560 research outputs found
Zitterbewegung as purely classical phenomenon
Nonrelativistic formalism is developed, which allows describing systems with
internal degrees of freedom in the scalar potential field , which is a
function both on relative coordinates and time, and on relative speed and
accelerations. The equation for energy, which is an integral of motion when
satisfies to certain differential condition, is derived for the general case.
For a free mass point all solutions of the equations of motion in the
center-of-inertia reference frame, moving with constant velocity, are found. As
a result, the center of mass follows along helical and more complicated
trajectories round a direction of motion of the center of inertia. This motion
can be interpreted as trembling movement (Zitterbewegung). On this basis a
conclusion is done that Zitterbewegung has purely classical origin, arising
even in a nonrelativistic case if internal degrees of freedom are taken into
account. The general equation of motion for a spin which can be interpreted
from positions of a classical mechanics is written down. Application of the
obtained results to the electron leads to new conception of electric charge,
sign of which corresponds to a sign of spin polarization.Comment: 24 pages, 4 figure
Mathematical models of optimal allocation of logistical capacities in the regional system of commodity flows ΠΠ°ΡΠ΅ΠΌΠ°ΡΠΈΡΠ΅ΡΠΊΠΈΠ΅ ΠΌΠΎΠ΄Π΅Π»ΠΈ ΠΎΠΏΡΠΈΠΌΠ°Π»ΡΠ½ΠΎΠ³ΠΎ ΡΠ°Π·ΠΌΠ΅ΡΠ΅Π½ΠΈΡ Π»ΠΎΠ³ΠΈΡΡΠΈΡΠ΅ΡΠΊΠΈΡ ΠΌΠΎΡΠ½ΠΎΡΡΠ΅ΠΉ Π² ΡΠ΅Π³ΠΈΠΎΠ½Π°Π»ΡΠ½ΠΎΠΉ ΡΠΈΡΡΠ΅ΠΌΠ΅ ΡΠΎΠ²Π°ΡΠ½ΡΡ ΠΏΠΎΡΠΎΠΊΠΎΠ²
The article considers tasks of allocation and innovation development of logistical capacities. The essence of the first task lies in optimal (from the position of the criterion of minimum expenditures) formation of the regional system of logistical capacities of various types and purposes in possible places of their location. These capacities should treat the set commodity flows, which possess a complex structure and are produced by different centres. The original conditions of the second task are a certain, already existing, system of logistical capacities. It is necessary to conduct its modification by a rational method using methods with various degree of innovation: simple accumulation of capacities of original productivity, modernisation of existing technological units and introduction of new more efficient capacities. In order to solve the specified tasks, the article built optimisation models, which correspond with setting mathematical tasks of partially integer-valued linear programming.Π ΡΡΠ°ΡΡΠ΅ ΡΠ°ΡΡΠΌΠ°ΡΡΠΈΠ²Π°ΡΡΡΡ Π·Π°Π΄Π°ΡΠΈ ΡΠ°Π·ΠΌΠ΅ΡΠ΅Π½ΠΈΡ ΠΈ ΠΈΠ½Π½ΠΎΠ²Π°ΡΠΈΠΎΠ½Π½ΠΎΠ³ΠΎ ΡΠ°Π·Π²ΠΈΡΠΈΡ Π»ΠΎΠ³ΠΈΡΡΠΈΡΠ΅ΡΠΊΠΈΡ
ΠΌΠΎΡΠ½ΠΎΡΡΠ΅ΠΉ. Π‘ΡΡΡ ΠΏΠ΅ΡΠ²ΠΎΠΉ Π·Π°Π΄Π°ΡΠΈ ΡΠΎΡΡΠΎΠΈΡ Π² ΠΎΠΏΡΠΈΠΌΠ°Π»ΡΠ½ΠΎΠΌ (Ρ ΠΏΠΎΠ·ΠΈΡΠΈΠΉ ΠΊΡΠΈΡΠ΅ΡΠΈΡ ΠΌΠΈΠ½ΠΈΠΌΡΠΌΠ° Π·Π°ΡΡΠ°Ρ) ΡΠΎΡΠΌΠΈΡΠΎΠ²Π°Π½ΠΈΠΈ ΡΠ΅Π³ΠΈΠΎΠ½Π°Π»ΡΠ½ΠΎΠΉ ΡΠΈΡΡΠ΅ΠΌΡ Π»ΠΎΠ³ΠΈΡΡΠΈΡΠ΅ΡΠΊΠΈΡ
ΠΌΠΎΡΠ½ΠΎΡΡΠ΅ΠΉ ΡΠ°Π·Π»ΠΈΡΠ½ΠΎΠ³ΠΎ Π²ΠΈΠ΄Π° ΠΈ Π½Π°Π·Π½Π°ΡΠ΅Π½ΠΈΡ Π² Π²ΠΎΠ·ΠΌΠΎΠΆΠ½ΡΡ
ΠΏΡΠ½ΠΊΡΠ°Ρ
ΠΈΡ
Π΄ΠΈΡΠ»ΠΎΠΊΠ°ΡΠΈΠΈ. ΠΠ°Π½Π½ΡΠ΅ ΠΌΠΎΡΠ½ΠΎΡΡΠΈ Π΄ΠΎΠ»ΠΆΠ½Ρ ΠΎΠ±ΡΠ°Π±Π°ΡΡΠ²Π°ΡΡ Π·Π°Π΄Π°Π½Π½ΡΠ΅ ΡΠΎΠ²Π°ΡΠ½ΡΠ΅ ΠΏΠΎΡΠΎΠΊΠΈ, ΠΊΠΎΡΠΎΡΡΠ΅ ΠΎΠ±Π»Π°Π΄Π°ΡΡ ΡΠ»ΠΎΠΆΠ½ΠΎΠΉ ΡΡΡΡΠΊΡΡΡΠΎΠΉ ΠΈ ΠΏΡΠΎΠ΄ΡΡΠΈΡΡΡΡΡΡ ΡΠ°Π·Π»ΠΈΡΠ½ΡΠΌΠΈ ΡΠ΅Π½ΡΡΠ°ΠΌΠΈ. ΠΡΡ
ΠΎΠ΄Π½ΡΠΌΠΈ ΡΡΠ»ΠΎΠ²ΠΈΡΠΌΠΈ Π²ΡΠΎΡΠΎΠΉ Π·Π°Π΄Π°ΡΠΈ ΡΠ²Π»ΡΠ΅ΡΡΡ Π½Π΅ΠΊΠΎΡΠΎΡΠ°Ρ ΡΠΆΠ΅ Π΄Π΅ΠΉΡΡΠ²ΡΡΡΠ°Ρ ΡΠΈΡΡΠ΅ΠΌΠ° Π»ΠΎΠ³ΠΈΡΡΠΈΡΠ΅ΡΠΊΠΈΡ
ΠΌΠΎΡΠ½ΠΎΡΡΠ΅ΠΉ. ΠΠ΅ΠΎΠ±Ρ
ΠΎΠ΄ΠΈΠΌΠΎ ΡΠ°ΡΠΈΠΎΠ½Π°Π»ΡΠ½ΡΠΌ ΠΎΠ±ΡΠ°Π·ΠΎΠΌ Π²ΡΠΏΠΎΠ»Π½ΠΈΡΡ Π΅Π΅ ΠΌΠΎΠ΄ΠΈΡΠΈΠΊΠ°ΡΠΈΡ, ΠΈΡΠΏΠΎΠ»ΡΠ·ΡΡ ΠΌΠ΅ΡΠΎΠ΄Ρ ΡΠ°Π·Π»ΠΈΡΠ½ΠΎΠΉ ΡΡΠ΅ΠΏΠ΅Π½ΠΈ ΠΈΠ½Π½ΠΎΠ²Π°ΡΠΈΠΎΠ½Π½ΠΎΡΡΠΈ: ΠΏΡΠΎΡΡΠΎΠ΅ Π½Π°ΡΠ°ΡΠΈΠ²Π°Π½ΠΈΠ΅ ΠΌΠΎΡΠ½ΠΎΡΡΠ΅ΠΉ ΠΈΡΡ
ΠΎΠ΄Π½ΠΎΠΉ ΠΏΡΠΎΠ΄ΡΠΊΡΠΈΠ²Π½ΠΎΡΡΠΈ, ΠΌΠΎΠ΄Π΅ΡΠ½ΠΈΠ·Π°ΡΠΈΡ ΠΈΠΌΠ΅ΡΡΠΈΡ
ΡΡ ΡΠ΅Ρ
Π½ΠΎΠ»ΠΎΠ³ΠΈΡΠ΅ΡΠΊΠΈΡ
Π±Π»ΠΎΠΊΠΎΠ² ΠΈ Π²Π²ΠΎΠ΄ Π² Π΄Π΅ΠΉΡΡΠ²ΠΈΠ΅ Π½ΠΎΠ²ΡΡ
, Π±ΠΎΠ»Π΅Π΅ ΡΡΡΠ΅ΠΊΡΠΈΠ²Π½ΡΡ
ΠΌΠΎΡΠ½ΠΎΡΡΠ΅ΠΉ. ΠΠ»Ρ ΡΠ΅ΡΠ΅Π½ΠΈΡ ΡΠΊΠ°Π·Π°Π½Π½ΡΡ
Π·Π°Π΄Π°Ρ ΠΏΠΎΡΡΡΠΎΠ΅Π½Ρ ΠΎΠΏΡΠΈΠΌΠΈΠ·Π°ΡΠΈΠΎΠ½Π½ΡΠ΅ ΠΌΠΎΠ΄Π΅Π»ΠΈ, ΠΊΠΎΡΠΎΡΡΠ΅ ΡΠΎΠΎΡΠ²Π΅ΡΡΡΠ²ΡΡΡ ΠΌΠ°ΡΠ΅ΠΌΠ°ΡΠΈΡΠ΅ΡΠΊΠΈΠΌ ΠΏΠΎΡΡΠ°Π½ΠΎΠ²ΠΊΠ°ΠΌ Π·Π°Π΄Π°Ρ ΡΠ°ΡΡΠΈΡΠ½ΠΎ ΡΠ΅Π»ΠΎΡΠΈΡΠ»Π΅Π½Π½ΠΎΠ³ΠΎ Π»ΠΈΠ½Π΅ΠΉΠ½ΠΎΠ³ΠΎ ΠΏΡΠΎΠ³ΡΠ°ΠΌΠΌΠΈΡΠΎΠ²Π°Π½ΠΈΡ
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