups at C(2) and C(3) of D-galactopyranuronic acid units influences the initial rate of degradation, degree of degradation and its maximum rate, the differences being found also in the time of limit degradations of the individual O-acetyl derivatives. Value of the apparent Michaelis constant increases with increase of substitution and value of Vmax changes. O-Acetyl derivatives act as a competitive inhibitor of degradation of D-galacturonan. The extent of the inhibition effect depends on the degree of substitution. The only product of enzymic reaction is D-galactopyranuronic acid, what indicates that no degradation of the terminal substituted unit of O-acetyl derivative of pectic acid takes place. Substitution of hydroxyl groups influences the affinity of the enzyme towards the modified substrate. The results let us presume that hydroxyl groups at C(2) and C(3) of galacturonic unit of pectic acid are essential for formation of the enzyme-substrate complex.

scholarly journals Statistical Uncertainty of DNS in Geometries without Homogeneous Directions

2021 ◽  
Vol 11 (4) ◽  
pp. 1399
Author(s):  
Jure Oder ◽  
Cédric Flageul ◽  
Iztok Tiselj
Keyword(s):  
Channel Flow ◽  
A Priori ◽  
Time Integration ◽  
Navier Stokes ◽  
Statistical Uncertainty ◽  
Time Step ◽  
Order Of Magnitude ◽  
Averaging Time ◽  
The Individual ◽  
Time Required

In this paper, we present uncertainties of statistical quantities of direct numerical simulations (DNS) with small numerical errors. The uncertainties are analysed for channel flow and a flow separation case in a confined backward facing step (BFS) geometry. The infinite channel flow case has two homogeneous directions and this is usually exploited to speed-up the convergence of the results. As we show, such a procedure reduces statistical uncertainties of the results by up to an order of magnitude. This effect is strongest in the near wall regions. In the case of flow over a confined BFS, there are no such directions and thus very long integration times are required. The individual statistical quantities converge with the square root of time integration so, in order to improve the uncertainty by a factor of two, the simulation has to be prolonged by a factor of four. We provide an estimator that can be used to evaluate a priori the DNS relative statistical uncertainties from results obtained with a Reynolds Averaged Navier Stokes simulation. In the DNS, the estimator can be used to predict the averaging time and with it the simulation time required to achieve a certain relative statistical uncertainty of results. For accurate evaluation of averages and their uncertainties, it is not required to use every time step of the DNS. We observe that statistical uncertainty of the results is uninfluenced by reducing the number of samples to the point where the period between two consecutive samples measured in Courant–Friedrichss–Levy (CFL) condition units is below one. Nevertheless, crossing this limit, the estimates of uncertainties start to exhibit significant growth.

A dynamic programming approach to evolutionarily stable strategy theory

1980 ◽  
Vol 12 (1) ◽  
pp. 3-5 ◽  
Author(s):  
C. Cannings ◽  
D. Gardiner
Keyword(s):  
Dynamic Programming ◽  
Density Function ◽  
Evolutionarily Stable Strategy ◽  
Programming Approach ◽  
Stable Strategy ◽  
War Of Attrition ◽  
Dynamic Programming Approach ◽  
The Individual ◽  
Image Position ◽  
Evolutionarily Stable

In the war of attrition (wa), introduced by Maynard Smith (1974), two contestants play values from [0, ∞), the individual playing the longer value winning a fixed prize V, and both incurring a loss equal to the lesser of the two values. Thus the payoff, E(x, y) to an animal playing x against one playing y, is A more general form (Bishop and Cannings (1978)) has and it was demonstrated that with and there exists a unique evolutionarily stable strategy (ess), which is to choose a random value from a specified density function on [0, ∞). Results were also obtained for strategy spaces [0, s] and [0, s).

The theory of weak functions. I

1963 ◽  
Vol 276 (1365) ◽  
pp. 149-167 ◽  
Keyword(s):  
Continuous Function ◽  
Weak Convergence ◽  
Fourier Transforms ◽  
Generalized Functions ◽  
One Dimension ◽  
Test Functions ◽  
Weak Derivative

This paper develops the theory of distributions or generalized functions without any reference to test functions and with no appeal to topology, apart from the concept of weak convergence. In the calculus of weak functions, which is so obtained, a weak function is always a weak derivative of a numerical continuous function, and the fundamental techniques of multiplication, division and passage to a limit are considerably simplified. The theory is illustrated by application to Fourier transforms. The present paper is restricted to weak functions in one dimension. The extension to several dimensions will be published later.

Word problems related to derivatives of the displacement map

1991 ◽  
Vol 110 (3) ◽  
pp. 569-579 ◽  
Author(s):  
J. Devlin
Keyword(s):  
Periodic Solutions ◽  
Word Problem ◽  
Word Problems ◽  
Abstract Word ◽  
Local Problem ◽  
Image Position ◽  
Derivatives Of

In [6], we considered the equationwhere z ∈ ℂ and the pi are real-valued functions; abstract word-problem concepts and techniques were applied to the local problem of the bifurcation of periodic solutions out of the solution Z ≡ 0. This paper is a sequel to [6]; we present an extension of certain concepts given in that paper, and give a global version of some of our word-problem results.

scholarly journals On a Set of Conform-Invariant Equations of the Gravitational Field

1953 ◽  
Vol 10 (1) ◽  
pp. 16-20 ◽  
Author(s):  
H. A. Buchdahl
Keyword(s):  
Gravitational Field ◽  
Linear Combination ◽  
Curvature Tensor ◽  
Wide Class ◽  
Ricci Tensor ◽  
Empty Space ◽  
Image Position ◽  
Derivatives Of

Eddington has considered equations of the gravitational field in empty space which are of the fourth differential order, viz. the sets of equations which express the vanishing of the Hamiltonian derivatives of certain fundamental invariants. The author has shown that a wide class of such equations are satisfied by any solution of the equationswhere Gμν and gμν are the components of the Ricci tensor and the metrical tensor respectively, whilst λ is an arbitrary constant. For a V4 this applies in particular when the invariant referred to above is chosen from the setwhere Bμνσρ is the covariant curvature tensor. K3 has been included since, according to a result due to Lanczos3, its Hamiltonian derivative is a linear combination of and , i.e. of the Hamiltonian derivatives of K1 and K2. In fact

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