CC BY
10
7. Taussky O. A recurring theorem on determinants / O. Taussky // The American Mathematical Monthly. 1949. V. 56. N 10. P. 672-676.
8. Ahlberg J. H. Convergence properties of the spline fit / J. H. Ahlberg, E.N. Nilson // J.SIAM. 1963. V. 11. N 1. P. 95-104.
Сведения об авторе
Марина Степановна Апанович
канд. физ.-мат. наук, доцент кафедры медицинской кибернетики и информатики «Красноярский государственный медицинский университет имени профессора В. Ф.Войно-Ясе-нецкого» Министерства здравоохранения РФ Россия, Красноярск Эл. почта: rogozina.marina@mail.ru
About the author
Marina Stepanovna Apanovich
Candidate of Sciences (Physics and Mathematics), Associate Professor of the Department of Medical Cybernetics and Informatics
Krasnoyarsk State Medical University named after
Prof. V.F. Voino-Yasenetsky
Russia, Krasnoyarsk
Е-mail: rogozina. marina@mail. ru
УДК 517.1/.18 V.M. Belov1, A.M. Sukhotin2, Ya.I. Shmakov2
1 Tomsk Agricultural College 2National Research Tomsk polytechnic university
ALTERNATRIVE ANALYSIS, CRITERIUM OF BIJECTIVITY AND SOME DOGMAS
In our paper we prove that any mapping f: N —A, with AdN, does not be an injective one, i. е. (AcN) -(A~N)). We proved Euclidean 8th axiom: "The Whole is more than its own Part". This theorem opens a new path to the solution of D. Gilbert's two parts first problem on the continuum. A generalization of the concept of k-countability of infinite sets completes our article. Key words: Continuum-hypothesis, Peano 's axioms, criterion of a bijectivity, sequence of natural numbers, countable set, exact-permutation, a-chain, an infinite set.
1. Introduction. Every mathematical discipline has two languages: the mathematical and the meta-language. By G. Cantor each set is determined by its "not equal and distinguishable" elements, therefore this (as in each alphabet of language) hasn't equal elements, by default. The non-strict inequalities <, > and the non-strict inclusions С, з contain, at least on one side, the variable elements. Let (А, В) be a pair of sets, then there exists [1, Sec. 8] a pair (F, G) of all functions, given on A, B, respectively, such that
Vf 6 F3(A^B): (D(f) С A)&E(f) С В, Vg 6 G3(B ^ A): (D(g) С B)&E(g) С A.
However, the concept of "one-to-one function" allows an incorrect replacement of non-strict inclusions with equalities (by default more often).We use known mathematical texts in this report and we follow the Paul Cohen's forecast about continuum-hypothesis (CH) [2, IV.13]: «A point of view which the author feels may eventually come to be accepted is that CH is obviously false». Linear independence (dependence) is the main concept of the linear space En. "If there is a finite quantity of vector in the basis, the space is said to be finite dimensional and its dimension is equal to the quantity of vectors in its basis. Otherwise, it is infinite dimensional. For an infinite dimension space a basis usually means a sequence of elements x1, x2,... such that every x is uniquely expressible in form x = ^^ a-i^i (meaning that the limit as n becomes infinite x — ^^ atxt is zero)" [3, p. 27]. The basis Bn of En contains n linearly independent vectors and the set with n + 1 vectors in En are linearly dependent and there does not exist any injective mapping of basis Bn+1 in En+1 into the basis Bn in En. Below we shall generalized easily those statements onto infinite-dimensional linear space Em. A mapping ф: A ^ В is said to be injective one, or an injection, if either
a = q holds ф(а) = p(q), or р(а)лр(д) ^a^q. (1)
Instead of "the f is injective one" one speaks [4, II.3.7] also, that "the f is one-to-one function" or "the f is 1-1-correcpondence" [5]. The mapping ф: A ^ В which has ф(^) = В is said to be either "as
mapping on", or "a surjective mapping", or, more shortly, "a suijection". Injective mapping ty: A ^ B is named as bijective mapping or bijection [6, I.6] if it is surjective too, i.e., when it is true (1) and ty(^) = B. In this case one speaks, that sets A and B either are bijective, or they have equal power [6, I.9] and we write A~B. Identical mapping id: A ^ A, with id(a) = a, is an example of the mapping for all noted above three types mappings. A bijection ty: A ^ A is named also a rearrangement, or a permutation, or a transformation of set A. Let symbols I(A,B), S(A,B) and B(A, B) designate the sets of injections, surjections and bijections from A in B or, accordingly, into B. At those notations we have the following equality
B(A, B) = I (A, B) n S(4, B). (2)
The equality (2) shows that we must confirm both two properties of a mapping ty: A^B:
1) the injectivity of ty and 2) its surjectivity ty(^) = B before we can say that the mapping ty: A ^ B is any bijection. Unfortunately, this requirement is ignored at any operating with the term 1-1-cor-recpondence either implicitly, or by default.
2. A bijectivity criterion. Bellow we need a criterion of bijectivity of a mapping ty: A^B that was demonstrated in [7, Th.3.10] and everyone can easily prove this criterion.
Theorem 1. Criterion of a bijectivity. The mapping 9: A ^ B is bijective one if and only if for every splitting of the set A = into nonintersecting subsets At, i E I^N, tree following conditions are fulfilled:
(1) Vi mapping tyi: At ^ B is any injection, where ty = ,
'Ai
(2) V(i, j: i ± j) BintBj = 0, where Bt = ),
(3) UiBt + C^C = B.
2. Peano's axioms and Galileo Galilei's paradox. Now we write down and use the Peano's axioms of the natural numbers set N [8, 3.1] by modern traditional symbols, in partiqular:
, 3!, A, c, =, ( ), & etc .
(P1) 0 e N.
(P2) Vx e N, 3! y = x'eN, the x' is said to be "immediately follows to x", the 3! excludes
branch points, and (P2) ^ VzeN3z': (z, z'), specifically, 3a A 0': (0,a), ^ (P4) y' =
x' holds x = y .
(P3) VzeN z' (P1)- (P3) ^ N = (0, a, a',...)
(P5) Let any attribute Q(x) be truth: Q(x) ^ x = x. If
(I) Q(0) ^ 0 = 0 &
(II) (VxeN Q(x) ^x = x) ^ (Q(x') ^ x' = x'), therefore
(III)VzeN Q(z) ^ z = z.
Axiom (P5) affirms the principle of mathematical induction. Further we use this mathematical induction technique every often. Galileo Galilei discovered on 1638 [9, c. 140-146] the paradox which is Va E N 3b = a2eN. G. Galilei has said about this as following: "The notions both either equality or inequality can be used only for finite qualities and not be applied for the infinite ones".
We find the correct boundary between finite and infinite using the concept of linear order. By definition each subset of the finite set (except trivial) has two boundary points. The set is called as an infinite set if there exits its subset, which has less two boundary points. Our methodology emphasizes the priority of the concept of "ordering" in comparison with the concept of "finite-infinite". However, on the end XIX - the beginnings of XX centuries there was appeared far-reaching with mistakes generalization of G. Galilee's opinion about his paradox. S. Kleene write down [5, IV.32] this point of view as following: " ... it is possible to establish 1-1-correcpondence between squares ofthe positive integers
numbers and itself integers, that conflicts with Euclidian axiom [10], according to which whole more than any of its own parts... ."
Now we shall give two new interpretation of G. Galilee's paradox. At the first, to be exact we prove the mapping ф: N ^ N, with ф(п) = n2, is not realizable on the all set N. We shall consider the square In x In, where In = (0,1,2,..., n). Let now S(n) be the following statement: "The ф(п) = n2 is not realizable on the all In ". We prove Vn E N the statement S(n) by mathematical induction technique.
(I) Let n=2, so I2 = (0,1,2) and ф(/2) = (0,1,4)-(с l2 = (0,1,2)). Then S(2) = truth. (II. 1) Let n=k and S(k) = truth, by induction.
(II.2) Let n=k+1, then ф(1к+1) = ф(1к )^ф(к + 1). Since ф(к + 1) = (к + 1)2 and with (II.1) ф(1к+1)-(С Ik+i). Therefore,
(III) Vne N, n > 1, S(n) = truth.
Secondly, let Q+ is the positive rational numbers amount. In the paper [11, p.108-109] we obtained an estimate of Q+ in the following form:Q+ « 0,6|N| • \N\.
Now, let both к = t-2 > 0, be any pozitiv parametr, q E Q+, x E R+ and q = f(x) = kx2. Thus we have x = t • Jq. So the domain of definition f(x) is an interval (0,V£>], at limq = |Q+| = 5o, with |oo| < |ot| (see, please, it.7-8).
3. Stephen C. Kleene and Cantor's diagonal method. Stephen C. Kleene had used [12, I, 1,2] the Cantor diagonal method for the proof an uncountability of all natural sequences sets. Namely, he has chosen the countable subset of natural sequences ff(k) from the set (faof all ones. Farther, S. Kleene wrote the sequences ff(k) in the infinite-dimensional square matrix H
Ifo (0) fo*(1) /о (2) ■ HA{fi(0) fi*(1) fi*(2) ••• I (3)
\ Я(0) fi (1) fz(2) ■
Further S. Kleene has defined one more sequence under the formula f*(k) = fj*(k) + 1. It is obvious, that the sequence f*(k) has not entered in the list-matrix Hof countable subsets of sequences. And consequently Stephen C. Kleene has formulated the following statement: "The set of all number natural sequences is not countable one." Realy, S. Kleene has proved much greater [13]. He proved that there not exists any bijection between the set H of sequences ft*(k) and set H* = H U iff (к)} . We write below some proofs of this statements.
Let Bn+1 = (b0, b1,..., bn) and Hn = (h0, h1, ..., hn-1) be bases of linear spaces En+1 and En, accordingly. We shall use the mathematical induction technique for proving the following statement.
Theorem 2.
VneN3(i, j, m): i Ф j, i + 1, j + 1, m< n:V (ф: Bn+1 ^ Нп)^ф(Ь1) = ф(Ь^ = hm. (I) Let n=1, then Bn+1 = B2 = (b0, b1) and Hn = H1 = (h0). Now we have following equflities: ф(Ь0) = ф(Ь1) = h0, so Th. 2=truth. (II. 1) Let n=k and Th. 2 = truth, by induction.
(II 2) Let n=k+1.Then Bn+i = Вк+2 = Oo, •", bk+i), Hn = Hk+i = (ho, Ьъ ■■■, bk ).
Now we have q(Bk+2) = q(Bk+i U {bk+J) = ip(Bk+i)vjip({bk+i })cHfc+i.Thus 3m, m<k + 1: ^(bk+1) = hm E Hk+1. At the second hand, by virtue ^(Bk+1)<^Hk+1, 31,1 < k + 1: ty(bt) = hm. Then we have q>(bt) = hm = ^(bk+1). That is Th. 2 = truth in this case too.
Therefore, (III) Vne N, n > 1,Th. 2 = truth.
4. New proof Euclidean 8th axiom. Let in (3) matrix Hbe identity matrix HE
HE =
/ 1 0 0 0 0 .\ 0 1 0 0 0 ...
0 0 . 01 0 . V ... !
The lines of matrix HE are canonic natural sequences fa = : к = 0, oo), i = 0, oo, here 8f - are Kron-ecker's symbols. Those sequences make up a basis B(F) of infinite-dimensional space Fthat is the set of all natural sequences (ft (&)fc=0)^0 , i, k6N [3, p. 349]. For example, if f* = (0 1 1 0 ...), thus f* = f1 + f2. Now we identifier every element fk of basis B(F) with the corresponding natural number k by means following bijection
ф: N^B(F), ф(к)= fk. (4)
Let F* = B(F) U {/*}. Now we shall prove the following main statement.
Теорема 3. There is not mapping y: F*—B(F) that is injective one. (1) Зт: ф(/*) = bm. (2) Yet the mapping y: B(F) ^ B(F)\{bm} is not injective by virtue of both Theorem 1 and Theorem 2.
Theorem 3 is equivalent by virtue of the bijection (4) with Euclidian 8th axiom for the set N of all natural numbers in following form:
N^N* = NU {a, a<£N}^(-(N*~N)).
The second example is more instructive: The bijective mapping exists by virtue of the bijection (4) between the set of all even natural numbers and the set, for example, those and only those vectors of the basis B(F) , which have even indexes. The legend about an existence of any bijective correspondence between of natural numbers set and the set of even numbers was constructed on any misunderstanding if to not say more.
5. E'xaci-permutation. In this item we use essentially the Axiom of Choice [14] and a notion of exact-permutation [7, 3.7] in our proofs. If f 6 F(A, A) and H С A, then not always f (H) Ф H .
A set Bex( A, A) of all exact rearrangements of a set A (exact-permutation or anti-cyclic permutation, a rearrangements without cycles [7, Ch. 3]) we define by the following equality
Bex (A, A) = {/: (f 6 В (A, A)&f(H) = H) ^H = A}. (5)
For example, the bijection f 6 Bex (I, I), I = [0,1] , can be defined with following formula:
fix) = {1
x + h, if 0 < x < 1 — h, — x, if 1 — h < x < 1,
where h<0,5 and h is a transcendental number.
The graph of this function in the obvious image testifies, that the given function f satisfies to a condition (5). It is easy to write down exact rearrangements on set of natural numbers but what would not be the first element in image at a mapping f E Bex (N, N), the last element there cannot be, that is obvious, by virtue of the potential nature of set N.
Now we shall choose a pair {a,y}, a E A and y E Bex (A, A) by means of the Axiom of a choice [15, 0.23-0.24] from sets A and Bex(N, N) . Further we define with the help of the pair {a,y} following
sequence of investments of subsets of set A, named ([3, Sec. 14], [15, 0.9; 0.23]) as a chain (a chain on an investment):
{а} с {a, b} с {a, b, с} с ■■■ с {a, b, с, ■■•, p} с {a, b, c, ■■•, p, q} с — с A. (6)
where b = y(a), с = y(b), ■■•, q = y(p), ■■• .
The remark 1. If the element ф-1(а) exists, then a decreasing chain on an investment
А з Л\{а} э^\{о, b} э^\{а, b, с} з ■■■ з A\{a, b, ■■•p, q,} з ■■■ з ф-1(а) (7)
will be defined by means of a full chain (6) by obvious image. We name the chain (7) as dual-chain for a-chain (6).
Theorem 4. There exists such mapping феВех(A, A) V(y, Y: yeYc A) that
y-chain P(y, ф, A)^>Y.
The proof of Theorem 4 uses the limitation of exact-permutation ф on the subset YcA, i.e., if o= ф|Y , then aeBex(Y, Y).
Let (A,v) be a set А with order v from it.5. Further let P(A) be a set of all subsets of set A. Now we consider the class containing all elements of the set P(A), the set of all a-chains and the
set of all dual-chains. We shall write down two definitions (on [15, 0.23]) for the family Element H of family P(A)) is said to be the maximal element of this family, if and only if no element G c^(P(A)) contains H as own part. Element G is said to be similar image the minimal element of family in only case, when no element S contains as own part in G. It is obvious,
that the minimal elements of family are single-element subsets of set A and only they. Simi-
larly, both the set {P(q, ^, A): q 6 A, ^6 B(A, A) } of all full q-chains and the set of dual-chains of set A form the set of all maximal elements of a family P(A)).
Now everyone can easily prove the following statement.
Theorem 5. Any chain V, being either some own part of the full q-chain or an own part of the dual-chain, has the least element.
6. Another proof Euclidean 8th axiom. Now we shall prove Euclidian 8th Axiom in the form of the following theorem.
Теорема 6. Let both ВсА and ф 6 F(A, B), then there exists such pair (a, q) of elements a and q into the set A, that
а * q and ф(а) = ф^). (8)
Proof. Let ф(^) = В. It will not break a generality of our reasoning. Now we shall assume opposite (8), i. е., <peI(A, В) * 0 at BcA and let further H = A\B, then Hn>B = 0. Now we have a circuit of equalities in those designations: В = ф(Л) = ф(В U H) = ф(В) U ф(Н). Therefore, ф(В) С В and ф(Н) С в. An inclusion ф(Н) С в means, that Vh 6 Н and b 6 В exists such that ф^) = b. On the other hand, if ф(В) = В, then an element geB exists for every beB by virtue of ye I (A, B) such, that ф® = b = ф^) at g^h, since Bn*H = 0. This proves the condition (8). In case ф(В) = B1cB and H1 = B\B1, then ф(В) = ф(В1 U Я1) = ф(В1) U р(Я1). Therefore, ф(В1) С B1 and ф(Н1) С в1. As well as above we prove, that either ф(В1) = B1 and the condition (8) is proved, or ф(В1) = В2сВ1, H2 = B1\B2 and so on. Thus, we shall receive following a decreasing sequence Z of investments: з.... Here ieJ and J is some set of indexes corresponding con-
structed chain Z. This chain Z contains by virtue either of Theorem 4 or lemma Kuratowski [15, 0.25. (d)] in some maximal chain. Therefore in our case the chain Z has any least element. It means that □ k □ J : ф(Вк-1~) = Bk and ф(^) = BkcBk-1. If as above, Hk = Bk-1\Bk * 0, then ф(Вк-1~) = ф(Вк) = ф(Вк)иф(Я&) = Bk. Therefore, ф(Вк) = Вк and ф^) С вк at ВкпЯ,, = 0 by
virtue of a choice of the set Hk. It means that the conclusion (8) of Theorem 6 is proved. Theorem 6 has following the canonical brief form: BcA^—A~B.
In particular, qgA^— (A~ (Au{q})), and we speak about this so: concerning equivalence the family of infinite sets (as well as finite sets) divides on classes "to within an element". Obviously in turn, this result opens a new way both of research a continuum-hypothesis [2] and the theory of infinite sets power [16, II, 4].
Now we have written down below without the proof only two of statements equivalent to the Theorem 6, which as well as Theorems 1 -6 are obvious to the finite sets.
Statement 1. If B^A and 9e/ (B, A), then 3f E I(A.= 9.
Statement 2. A~B^(Bex(B, B)~Bex(A, A))& (B(B, A)~B(A, A))
The second part of the Statement 2, namely the equivalence
(B(B, A)~B(A, A))& ((Bex(B, B)~Bex(A, A)),
characterizes the equivalent sets as the indistinguishable ones in the functional attitude.
7. On finite and infinite. We find the correct boundary between finite and infinite using the concept of linear order. By definition each subset of the finite set (except trivial) has two boundary
points. The set is called as an infinite set if there exits its subset, which has less two boundary points.
_ f
Let q E [q, A C R+ a R+ U {to}] be a real variable, /( A, B) be the set of injections A^B as above. Let, further,/ EC2 , VaEA f'(a) > 0, /'' (a) < 0, thus limf (x) = to. For example, f1(x) A ln(x), f2(x) A Vx, h(x) A kx, 0 < k < 1.
Theorem 7. There exist such number k, 0 < k <1, for each monotonic convex function f E /&( A, B) c /( A, B) that /(to) =h(a>).
The symbol to of infinity has unusual algebraic properties: a + to = to, a • to = to, to + to = to, to-to = to, toot = to etc.
Definition. We call the limit value of function f E /&( A, B) as an infinitely large number (ILN) with symbol fi(/).
Now we present new extension of the set of real numbers. Linear functions have limx^ h(x) = limx^ m(kx) A a)k A <xk at 0 < k <1, as usual we take to1 Aw.
Every functionf: R+^R+ with the condition /'(a) > 1, a E H c R+, grows almost everywhere on H faster than the linear function h (x) Akx and reaches its fmax=to1 at x = maxyER+f-1(y) < H.
Of course, this functionf: R+^R+ will be almost everywhere concave and its graph will lie almost everywhere above its tangent.
A generalization of the concept of both k-countability and (r, p, q, s) — countability [16, II.4] completes our article.
Let both H c {E, N, Z, Q, R, R , C, Rn etc} AM be some index subset and ty: H ^ M c M, and
kklimtE,9(t)^(t)} , (9)
LEn tPln(t) v '
where Lnq(x) A{ln{. {ln(x)} . }} is q- operator for 0 < k < to1. Now we can name the setM as H,k,p,q)-countable set. Let, for example, MAN, HAN, thus ty( t) = t, q=p=1 and k = 1. Now let MA Q+, HAN, by item 2, thus ty( t) = 0.6 • t2, q=1, p=2 and k = 1,2.
8. Discrete ordered of all decimals set. In 1997 we obtained [17, p. 82-85] an algorithm for constructing an infinite matrix whose lines are of the unit interval [0,1]. As is well known, the infinite decimal a has the following recording forms, for example a E [0.1]:
a A 0. %a2a3 ...an ... A 0 + %/10 + a2/102+... + an/10n+..., at E {0,1,2 ,3 , .,8,9}. (10)
Here the infinity in (10) is assumed to be a potential nature. Let now we have by definition 1iS = {0.0,0.1,0.2,0.3.....0.9},
^S = {0.01,0.02,0.03,... 0.31,... 0.91, .,0.99}.
Let further SZi10 = 10S U ^S, . •., 5d10 = U'k=1 ^S. Thus the 5d10 is full list of all d-valued decimal fractions of the interval [0, 1]. Now we complete the lists Sk10, k<d, with the necessary zeros on the right, thus a matrix S = (Sd,10) of the list Sd10 has a size (10d x (d + 1)). Let 5ы10 = timd6N (Sd,10), here we understand the limit as an unlimited possibility of a transition in the matrix S from the list 5d10 into the list Srf+x^: each of the 10 digits 0, 1, ..., 8, 9 -is assigned to each line of the matrix S. Thus we summarize both following equalities
|Sd+1,1o| = 10 • \Sd,1o\ , (OWo)) = <10ra, o) and 5ш,ю = {ак: ак = ^ aki • 10-, k, i 6 N }.
Finally, we call the qualitative results: 1) the matrices (Sd,10) and 5ы10 are nondiagonalizable, 2) The lists 5d10 and 5ы10 are discrete. The lines of the matrix Sw10 contain decimals of three types:
1) finite fractions (am), 2) periodic fractions (ak) and irrational fractions (algebraic and transcendental). 1) There exists for number am = ^ ami 10- such number i0: Vj > i0 amj = 0,
there exists for number ak = ^ aki 10- both a period (akioakio+i ... akio+q) and number j0,
i0 < j0 < i0 + q: akjo * 9. For the matrix Sw10 the author do not able to name the characteristic features of the third type numbers in view of the infinite orders of the corresponding algebraic equations.
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Information about authors
Сведения об авторах
Viktor Mikhilovich Belov
docent
Tomsk Agricultural College Е-mail: bvm@tpu.ru Russia, Tomsk
Alexander Mikhilovich Sukhotin
docent, unemployed prof. RAE National Research Tomsk polytechnic university Е-mail: asukhotin@yandex.ru Russia, Tomsk
Виктор Михайлович Белов
доцент
Томский сельскохозяйственный колледж Эл. почта: bvm@tpu.ru Россия, Томск
Сухотин Александр Михайлович, доцент, безработный проф.
Национальный исследовательский Томский политехнический университет Эл. почта: asukhotin@yandex.ru Россия, Томск Ярослав Игоревич Шмаков магистрант, студент,
Национальный исследовательский Томский государственный университет Эл. почта: y.s.unegound2@bk.ru Россия, Томск
Yaroslav Igorevich Shmakov
magistrant, student
National Research Tomsk government university Е-mail: y.s.undegound2@bk.ru Russia, Tomsk
УДК: 004.021; 614.2
А.Е. Власенко, Н.М. Жилина, Г.И. Чеченин, А.А. Кожевников
НГИУВ - филиал ФГБОУ ДПО РМАНПО Минздрава России
ИНФОРМАЦИОННОЕ ОБЕСПЕЧЕНИЕ СТРАТЕГИИ РАЗВИТИЯ ПРОМЫШЛЕННОГО РЕГИОНА (НА ПРИМЕРЕ НОВОКУЗНЕЦКОГО РАЙОНА)
Рассматривается один из возможных подходов к созданию алгоритма интеграции показателей, отличающийся простотой и удобством в практическом применении в различных социально-экономических отраслях. Показано применение алгоритма для выявления зон риска здоровья при разработке программы развития промышленного региона (на примере Новокузнецкого района).
Ключевые слова: риски здоровья, интеграция показателей, связи «среда-здоровье», классификация проблем, автоматизированный мониторинг.
INFORMATION SUPPORT FOR STRATEGY OF DEVELOPMENT OF INDUSTRIAL REGION (ON THE EXAMPLE OF NOVOKUZNETSK DISTRICT)
One of the possible approaches to the creation of the algorithm of indicators integration, which is simple and convenient in practical application in various socio-economic sectors, is considered. The application of the algorithm to identify health risk areas in the development of the program of development of the industrial region (for example, Novokuznetsk district).
Key words: health risks, integration of indicators, environment-health relations, classification ofprob-lems, automated monitoring.
Введение. Основная цель стратегии развития промышленного региона - определение путей повышения уровня и качества жизни населения, улучшение социально-экономических условий, обеспечение устойчивого развития территории, повышение привлекательности территории для проживания и инвестиций [2, 3, 5]. В связи с тем, что социально-экономическое развитие территории включает в себя множество сфер и отраслей, для разработки стратегии развития промышленного региона (на примере Новокузнецкого района) были привлечены группы специалистов в различных областях. В качестве экспертов и аналитиков в сфере охраны здоровья выступили сотрудники Новокузнецкого государственного института усовершенствования врачей [1,4,7]. Было необходимо проанализировать ситуацию и выработать конкретные управленческие мероприятия, наиболее приоритетные с позиции решения проблем в области общественного
A.E. Vlasenko, N.M. Zhilina, G.I. Chechenin, A.A. Kozhevnikov
NSIFTPh - Branch Campus of the FSBEI FPE RMACPE MOH
