In the present paper we introduce a generalization of complete invariance property (CIP) for metric spaces, which will call ε-approximated (ε-ACIP). For our goals, use so called degree nondensifiability (DND) which, roughly speaking, measures (in specified sense) distance from bounded space to its class Peano continua. Our main result relates ε-ACIP with DND and, in particular, proves that dens...

Denote by T (X) the semigroup of full transformations on a set X. For ε ∈ T (X), the centralizer of ε is a subsemigroup of T (X) defined by C(ε) = {α ∈ T (X) : αε = εα}. It is well known that C(idX) = T (X) is a regular semigroup. By a theorem proved by J.M. Howie in 1966, we know that if X is finite, then the subsemigroup generated by the idempotents of C(idX) contains all non-invertible trans...

and Applied Analysis 3 Proposition 3. For any constant R ≥ 0 there exists a positive constant K = K(R) such that, for any initial data u 1 (0), u 2 (0) with ‖u i (0)‖ Hη,ε ≤ R, i = 1, 2 one has 󵄩󵄩󵄩󵄩 S ε,η (t)u 1 (0) − S ε,η (t)u 2 (0) 󵄩󵄩󵄩󵄩Hη,ε ≤ e (K 2 /ε 2 )t󵄩󵄩󵄩󵄩u1 (0) − u2 (0) 󵄩󵄩󵄩Hη,ε , (20) where S ε,η (t) is the solution semigroup of the problem (1). Proof. Let u 1 , u 2 , two solutions of ...

We study the set of solutions of the complex Ginzburg-Landau equation in R, d < 3. We consider the global attracting set (i.e., the forward map of the set of bounded initial data), and restrict it to a cube Q L of side L. We cover this set by a (minimal) number N QL (ε) of balls of radius ε in L∞(Q L ). We show that the Kolmogorov ε-entropy per unit length, H ε = lim L→∞ L −d log N QL (ε) exist...

ε-support vector regression (ε-SVR) can be converted into an unconstrained convex and non-smooth quadratic programming problem. It is not solved by the traditional algorithm. In order to solve this non-smooth problem, a class of piecewise smooth functions is introduced to approximate the ε-insensitive loss function of ε-SVR, which generates a ε-piecewise smooth support vector regression (ε-dPWS...

There is a constant c > 0 such that for every ε ∈ (0, 1) and n > 1/ε, the following holds. Any mapping from the n-point star metric into l1 with bi-Lipschitz distortion 1 + ε requires dimension d > c log n ε log(1/ε) .

Dynamic recrystallization, DRX, behaviour of a precipitation hardened, PH, stainless steel was studied in connection with microstructural developments in a compression test. The experimental results showed that the dominant mechanism of softening is DRX, but at high strain rates and low temperatures, ie, high Zener-Holman parameter, Z, work hardening and dynamic recovery, DRV, produced a pancke...

The introduction of high-speed circuits to realize an arithmetic function f as a piecewise linear approximation has created a need to understand how the number of segments depends on the interval a ≤ x < b and the desired approximation error ε. For the case of optimum non-uniform segments, we show that the number of segments is given as s(ε) ∼ c √ ε , (ε → 0), where c = 1 4 ∫ b a √ |f ′′(x)|dx....

Claim 1 Let q and q′ be any two points in Σ so that d(q, q′) ≤ εmin{f(q), f(q′)} for ε ≤ 1 3 . Then, ∠nq, nq′ ≤ ε 1−3ε . Unfortunately, the proof of this claim as given in Amenta and Bern [1] is wrong; it also appears in the book by Dey [2]. In this short note, we provide a correct proof with an improved bound of ε 1−ε . Theorem 2 Let q and q′ be two points in Σ with d(q, q′) ≤ εf(q) where ε ≤ ...

We show that for every ε > 0 there exists an r0 = r0(ε) such that for all integers r ≥ r0 every graph of average degree at least r + ε and girth at least 1000 contains a subdivision of Kr+2. Combined with a result of Mader this implies that for every ε > 0 there exists an f(ε) such that for all r ≥ 2 every graph of average degree at least r + ε and girth at least f(ε) contains a subdivision of ...