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1

Hyers-Ulam stability of quadratic forms in 2-normed spaces

Park Won-Gil, Bae Jae-Hyeong

Demonstratio Mathematica

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2019

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Vol. 52, nr 1

496--502

EN

In this paper, we obtain Hyers-Ulam stability of the functional equations f(x+y, z+w) + f(x-y, z-w) = 2f(x, z) + 2f(y, w), f(x+y, z-w) + f(x-y, z+w) = 2f(x, z) + 2f(y, w) and f(x+y, z-w) + f(x-y, z+w) = 2f(x, z) - 2f(y, w) in 2-Banach spaces. The quadratic forms ax2+bxy+cy2, ax2+by2 and axy are solutions of the above functional equations, respectively.

2

One some cancellation algorithms II

Zakarczemny M.

Technical Transactions

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2017

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Vol. 5(114)

97--103

EN

We define bƒ(n) to be the smallest integer (a natural number) d such that numbers ƒ(n1 , n2, ..., nm), where n1+n2+ ... + nm ≤ n are not divisible by d. For the given functions ƒ : ℕm → ℕ, we will obtain the asymptotic characterisation of the sequence of the least non canceled numbers (bƒ(n)) n∈ℕ. In the case ƒ : ℕ2ͽ (k, l)→k3+l3∈ℕ, this characterisation can be rewritten in the terms of the permutations polynomials of finite commutative quotient ring ℤ/mℤ. There are situations in which we cannot expect formula for bƒ(n) to be simple, but we can provide the upper and lower bounds of it.

PL

Definiujemy bƒ(n) jako najmniejszą d∈ℕ, taką że liczby ƒ(n1 , n2, ..., nm), gdzie n1+n2+ ... + nm ≤ n są niepodzielne przez d. Dla wybranych funkcji ƒ : ℕm → ℕ, znajdziemy wartości elementów ciągu (bƒ(n)) n∈ℕ, lub podamy inną charakteryzacje. Dla funkcji ƒ : ℕ2ͽ (k, l)→k3+l3∈ℕ, Charakteryzacja ciągu (bƒ(n)) n∈ℕ może być podana z użyciem wielomianów permutacyjnych skończonego, przemiennego, pierścienia ilorazowego ℤ/mℤ. W szczególnych przypadkach funkcji f podamy dolne i górne ograniczenia na wartości ciągubƒ(n).

3

Equivalence of second order optimality conditions for bang-bang control problems. Part 1: Main results

Osmolovskii N. P., Maurer H.

Control and Cybernetics

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2005

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Vol. 34, no. 3

927-950

EN

Second order optimality conditions have been derived in the literature in two different forms. Osmolovskii (1988a, 1995, 2000, 2004) obtained second order necessary and sufficient conditions requiring that, a certain quadratic form be positive (semi)-definite on a critical cone. Agrachev, Stefani, Zezza (2002) first, reduced the bang-bang control problem to finite-dimensional optimization and then show that well-known sufficient optimality conditions for this optimization problem supplemented by the strict bang-bang property furnish sufficient conditions for the bang-bang control problem. In this paper, we establish the equivalence of both forms of sufficient conditions and give explicit relations between corresponding Lagrange multipliers and elements of critical cones. Part 1 summarizes the main results while detailed proofs will be given in Part 2.

4

Second-order sufficient conditions for an extremum in optimal control

Osmolovskij N.

Control and Cybernetics

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2002

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Vol. 31, no. 3

803-831

EN

Suffcient quadratic optimality conditions for a weak and a strong minimum are stated in an optimal control problem on a fixed time interval with mixed state-control constraints, under the assumption that the gradients of all active mixed constraints with respect to control are linearly independent. The conditions are stated for the cases of both continuous and discontinuous controls and guarantee in each case a lower bound of the cost function increase at t1e reference point. They are formulated in terms of an accessory problem with quadratic form, which must be positive-definite on the so-called critical cone. In the case of discontinuous control the quadratic form has some new terms related to the control discontinuity.