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Mathematical solution
In mathematics, a weak solution (also called a generalized solution) to an ordinary or partial differential equation is a function for which the derivatives
Weak_solution
Game whose outcome can be correctly predicted
simple enough to remember (e.g., Maharajah and the Sepoys). An ultra-weak solution (e.g., Chomp or Hex on a sufficiently large board) generally does not
Solved_game
Type of differential equation
example for a weak solution is as follows: ϕ ( x ) = 1 4 π 1 | x | {\displaystyle \phi (x)={\frac {1}{4\pi }}{\frac {1}{|x|}}} is a weak solution satisfying
Partial_differential_equation
Mathematical tools
}fv\,dx.} This is what is usually called the weak formulation of Poisson's equation. Functions in the solution space V {\displaystyle V} must be zero on
Weak_formulation
continuous, then there exists a unique solution u ∈ H 0 1 ( Ω ) {\displaystyle u\in H_{0}^{1}(\Omega )} to the weak problem (*). If further A ( u , φ ) {\displaystyle
Elliptic boundary value problem
Elliptic_boundary_value_problem
On weak solutions of differential equations
classical solution satisfying the PDE in regular sense, so we search for a weak solution at first, and then find out whether the weak solution is smooth
Regularity_theory
Chemical compound
at 14%, 20 °C. Stoichiometric solutions are fairly alkaline, with pH 11 or higher since the hypochlorite ion is a weak base: OCl− + H2O ⇌ HOCl + OH− The
Sodium_hypochlorite
Measure of the tendency of an acid to dissociate
present, in solution, in equilibrium with each other. HA ⇌ H+ + A− Acetic acid (CH3COOH) is an example of a weak acid. The strength of a weak acid is quantified
Acid_strength
Geometric flow
a proper and locally Lipschitz function u on M which is a positive weak solution on the complement of U and which is nonpositive on U; moreover such
Inverse_mean_curvature_flow
2022 South Korean television series
Weak Hero (Korean: 약한영웅) is a South Korean television series written and directed by Yoo Soo-min with Kim Jin-seok and Park Dan-hee, starring Park Ji-hoon
Weak_Hero
Second-order partial differential equation
(see weak solution). It is common to take a different sign convention for this equation than one typically does when defining fundamental solutions. This
Laplace's_equation
Aqueous solution of a weak acid and its conjugate base
acts as a buffer in the ocean. Buffer solutions resist pH change because of a chemical equilibrium between the weak acid HA and its conjugate base A−: HA
Buffer_solution
Solution to partial differential equation
(defined below), one can define a type of weak solution called viscosity solution. Under the viscosity solution concept, u does not need to be everywhere
Viscosity_solution
Theorem in probability theory
class of stochastic differential equations a weak solution with pathwise uniqueness implies a strong solution and uniqueness in distribution. In its original
Yamada–Watanabe_theorem
Mathematical theorem
bilinear form can be "inverted" to show the existence and uniqueness of a weak solution to a given boundary value problem. The result was proved by J. Nečas
Babuška–Lax–Milgram_theorem
Concept in optimization
always less than or equal to the solution to the dual minimization problem. So, in short: weak duality states that any solution feasible for the dual problem
Weak_duality
Boundary condition for generalized functions
with prescribed boundary conditions (boundary value problems), where weak solutions may not be regular enough to satisfy the boundary conditions in the
Trace_operator
American chemist, engineer, scientist
bulbs with another solution of the acid, but in a weaker solution. If he left the filled bulb for a while with this weaker solution it would clean out
Marvin_Pipkin
Elliptic partial differential operator
as weak solutions. For example, we say that a function u belonging to the Sobolev space W 1 , p ( Ω ) {\displaystyle W^{1,p}(\Omega )} is a weak solution
P-Laplacian
Photographic process
paper with a weak solution of ordinary table salt (sodium chloride), blotting and drying it, then brushing one side with a strong solution of silver nitrate
Salt_print
Base that does not associate completely with hydrogen ions
acid. Upon dissolution in water, a weak base does not dissociate completely, so that the resulting aqueous solution contains only a small proportion of
Weak_base
Auxiliary functions used to probe equations, distributions, and weak formulations
Schwartz function is also a Schwartz function. In the theory of weak formulations and weak solutions of partial differential equations, the term test function
Test_function
Differential equations involving stochastic processes
what constitutes a solution to the SDE. There are two main definitions of a solution to an SDE, a strong solution and a weak solution Both require the existence
Stochastic differential equation
Stochastic_differential_equation
Concept in mathematics
unconditional existence and uniqueness for weak solutions of the harmonic map heat flow. Moreover, he found that his weak solutions are smooth away from finitely many
Harmonic_map
Type of differential operator
lemma and Fredholm alternative, states the sufficient condition for a weak solution u to exist in the Sobolev space Hk. For example, for a second-order
Elliptic_operator
Generalisation of the derivative of a function
products of functions also hold for the weak derivative. This concept gives rise to the definition of weak solutions in Sobolev spaces, which are useful for
Weak_derivative
Solution in which the solvent is water
characteristic of an aqueous strong electrolyte solution; the solutes in a weaker electrolyte solution are present as ions, but only to a small degree
Aqueous_solution
Mathematical equation
that every weak solution of Laplace's equation is a smooth solution. This contrasts with the wave equation, for example, which has weak solutions that are
Weyl's lemma (Laplace equation)
Weyl's_lemma_(Laplace_equation)
Antiseptic solution rubbed on skin before surgical operations
Tincture of iodine, iodine tincture, or weak iodine solution is an antiseptic tincture containing iodine, ethanol and water. It is usually 2% elemental
Tincture_of_iodine
Oxidizing acid mixture containing sulfuric acid and hydrogen peroxide
container containing even a weak or depleted piranha solution should be labelled appropriately to prevent this. The solution should be allowed to cool,
Piranha_solution
3-dimensional Navier–Stokes equations is not known to hold for weak solutions, since a weak solution u {\displaystyle u} is only known to satisfy u ∈ L 2 ( [
Lions–Magenes_lemma
Substance whose dissolved ions conduct electricity
weak. The properties of electrolytes may be exploited using electrolysis to extract constituent elements and compounds contained within the solution.
Electrolyte
coerciveness, solution can be shown to exist in a weak sense, so-called "weak solutions". A general regularity theorem than implies that the solutions of the
Sobolev spaces for planar domains
Sobolev_spaces_for_planar_domains
Measure of the level of acidity or basicity of an aqueous solution
equilibrium of solutions of sodium hydroxide at higher concentrations must also be considered. A weak acid or the conjugate acid of a weak base can be treated
PH
example of a stochastic differential equation which admits a weak solution but has no strong solution. It is named after the Japanese mathematician Hiroshi Tanaka
Tanaka_equation
Chemical compound giving a proton or accepting an electron pair
the solution to cause the acid to remain in its protonated form. Solutions of weak acids and salts of their conjugate bases form buffer solutions. To
Acid
equation) is a stochastic differential equation which has a weak solution but no strong solution. It is therefore a counter-example and named after its discoverer
Tsirelson's stochastic differential equation
Tsirelson's_stochastic_differential_equation
Theorem limiting types of conformal mappings in Euclidean space of dimension > 2
derivative, T is the matrix transpose, and I is the identity matrix. A weak solution of this system is defined to be an element f of the Sobolev space W1
Liouville's theorem (conformal mappings)
Liouville's_theorem_(conformal_mappings)
Functional analysis theorem
function can be "inverted" to show the existence and uniqueness of a weak solution to a given boundary value problem. The result is named after the mathematicians
Lions–Lax–Milgram_theorem
Type of partial differential equation
existence of a weak solution of a differential equation with reasonable ease using some functional analysis arguments. However, the weak solution exhibited
Free_boundary_problem
Type of fuel cell
diffuses through the membrane without reacting, methanol is fed as a weak solution: this decreases efficiency significantly, since crossed-over methanol
Direct_methanol_fuel_cell
Chemicals used to whiten or disinfect
solution, 3–6%, (common household bleach) is typically diluted for safe use when disinfecting surfaces and when used to treat drinking water. A weak solution
Bleach
Board game
in a draw. After eighteen years, they have computationally proven a weak solution to the game of checkers. Using between two hundred desktop computers
English_draughts
Vector space of functions in mathematics
that weak solutions of some important partial differential equations exist in appropriate Sobolev spaces, even when there are no strong solutions in spaces
Sobolev_space
Unequal size of the eyes' pupils
stimulus, segmental iris constriction, and sensitivity of pupil to a weak solution of pilocarpine. Oculomotor nerve palsy: Ischemia, intracranial aneurysm
Anisocoria
Species of flowering plant
relief from a normal reaction can be found with the application of a weak solution of ammonia, "Windex is a good choice" or when in remote areas that urine
Cnidoscolus_texanus
Principle in mathematical optimization
The optimal solution to the dual program is a lower bound for the optimal solution of the original (primal) program; this is the weak duality principle
Duality_(optimization)
Model in finance
explicit solution of the Heston price equation in terms of the volatility was developed by Kouritzin. This can be combined with known weak solutions for the
Heston_model
Inequality for Harmonic Functions
to solutions of elliptic or parabolic partial differential equations. Such results can be used to show the interior regularity of weak solutions. Perelman's
Harnack's_inequality
system of conservation laws converges, then it converges towards a weak solution. Lax–Wendroff method Godunov's scheme Randall J. LeVeque, Numerical
Lax–Wendroff_theorem
Solute that (almost) completely ionizes or dissociates in solution
completely, ionizes or dissociates in a solution. These ions are good conductors of electric current in the solution. Originally, a "strong electrolyte" was
Strong_electrolyte
Type of photographic print
chlorate as the restrainer (which is ineffective for palladium), a weak solution of sodium chloroplatinate is used instead. Sodium chloroplatinate, in
Platinum_print
H-principle Atiyah–Singer index theorem Bäcklund transform Viscosity solution Weak solution Loewy decomposition of linear differential equations Broer–Kaup
List of partial differential equation topics
List_of_partial_differential_equation_topics
Solution with prescribed norm
norm. Thus, we can find the weak solution of the equation. Moreover, if it satisfies the constraint, it's a normalized solution. On a Euclidean space R 3
Normalized solution (mathematics)
Normalized_solution_(mathematics)
Colorless flammable liquid with an ammonia-like odor
nuclear reactor waste. Some colour photographic processes also use a weak solution of hydrazine as a stabilising wash, as it scavenges dye coupler and
Hydrazine
Homeomorphism between plane domains
is required to be a weak solution of (1). When μ is zero almost everywhere, any homeomorphism in W1,2(D) that is a weak solution of (1) is conformal.
Quasiconformal_mapping
Selection technique used in evolutionary algorithms
a chance some weaker solutions may survive the selection process. This is because even though the probability that the weaker solutions will survive is
Fitness proportionate selection
Fitness_proportionate_selection
Solution used for paper pulping
White liquor is a strongly alkaline solution composed mainly of sodium hydroxide and sodium sulfide. It is used in the first stage of the Kraft process
White_liquor
Chemical compound
chlorine gas through a solution of potash lye. The resulting liquid, known as "Eau de Javel" ("Javel water"), was a weak solution of potassium hypochlorite
Potassium_hypochlorite
the function u can simply be interpreted as a weak solution in the usual sense. Multipeakon solutions are formed by taking a linear combination of several
Peakon
Shock wave that is inclined with respect to the incident upstream flow direction
1, in oblique shock M2 can be supersonic (weak shock wave) or subsonic (strong shock wave). Weak solutions are often observed in flow geometries open
Oblique_shock
Technique used on apparel for after wash effects
1986. They accidentally tumbled jeans and pumice stones wetted with a weak solution of bleach in a washing machine without water. American Garment Finishers
Stone_washing
Form of artificial intelligence
Weak artificial intelligence (weak AI) is artificial intelligence that implements a limited part of the mind, or, as narrow AI, artificial narrow intelligence
Weak_artificial_intelligence
Class of partial differential equations
is subject to various notions of weak solutions, i.e., reformulating the equations in a way that allows for solutions with various irregularities (e.g
Elliptic partial differential equation
Elliptic_partial_differential_equation
Scottish inventor, engineer and chemist (1736–1819)
cheaper method. He passed the chlorine into a weak solution of alkali, and obtained a turbid solution that appeared to have good bleaching properties
James_Watt
Method of chemical quantitative analysis
to form a neutral (pH = 7) solution. A strong acid will react with a weak base to form an acidic (pH < 7) solution. A weak acid will react with a strong
Acid–base_titration
Chemical compound of hydrogen and oxygen
property is relied upon by all vascular plants, such as trees. Water is a weak solution of hydronium hydroxide –there is an equilibrium 2H 2O ⇌ H 3O+ + OH−
Water
Chemical element with atomic number 17 (Cl)
chlorine gas through a solution of sodium carbonate. The resulting liquid, known as "Eau de Javel" ("Javel water"), was a weak solution of sodium hypochlorite
Chlorine
Section 2.4 in Bensoussan is based on weak solutions of the stochastic differential equation. Considering such solutions of d x = A ( t ) x ( t ) d t + B 1
Separation principle in stochastic control
Separation_principle_in_stochastic_control
Castle in Perth and Kinross, Scotland
Pillans Laurie of Heriot-Watt University in 1912. He advised using a weak solution of gelatine to fix the flaking paint pigment. Huntingtower is said to
Huntingtower_Castle
Refrigerator that uses a heat source
the absorber, this mixture of gases contacts water (technically, a weak solution of ammonia in water). The gaseous ammonia dissolves in the water, while
Absorption_refrigerator
Human disease
ablution; he should even wash his hands with some disinfecting fluid, a weak solution of chlorine for instance: he should avoid going in the same dress to
Postpartum_infections
Italian mathematician
variational solutions via the calculus of variations, J. Differential Equations, 256, 2014, 3912–3942. Growth conditions and regularity for weak solutions to nonlinear
Paolo_Marcellini
Form of electromagnetic radiation
176: "As to the infra-red rays, they may be absorbed by means of a weak solution of sulphate of copper, ..." See also: Rosenberg, Gary (2012). "Letter
Infrared
Type of chemical substance
acid neutralize exactly, leaving only NaCl, effectively table salt, in solution. Weak bases, such as baking soda or egg white, should be used to neutralize
Base_(chemistry)
Informal group of fungi
soon after production. A weak solution of water and sugar can be used to determine whether yeast is expired. In the solution, active yeast will foam and
Yeast
Used in hydrology
denotes convolution with respect to x. In this formulation, it admits weak solutions with a very low degree of regularity, even discontinuous ones (shock
Degasperis–Procesi_equation
solution (sometimes also called weak solution) of such an embedding problem is a continuous homomorphism γ : F → H such that φ = f γ. If the solution
Embedding_problem
Type of vector space in math
possible to consider a generalized solution (known as a weak solution) by enlarging the class of functions. Many weak formulations involve the class of
Hilbert_space
Soft, brown, combustible, sedimentary rock
to a fine powder by trituration, and if submitted to the action of a weak solution of potash, it yields a considerable quantity of humic acid. Leonardite
Lignite
Solution of hydrogen fluoride in water
In dilute aqueous solution, hydrogen fluoride behaves as a weak acid. Infrared spectroscopy has been used to show that, in solution, dissociation is accompanied
Hydrofluoric_acid
Measure of the ability of a solution containing electrolytes to conduct electricity
Instead, the solution becomes ever more fully dissociated at weaker concentrations, and for low concentrations of "well behaved" weak electrolytes, the
Conductivity_(electrolytic)
Duality (optimization) Weak duality — dual solution gives a bound on the primal solution Strong duality — primal and dual solutions are equivalent Shadow
List of numerical analysis topics
List_of_numerical_analysis_topics
Process by which the human body regulates pH
concentrations of the weak acid to the weak base. The higher the concentration of the weak acid in the solution (compared to the weak base) the lower the
Acid–base_homeostasis
Russian mathematician (1908-1989)
known as distributions) were first introduced by Sobolev in 1935 for weak solutions, and further developed by Laurent Schwartz. Sobolev abstracted the classical
Sergei_Sobolev
Chemical compound
a solution becoming less acidic, methyl orange changes from red to orange and, finally, to yellow—with the reverse process occurring in a solution of
Methyl_orange
In mathematics, a function f {\displaystyle f} is weakly harmonic in a domain D {\displaystyle D} if ∫ D f Δ g = 0 {\displaystyle \int _{D}f\,\Delta g=0}
Weakly_harmonic_function
Italian mathematician (born 1984)
with very rough potentials, and to study the Lagrangian structure of weak solutions to the Vlasov–Poisson equation. More recently, in collaboration with
Alessio_Figalli
Method of function approximation and data interpolation
needed. Inserting these quantities into the E−L equation shows that A weak solution f ( x ) {\displaystyle f(\mathbf {x} )} of (4) satisfies for all smooth
Polyharmonic_spline
Method in Itô calculus
simply called the Euler method) is a method for the approximate numerical solution of a stochastic differential equation (SDE). It is an extension of the
Euler–Maruyama_method
Chemical compound
food service and water distribution, specialized equipment to generate weak solutions of HClO from water and salt is sometimes used to generate adequate quantities
Hypochlorous_acid
Topics referred to by the same term
groups over global fields Weak weak approximation, a form of weak approximation for varieties Weak-field approximation, a solution in general relativity This
Weak_approximation
American mathematician (1953–2026)
Caffarelli in which they obtained partial results about the regularity of weak solutions of the Navier–Stokes equations. Kohn received a Sloan Research Fellowship
Robert_V._Kohn
Equation used to estimate pH of a weak acid or base solution
In chemistry and biochemistry, the pH of weakly acidic chemical solutions can be estimated using the Henderson-Hasselbalch Equation: pH = p K a + log 10
Henderson–Hasselbalch equation
Henderson–Hasselbalch_equation
Russian–Israeli mathematician (1950–2020)
stochastic differential equations, is an SDE which has a weak solution but no strong solution. The Gaussian isoperimetric inequality (proved by Vladimir
Boris_Tsirelson
Machine that removes heat from a liquid coolant
The concentrated lithium bromide solution moves down to the heat exchanger, where it is cooled by the weak solution being pumped up to the generator.
Chiller
Theorem in analysis
standard argument in the calculus of variations says that y must be a weak solution of the Euler–Lagrange equation y′′(x) + y(x) = 0 with y′(0) = y′(1)
Wirtinger's inequality for functions
Wirtinger's_inequality_for_functions
doi:10.4310/jdg/1442364652, MR 3399098. Seregin, Gregory A. (2010), "Weak solutions to the Navier-Stokes equations with bounded scale-invariant quantities"
Ancient_solution
volumes of submanifolds with the given boundary. It produced a candidate weak solution to Plateau's problem. Federer, Herbert (1969), Geometric measure theory
Flat_convergence
Manually applying colour to black-and-white photographs
photographs in Germany in the 1860s. When hand-colouring with dyes, a weak solution of dye in water is preferred, and colours are often built up with repeated
Hand-colouring_of_photographs
WEAK SOLUTION
WEAK SOLUTION
Boy/Male
Arabic, Muslim
Weak
Boy/Male
Spanish
Weak.
Boy/Male
Hindu
Peak
Surname or Lastname
English
English : variant of Week.
Boy/Male
Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Sindhi, Tamil, Telugu
Peak
Boy/Male
Arabic
One who has Weak Eyes
Surname or Lastname
English
English : variant spelling of Leake.
Boy/Male
Biblical
Weak, slacked.
Girl/Female
Hindu, Indian, Traditional
Peak
Boy/Male
Tamil
Peak
Surname or Lastname
English
English : variant of Wick, specifically a habitational name from any of various places called Week or Weeke, notably in Cornwall, Hampshire, and Somerset.Americanized spelling of Norwegian or Swedish Vik.
Biblical
weak; slacked
Surname or Lastname
English (Northumbria)
English (Northumbria) : topographic name for someone who lived by the Wear river in northern England. The river name is ancient, occuring in the form Vedra in Ptolemy’s Geographia; it is probably a Celtic word meaning ‘water’.English (Northumbria) : topographic name for someone who lived near a dam or weir, a variant spelling of Ware 1, or a habitational name from a place called Weare, in Devon and Somerset, from Old English wær, wer ‘weir’.
Girl/Female
Indian
Peak
Girl/Female
Australian, Christian, French, German, Hebrew
Hair; Lovelorn; Delicate; Weak
Boy/Male
Arabic, Muslim
Weak
Boy/Male
Hindu, Indian
Week
Surname or Lastname
English
English : topographic name for someone living by a pointed hill (or regional name from the Peak District (Old English Pēaclond) in Derbyshire), named with Old English pēac ‘peak’, ‘pointed hill’ (found only in place names). This word is not directly related to Old English pīc ‘point’, ‘pointed hill’, which yielded Pike; there is, however, some evidence of confusion between the two surnames.Possibly also Irish : reduced form of McPeak.Major concentrations of the surname Peak are found in Staffordshire and the West Country of England. Among the earliest known bearers are Richard del Pech or del Pek (d. 1196), son of Rannulf, sheriff of Nottingham, and Willielmus Piec (Winchester 1194). A century later, c.1284, a certain Richard del Peke settled in Denbighshire (now part of Clwyd), Wales, receiving lands from Henry de Lacey, earl of Lincoln, in return for helping to control the region. His descendants, who bear the name Peak(e), can be traced to the present day, and are found in New Zealand and Canada as well as in Britain. Peake is also the name of a family descended from John Pyke, who paid rent to the abbot of Leicester in 1477. The name took various forms, such as Peke and Pick, eventually becoming established as Peak in the 17th century.
Boy/Male
Australian, Hindu, Indian
Peak
Girl/Female
Tamil
Peak
WEAK SOLUTION
WEAK SOLUTION
Boy/Male
American, Australian, British, Chinese, English, German, Teutonic
From the Hedged in Valley
Girl/Female
Indian
Good Smell
Boy/Male
Indian
Blessing
Boy/Male
Hindu, Indian, Rajasthani, Traditional
Brave Lord; Similar to Virendra
Boy/Male
Hindu
Protector of Dharma
Girl/Female
Tamil
Cinthana | சீநà¯à®¤à®¾à®¨à®¾Â
Always smiling
Boy/Male
Hindu
Surname or Lastname
English
English : patronymic from Bill.
Girl/Female
Arabic
Benefit
Boy/Male
Muslim
Unique of the religion (Islam)
WEAK SOLUTION
WEAK SOLUTION
WEAK SOLUTION
WEAK SOLUTION
WEAK SOLUTION
v. i.
Wanting in power to influence or bind; as, weak ties; a weak sense of honor of duty.
v. i.
Not thoroughly or abundantly impregnated with the usual or required ingredients, or with stimulating and nourishing substances; of less than the usual strength; as, weak tea, broth, or liquor; a weak decoction or solution; a weak dose of medicine.
v. i.
Tending towards lower prices; as, a weak market.
v. i.
Lacking ability for an appropriate function or office; as, weak eyes; a weak stomach; a weak magistrate; a weak regiment, or army.
v. t.
To cause or make by friction or wasting; as, to wear a channel; to wear a hole.
n.
The upper aftermost corner of a fore-and-aft sail; -- used in many combinations; as, peak-halyards, peak-brails, etc.
a.
Having weak knees; hence, easily yielding; wanting resolution.
v. i.
Wanting in point or vigor of expression; as, a weak sentence; a weak style.
v. i.
Not able to sustain a great weight, pressure, or strain; as, a weak timber; a weak rope.
v. i.
Not able to resist external force or onset; easily subdued or overcome; as, a weak barrier; as, a weak fortress.
v. i.
Lacking in elements of political strength; not wielding or having authority or energy; deficient in the resources that are essential to a ruler or nation; as, a weak monarch; a weak government or state.
v. i.
Feeble of mind; wanting discernment; lacking vigor; spiritless; as, a weak king or magistrate.
a.
Having a weak mind, either naturally or by reason of disease; feebleminded; foolish; idiotic.
v.
A crack, crevice, fissure, or hole which admits water or other fluid, or lets it escape; as, a leak in a roof; a leak in a boat; a leak in a gas pipe.
v. i.
Not firmly united or adhesive; easily broken or separated into pieces; not compact; as, a weak ship.
v. i.
Not having power to convince; not supported by force of reason or truth; unsustained; as, a weak argument or case.
v. i.
Not able to withstand temptation, urgency, persuasion, etc.; easily impressed, moved, or overcome; accessible; vulnerable; as, weak resolutions; weak virtue.
v. i.
To rise or extend into a peak or point; to form, or appear as, a peak.
v. i.
Not stiff; pliant; frail; soft; as, the weak stalk of a plant.
a.
To make or become weak; to weaken.