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The paper deals with game-theoretical control problem for the continuity equation. It is assumed that all agents of a multiagent system are influenced by the same controls of both players depending only on current time and current distribution of the agents. We extend the notion of - and -stability and the Krasovskii–Subbotin extremal shift rule to a given case and construct suboptimal strategies of players. Also, the existence theorem for a value function is proved. [ABSTRACT FROM AUTHOR]

The paper continues the research of the author begun in 2003–2021. Quantities of the type of modulus of continuity of functions defined on the sphere in the space are studied. These quantities are generated by a family of operators of multiplier type. Their equivalence to analogs of -functionals is established. [ABSTRACT FROM AUTHOR]

In this paper, the continuity of the set-valued map , , is proved where is the closed ball of radius in the space centered at the origin, is a finite and positive measure space, and is a separable Banach space. An application to input-output systems described by Urysohn type integral operators is discussed. [ABSTRACT FROM AUTHOR]

The aim of this paper is to obtain some common fixed point results on an S-metric space. For this purpose, we prove new generalized common fixed point theorems using the notions of weak commutation, compatibility, and orbital continuity for two and four continuous self-mappings. Our main results generalize the known common fixed point theorems on metric spaces. [ABSTRACT FROM AUTHOR]

The paper discusses the existence of a continuous extension of functions that are defined on subsets of ℝ and whose values are convex bodies in ℝ. This problem arose in convex geometry in connection with the notion, recently introduced in algebraic geometry, of convex Newton-Okounkov bodies. [ABSTRACT FROM AUTHOR]

We obtain sharp Jackson-Stechkin type inequalities for moduli of continuity of kth order Ω in which, instead of the shift operator T f, the Steklov operator S( f) is used. Similar smoothness characteristic of functions were studied earlier in papers of Abilov, Abilova, Kokilashvili, and others. For classes of functions defined by these characteristics, we calculate the exact values of certain n-widths. [ABSTRACT FROM AUTHOR]

We study a parabolic system of the form ∂ t u = div x A( x, t, ▿ x u) in a bounded cylinder Q T = Ω × (0, T) ⊂ ℝ. Here the matrix function A( x, t, ζ) is subject to the conditions of power growth in the variable ζ and coercitivity with variable exponent p( x, t). It is assumed that p( x, t) has a logarithmic modulus of continuity and satisfies the estimate For the weak solution of the system, estimates of the higher integrability of the gradient are obtained inside the cylinder Q T. The method of a solution is based on a localization of a special kind and a local variant (adapted for parabolic problems) of Gehring’s lemma with variable exponent of integrability proved in the paper. [ABSTRACT FROM AUTHOR]

The main results of the paper are contained in Theorems 1 and 2. Theorem 1 presents necessary and sufficient conditions for a sequence of functions h n: 〈 c, d〉 → 〈 a, b〉, n = 1, 2, ..., to have bounded sequences of Ψ-variations { VΨ (〈 c, d〉; f ◦ h n)} evaluated for the compositions of an arbitrary function f: 〈 a, b〉 → ℝ with finite Φ-variation and the functions h n. In Theorem 2, the same is done for a sequence of functions h n: ℝ → ℝ, n = 1, 2, ..., and the sequence of Ψ-variations { VΨ(〈 a, b〉; h n ◦ f)}. [ABSTRACT FROM AUTHOR]

Let C[-1, 1] be the Banach space of continuous complex functions f on the interval [-1, 1] equipped with the standard maximum norm ¦¦f¦¦; let ω(·) = ω(·, f) be the modulus of continuity of f; and let Rn = Rn (f) be the best uniform approximation of f by rational functions (r.f.) whose degrees do not exceed n = 1, 2, …. The space C[-1, 1] is also regarded as a pre-Hilbert space with respect to the inner product given by (f,g) (1/π) &integ;-1¹ f(x)&g(x)macr; (1 - x²)-½ dx. Let zn = [z1, z2, …, zn] be a set of points located outside the interval [-1, 1]. By F(·, f, zn) we denote an orthoprojection operator acting from the pre-Hilbert space C[-1, 1] onto its (n + 1)-dimensional subspace consisting of rational functions whose poles (with multiplicity taken into account) can only be points of the set zn. In this paper, we show that if f is not a rational function of degree ≤ n, then we can find a set of points zn = zn (f) such that ¦¦f(·) - F(·, f, zn)¦¦ ≤ 12Rnln 3 /ω-1(Rn/3). [ABSTRACT FROM AUTHOR]

Questions concerning the density of the sets of points of continuity of metric functions and metric projection onto sets in asymmetric spaces are studied. [ABSTRACT FROM AUTHOR]

The article studies the continuity of left metric functions and the upper semicontinuity of left metric projections onto boundedly sequential left-compact sets in asymmetric spaces. Relationships between the properties of approximative stability and approximative compactness are studied. [ABSTRACT FROM AUTHOR]

To a function $f\backslash in\; L\_2[-\backslash pi,\backslash pi]$ and a compact set $Q\backslash subset[-\backslash pi,\backslash pi]$ we assign the supremum $\backslash omega(f,Q)\; =\backslash sup\_\{t\backslash in\; Q\}\backslash |f(\backslash ,\backslash cdot\backslash ,+\; t)-f(\backslash ,\backslash cdot\backslash ,)\backslash |\_\{L\_2[-\backslash pi,\backslash pi]\}$, which is an analog of the modulus of continuity. We denote by $K(n,Q)$ the least constant in Jackson's inequality between the best approximation of the function $f$ by trigonometric polynomials of degree $n-\; \backslash nomathbreak\; 1$ in the space $L\_2[-\backslash pi,\backslash pi]$ and the modulus of continuity $\backslash omega(f,Q)$. It follows from results due to Chernykh that $K(n,Q)\backslash ge1/\backslash sqrt2$ and $K(n,[0,\backslash pi/n])=1/\backslash sqrt2$. On the strength of a result of Yudin, we show that if the measure of the set $Q$ is less than $\backslash pi/n$, then $K(n,Q)>1/\backslash sqrt2$. [ABSTRACT FROM AUTHOR]

Conditions for the constants of best approximation in the metric of the spaces Lp(B) to be continuous or semicontinuous as functions of the center of a ball B of fixed radius in a metric space with Borel measure are studied. [ABSTRACT FROM AUTHOR]

For the classes of functions W r (ω ℳ , Φ) : = { f ∈ L 2 r (ℝ) : ω ℳ (f (r) , t) ≤ Φ (t) ∀ t ∈ (0 , ∞) } , where Φ is a majorant and r ∈ ℤ+, lower and upper bounds for the Bernstein, Kolmogorov, and linear mean ν-widths in the space L2(ℝ) are obtained. A condition on the majorant Φ under which the exact values of these widths can be calculated is indicated. Several examples illustrating the results are given. [ABSTRACT FROM AUTHOR]

For any t ∈ [0, 1], we obtain the exact value of the modulus of continuity , where L* is the dual Orlicz space with Luxemburg norm and D is the operator of differentition at the point t. As an application, we state necessary and sufficient conditions in the Kolmogorov problem for three numbers. Also we solve the Stechkin problem, i.e., the problem of approximating an unbounded operator of differentition D by bounded linear operators for the class of functions x such that $\left\| {x''} \right\|_{L_N^* [0,1]} \leqslant 1$. [ABSTRACT FROM AUTHOR]

We obtain order-sharp estimates of best approximations for the classes B of periodic functions of several variables by trigonometric polynomials whose spectra are generated by the level surfaces of the function Ω( t). [ABSTRACT FROM AUTHOR]

We obtain sharp inequalities of Jackson-Stechkin type in which the modulus of continuity of the function is defined in terms of the Steklov function. For classes of functions given by this characteristic, we obtain sharp values of the n-widths. [ABSTRACT FROM AUTHOR]

We obtain exact values of best L1-approximations for the classes W r F, r ∈ ℕ, of periodic functions whose rth derivative belongs to a given rearrangement-invariant set F as well as for the classes W r H ω of periodic functions whose rth derivative has a given convex (up) majorant ω( t) of the modulus of continuity by subspaces of polynomial splines of order m ≥ r + 1 of deficiency 2 with nodes at the points 2 kπ/n, n ∈ ℕ, k ∈ ℤ. It is shown that these subspaces are extremal for the Kolmogorov widths of the corresponding function classes. [ABSTRACT FROM AUTHOR]

It is proved that, in the asymptotic formulas for the deviations of Bernstein polynomials from functions at the points of discontinuity of the first kind of the highest even-order derivative, the value of such a derivative can be replaced by the half-sum of its limits on the right and on the left. [ABSTRACT FROM AUTHOR]

We consider a class of convex bounded subsets of a separable Banach space. This class includes all convex compact sets as well as some noncompact sets important in applications. For sets in this class, we obtain a simple criterion for the strong CE-property, i.e., the property that the convex closure of any continuous bounded function is a continuous bounded function. Some results are obtained concerning the extension of functions defined at the extreme points of a set in this class to convex or concave functions defined on the entire set with preservation of closedness and continuity. Some applications of the results in quantum information theory are considered. [ABSTRACT FROM AUTHOR]

A Banach space $X$ possesses the PC (point of continuity) property if for any $w$-closed bounded subset $A\backslash subset\; X$ the identity map $(A,w)\backslash to(A,\backslash |\backslash ,\backslash cdot\backslash ,\backslash |)$ has a point of continuity ($w$ is the weak topology in $X$). We deduce some criteria for Banach spaces to have the PC property and describe (for dual Banach spaces) relationships between spaces possessing the PC property and spaces possessing the RN or the WRN property. [ABSTRACT FROM AUTHOR]

Conditions for the uniform continuity of a family of weakly regular set functions defined on an algebra $\backslash Sigma\backslash supset\; \backslash eta$ of subsets of a $\backslash sigma$-topological space $(T,\backslash eta)$ and taking values in an arbitrary topological space are found. [ABSTRACT FROM AUTHOR]

The article presents an analysis on the continuity of the output entropy of a positive linear map of nuclear operators' spaces on the set of input states. It discusses the von Neumann output entropy of a quantum channel as a discontinuous function on the set of input states. It stresses the output entropy H ࢪ Ф of a positive linear map which is a concave lower semicontinuous function. It describes the theorem showing that the function is continuous if it takes only finite values.