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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
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
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
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
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)
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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)
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
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
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
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
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
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
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
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
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
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
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
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
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
Method of producing printing blocks
now drawn reversed on the prepared block with an ink consisting of a weak solution of glue colored by lamp black or some other pigment. In drawing on the
Graphotype
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
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
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
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
Russian mathematician
MR 0117426, Zbl 0096.06603. An important paper where the author constructs a weak solution in BV for a nonlinear partial differential equation with the method
Olga_Oleinik
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
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
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)
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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)
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
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
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
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
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)
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
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
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
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
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
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
Species of flowering plant
frequently, as they could rot. The plant should be fed monthly with a weak solution (water-soluble fertilizers, diluted to half strength, are best) during
Nertera_granadensis
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)
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
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
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 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
Argentine mathematician
of his most cited results regards the Partial regularity of suitable weak solutions of the Navier–Stokes equations; it was obtained in 1982 in collaboration
Luis_Caffarelli
Infection following childbirth
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
Canadian-American mathematician (1925–2020)
seminal achievement in the 1930s, formulated an influential notion of weak solution for the equations and proved their existence. His work was later put
Louis_Nirenberg
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
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
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
Mathematical function often applied to matrices
cannot be represented). The theory can be extended to Sobolev norms and weak solutions, e.g. allowing "point loads" (Dirac functions in the right hand side)
Logarithmic_norm
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
1967 film by Robert Siodmak
in Washington—and announced an impending honour for Custer. It was a weak solution, but it saved a lot of money. That's show business for you. The film
Custer_of_the_West
Study of geometric properties of sets through measure theory
{\displaystyle \mathbb {R} ^{n}} . This has applications in the study of weak solutions to the Monge–Ampère equation. It is a nonnegative, locally finite, Borel
Geometric_measure_theory
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
Millennium Prize Problem
_{0}(x)} is sufficiently small. He also proved the existence of so-called weak solutions to the Navier–Stokes equations, which may not satisfy the equations
Navier–Stokes existence and smoothness
Navier–Stokes_existence_and_smoothness
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
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
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
Method for applying patterns to cloth using printing techniques
by steaming only, consists in passing the bleached calico through a weak solution of sulphated or turkey red oil containing 2.5 to 5 percent fatty acid
Textile_printing
French chemist (1748–1822)
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
Claude_Louis_Berthollet
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