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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
a formal integral by parts. An example for a weak solution is as follows: ϕ(x)=14π1|x| is a weak solution satisfying ∇2ϕ=δ in R3 in distributional sense
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
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
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
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
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
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
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)
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
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
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
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
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)
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
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
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
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
system of conservation laws converges, then it converges towards a weak solution. Lax–Wendroff method Godunov's scheme LeVeque, Randall J. (1992). Numerical
Lax–Wendroff_theorem
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
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
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
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
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
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
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
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
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
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 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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
Soft, brown, combustible, sedimentary rock
is reducible to a fine powder by trituration, and if submitted to a weak solution of potash, it yields a considerable quantity of humic acid. Leonardite
Lignite
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)
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
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
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
𝐯0(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
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
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)
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)
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)
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
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
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
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
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
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
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
Active fire protection device
not new, however. From about 1880 glass "fire grenades" filled with a weak solution of common salt and ammonium chloride in water were popular. The addition
Fire_extinguisher
Conductivity per molar concentration of electrolyte
The molar conductivity of an electrolyte solution is defined as its conductivity divided by its molar concentration: Λ m = κ c , {\displaystyle \Lambda
Molar_conductivity
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
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
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
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
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
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
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
Concrete with rebar
produce a weak solution of hydrochloric acid due to the presence of chlorides in the magnesite. Over a period of time (typically decades), the solution causes
Reinforced_concrete
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
Separation of chemical species from other acidic or basic compounds
weak and has a pKa value of ~5 (such as a carboxylic acid), adding additional acid can further improve separation by lowering the pH of the solution.
Acid–base_extraction
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
Chemical reaction in which an acid and a base react quantitatively
aqueous solution at room temperature. A chemical compound may behave as a strong acid in solution when its concentration is low and as a weak acid when
Neutralization_(chemistry)
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 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
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
Formula relating stochastic processes to partial differential equations
the above stochastic differential equation has a unique (strong or weak) solution. This is for example the case, when μ {\displaystyle \mu } and σ {\displaystyle
Feynman–Kac_formula
Measure of acidity
{c_{\rm {BH}}}}} Here BH is a weak acid used as an acid-base indicator, and B− is its conjugate base. In dilute aqueous solution, the predominant acid species
Acidity_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
Objects that generalize functions
Pseudodifferential operator Riesz representation theorem Vague topology Weak solution Distribution on a linear algebraic group Limit of a distribution Schwartz-Bruhat
Distribution (mathematical analysis)
Distribution_(mathematical_analysis)
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
Differential equation solution with a 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 solutions (nonlinear Schrödinger equation)
Normalized_solutions_(nonlinear_Schrödinger_equation)
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
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
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
Equation in fluid dynamics
Shuyin; Yin, Zhaoyang (2006), "Blow-up, blow-up rate and decay of the solution of the weakly dissipative Camassa–Holm equation", J. Math. Phys., 47 (1): 013504
Camassa–Holm_equation
English photography pioneer (1800–1877)
writing paper bathed in a weak solution of ordinary table salt (sodium chloride), dried, then brushed on one side with a strong solution of silver nitrate, which
Henry_Fox_Talbot
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
French mathematician (born 1962)
Saint-Raymond in 2004 he showed a connection of the weak solutions of the Boltzmann equation with the Leray solutions of the incompressible Navier-Stokes equations
François_Golse
travel, tourism, insurance
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travel, tourism, insurance