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Mathematical tools
Weak formulations are tools for the analysis of mathematical equations that permit the transfer of concepts of linear algebra to solve problems in other
Weak_formulation
Mathematical solution
equation show up (the new form is called the weak formulation, and the solutions to it are called weak solutions). Somewhat surprisingly, a differential
Weak_solution
Method for solving continuous operator problems (such as differential equations)
continuous operator problem, such as a differential equation, commonly in a weak formulation, to a discrete problem by applying linear constraints determined by
Galerkin_method
Finite element method for Navier-Stokes equations
streamline upwind Petrov–Galerkin pressure-stabilizing Petrov–Galerkin formulation for incompressible Navier–Stokes equations can be used for finite element
Streamline_upwind_Petrov–Galerkin_pressure-stabilizing_Petrov–Galerkin_formulation_for_incompressible_Navier–Stokes_equations
Numerical method for solving physical or engineering problems
computer. The first step is to convert P1 and P2 into their equivalent weak formulations. If u {\displaystyle u} solves P1, then for any smooth function v
Finite_element_method
Topics referred to by the same term
Weak form and strong form may refer to: Weaker and stronger versions of a hypothesis, theorem or physical law Weak formulations and strong formulations
Weak_form_and_strong_form
Certain vector fields are the sum of an irrotational and a solenoidal vector field
Helmholtz theorem. However, the compactness restriction in the usual formulation of the Hodge decomposition can be replaced by suitable decay assumptions
Helmholtz_decomposition
Class of numerical simulation algorithms
methods, where the so-called weakened weak (W2) formulation based on the G space theory were developed. The W2 formulation offers possibilities for formulate
Smoothed finite element method
Smoothed_finite_element_method
functional. The functional Φ {\displaystyle \Phi } is said to satisfy the weak Palais–Smale condition if for each sequence { x n } ⊂ X {\displaystyle \{x_{n}\}\subset
Palais–Smale compactness condition
Palais–Smale_compactness_condition
Mathematical method
introduced as follows. Let us consider an abstract problem posed as a weak formulation on a pair of Hilbert spaces V {\displaystyle V} and W {\displaystyle
Petrov–Galerkin_method
Mathematical theorem
u ∈ U such that Λ u = f . {\displaystyle \Lambda u=f.} However, in the weak formulation, this equation is only required to hold when "tested" against all other
Babuška–Lax–Milgram_theorem
Type of vector space in math
consider a generalized solution (known as a weak solution) by enlarging the class of functions. Many weak formulations involve the class of Sobolev functions
Hilbert_space
Initial result in using test functions to find extremum
condition of extremum (functional derivative equal zero) appears in a weak formulation (variational form) integrated with an arbitrary function δf. The fundamental
Fundamental lemma of the calculus of variations
Fundamental_lemma_of_the_calculus_of_variations
Mathematical theorem
{\displaystyle X} and X ∗ {\displaystyle X^{*}} are endowed with strong topology and weak* topology respectively. There exists r > 0 {\displaystyle r>0} such that
Mountain_pass_theorem
American mathematician and Nobel Laureate (1928–2015)
based primarily on the judicious choice of test functions in the weak formulation of partial differential equations, is in strong contrast to Nash's
John_Forbes_Nash_Jr.
Extremes of a linear function over a convex polygonal region occur at the region's corners
optimization, the fundamental theorem of linear programming states, in a weak formulation, that the maxima and minima of a linear function over a convex polygonal
Fundamental theorem of linear programming
Fundamental_theorem_of_linear_programming
Formulation of quantum mechanics
The path-integral formulation of quantum mechanics generalizes the action principle of classical mechanics. It replaces the classical notion of a single
Path-integral_formulation
Theorem in vector calculus
a non-Lipschitz surface. One (advanced) technique is to pass to a weak formulation and then apply the machinery of geometric measure theory; for that
Stokes'_theorem
Risk management strategy
plausible: weak formulations (which hold that precaution in the face of uncertain harms is permissible) are trivial, while strong formulations (which hold
Precautionary_principle
Phenomenological ionic model for human ventricular cells
{W}}=[L^{2}(\Omega )]^{3}} be two generic test functions. To obtain the weak formulation: multiply by p ∈ U {\displaystyle p\in U} the first equation of the
Bueno-Orovio–Cherry–Fenton model
Bueno-Orovio–Cherry–Fenton_model
Conjecture about prime numbers, proof under review
In number theory, Goldbach's weak conjecture, also known as the odd Goldbach conjecture, the ternary Goldbach problem, or the 3-primes problem, is the
Goldbach's_weak_conjecture
Elliptic partial differential equation
equation Helmholtz equation Uniqueness theorem for Poisson's equation Weak formulation Harmonic function Heat equation Potential theory Jackson, Julia A.;
Poisson's_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
Equations of motion for viscous fluids
_{\Gamma _{N}}\mathbf {h} \cdot \mathbf {v} .} Having this in mind, the weak formulation of the Navier–Stokes equations is expressed as: find u ∈ L 2 ( R +
Navier–Stokes_equations
Axiomatic set theories based on the principles of mathematical constructivism
Separation is adopted in a weak predicative form while Infinity is stated in a strong formulation. Akin to the classical formulation, this Separation axiom
Constructive_set_theory
Analysis and solving of problems that involve fluid flows
mathematical problem (the partial differential equation) to be cast in a weak formulation. This is typically done by multiplying the differential equation by
Computational_fluid_dynamics
Australian and American mathematician (born 1975)
law, which had previously been proved in weaker formulations by many other authors. In Tao and Vu's formulation, the circular law becomes an immediate consequence
Terence_Tao
Ensemble learning method
recursive majority gate formulation), and Yoav Freund (boost by majority), were not adaptive and could not take full advantage of the weak learners. Schapire
Boosting_(machine_learning)
Interpretation of the US Constitution regarding presidential power
officials, structurally centralizing control within a unified hierarchy. Weaker formulations are less controversial than stronger versions, which face constitutional
Unitary_executive_theory
Ways of writing certain laws of physics
The covariant formulation of classical electromagnetism refers to ways of writing the laws of classical electromagnetism (in particular, Maxwell's equations
Covariant formulation of classical electromagnetism
Covariant_formulation_of_classical_electromagnetism
Matrix used in finite element analysis
associated a bilinear form B on the Sobolev space Hk, so that the weak formulation of the equation Lu = f is B [ u , v ] = ( f , v ) {\displaystyle B[u
Stiffness_matrix
Numerical analysis method
{\text{X}}} in a weak sense inside the solid domain B {\displaystyle {\mathcal {B}}} . This requirement leads to the following weak formulation: { a f ( u
Fictitious_domain_method
Abelian charge found in electroweak theory
both lepton number and baryon number. Standard Model (mathematical formulation) Weak charge Donoghue, J.F.; Golowich, E.; Holstein, B.R. (1994). Dynamics
Weak_hypercharge
Numerical method for solving boundary value problems
applying a greedy algorithm, usually the fixed point algorithm, to the weak formulation of the problem. For each iteration i of the algorithm, a mode of the
Proper generalized decomposition
Proper_generalized_decomposition
Method for approximating eigenvalues
product ( ⋅ , ⋅ ) {\displaystyle (\cdot ,\cdot )} can be replaced by the weak formulation A ( ⋅ , ⋅ ) {\displaystyle {\mathcal {A}}(\cdot ,\cdot )} . If a subset
Rayleigh–Ritz_method
Quantum number related to the weak interaction
the resulting Z0 and γ0 likewise have zero weak isospin. Weak hypercharge Weak charge Mathematical formulation of the Standard Model Glashow, Sheldon L
Weak_isospin
Theorem in complex analysis
to be large relative to these bounds. Evans's book has a slightly weaker formulation, in which there is assumed to be a positive number λ which is a lower
Maximum_principle
Measurement of a quantum system which minimally disturbs it
In quantum mechanics (and computation and information), weak measurement is a type of quantum measurement that results in an observer obtaining very little
Weak_measurement
Mathematics of a particle physics model
under weak isospin SU(2) transformations the left-handed particles are weak-isospin doublets, whereas the right-handed are singlets – i.e. the weak isospin
Mathematical formulation of the Standard Model
Mathematical_formulation_of_the_Standard_Model
Weakened weak form (or W2 form) is used in the formulation of general numerical methods based on meshfree methods and/or finite element method settings
Weakened_weak_form
Concept of absolute rotation
least 21 formulations of Mach's principle are possible, some being considered more strongly Machian than others. A relatively weak formulation is the assertion
Mach's_principle
Hydrodynamic formulation of the Schrödinger equations
equations of quantum hydrodynamics, are Erwin Madelung's alternative formulation of the Schrödinger equation for a spinless non-relativistic particle
Madelung_equations
Doubt about God's existence
contrast between weak and strong agnosticism is sometimes framed as a claim about what is reasonable to believe. In this sense, weak agnosticism holds
Agnosticism
Methods for solving differential equations
Galerkin (DG) method is a finite element method formulated relative to a weak formulation of a particular model system. Unlike traditional CG methods that are
Discontinuous_Galerkin_method
Ethical principle; incorporates deontology into greatest-happiness thinking
places and times. SRU does not deteriorate into act utilitarianism like weak rule utilitarianism, but it shares weaknesses with similarly absolutist moral
Rule_utilitarianism
Bosons that mediate the weak interaction
mass List of particles Mathematical formulation of the Standard Model – Mathematics of a particle physics model Weak charge W′ and Z′ bosons – Hypothetical
W_and_Z_bosons
Hypothesis that inertial and gravitational masses are equivalent
equivalence of gravitational and inertial mass is a consequence of nature. The weak form, known for centuries, relates to masses of any composition in free fall
Equivalence_principle
Theory of forces and subatomic particles
describing three of the four known fundamental forces (electromagnetic, weak and strong interactions – excluding gravity) in the universe and classifying
Standard_Model
Type of partial differential equation
\mathrm {d} x\geq 0{\text{ for all }}v\leq \varphi .} This formulation permits the definition of a weak solution: using integration by parts on the last equation
Free_boundary_problem
Hypothesis about sapient life and the universe
different formulations of the anthropic principle. Philosopher Nick Bostrom counts thirty, but the underlying principles can be divided into "weak" and "strong"
Anthropic_principle
American mathematician (1943–2024)
manifold which satisfies a heat equation, giving both strong and weak formulations.[H86] Partly due to these foundational technical developments, Hamilton
Richard_S._Hamilton
Functions in mathematics
function, and so also is smooth. This is Weyl's lemma. There are other weak formulations of Laplace's equation that are often useful. One of which is Dirichlet's
Harmonic_function
Theoretical framework in physics
second major barrier came with QFT's apparent inability to describe the weak and strong interactions, to the point where some theorists called for the
Quantum_field_theory
Light bending by mass between source and observer
elliptical and the weak gravitational lensing signal is small, a very large number of galaxies must be used in these surveys. These weak lensing surveys
Gravitational_lens
Formulation of the quantum many-body problem
theory Feynman diagram History Background Field theory Electromagnetism Weak force Strong force Quantum mechanics Special relativity General relativity
Second_quantization
Types of atheism
Negative atheism, also called weak atheism and soft atheism, is any type of atheism where a person does not believe in the existence of any deities but
Negative_and_positive_atheism
possibilities directly with possible worlds. Possibility Haecceitism, a weaker formulation, proposes that there are distinct maximal possibilities, which are
Haecceitism
Relation between algebraic varieties and polynomial ideals
polynomial rings over algebraically closed fields is another common formulation of the weak Nullstellensatz. As another example of this correspondence and
Hilbert's_Nullstellensatz
Numerical resistivity — the same, with resistivity instead of diffusion Weak formulation — a functional-analytic reformulation of the PDE necessary for some
List of numerical analysis topics
List_of_numerical_analysis_topics
Generalization of the Dirac equation
theory Feynman diagram History Background Field theory Electromagnetism Weak force Strong force Quantum mechanics Special relativity General relativity
Dirac equation in curved spacetime
Dirac_equation_in_curved_spacetime
Elementary particle, fundamental constituent of matter
George Zweig in 1964. The proposal came shortly after Gell-Mann's 1961 formulation of a particle classification system known as the Eightfold Way—or, in
Quark
Binding of quarks in subatomic particles
approximately 100 times as strong as electromagnetism, 106 times as strong as the weak interaction, and 1038 times as strong as gravitation. In the context of atomic
Strong_interaction
Method for solving problems in continuum mechanics
also means that FPM is not based on a weak formulation like Galerkin's approach. Rather, FPM is a strong formulation which models differential equations
Finite_pointset_method
Logical formulation of graph properties
such as directed graphs, may involve different axioms, and logical formulations of multigraph properties require special handling such as having multiple
Logic_of_graphs
Unified description of electromagnetism and the weak interaction
interactions of nature: electromagnetism (electromagnetic interaction), and the weak interaction. Although these two forces appear very different at everyday
Electroweak_interaction
Elementary particle with extremely low mass
by the Greek letter ν) is an elementary particle that interacts via the weak interaction and gravity. The neutrino is so named because it is electrically
Neutrino
conditional existence statement for which there are many equivalent formulations, as discussed below. The principle is logically valid classically, but
Markov's_principle
Process in quantum mechanical theories
expansion over ħ of the commutator [A, B] expressed in the phase space formulation is iħ{A, B} . (Here, the curly braces denote the Poisson bracket. The
Canonical_quantization
Physics theory
Time-reversal symmetry Time-invariance Weak mode coupling (small perturbation of uncoupled modes) Energy conservation The formulation of the coupled mode theory is
Coupled_mode_theory
Liqueur
Greek Street, Soho, London, next to the Pillars of Hercules pub. The formulation of Baileys was motivated partly by the availability of alcohol from a
Baileys_Irish_Cream
Metaphysical theory
materialism." Physical causal closure has stronger and weaker formulations. The stronger formulations assert that no physical event has a cause outside the
Causal_closure
Theorem for reducing high-order derivatives
theory Feynman diagram History Background Field theory Electromagnetism Weak force Strong force Quantum mechanics Special relativity General relativity
Wick's_theorem
Species of elementary particle
the same way, if the (much weaker) electromagnetic interaction is neglected. Heisenberg noted that the mathematical formulation of this symmetry was in certain
Flavour_(particle_physics)
Concept within epistemology (philosophy)
Philosophy and Phenomenological Research, 74 (2007): 30–55. Huemer's weaker formulation of phenomenal conservatism. Hanna, Nathan. "Against Phenomenal Conservatism
Phenomenal_conservatism
Method in physics used to deal with infinities
γμ factors in this expression are gamma matrices as in the covariant formulation of the Dirac equation; they have to do with the spin of the electron
Renormalization
German-born theoretical physicist (1879–1955)
his special theory of relativity to a new idea of gravitation with the formulation of his equivalence principle, which asserts that an observer in a box
Albert_Einstein
Conjecture in physics
sufficient latitude for (at least) two independent formulations: a weak form, and a strong form. The weak and the strong cosmic censorship hypotheses are
Cosmic_censorship_hypothesis
Exactly solvable model of coupled oscillators
a model for the behavior of a large set of coupled oscillators. Its formulation was motivated by the behavior of systems of chemical and biological oscillators
Kuramoto_model
Formulation of quantum mechanics
Matrix mechanics is a formulation of quantum mechanics created by Werner Heisenberg, Max Born, and Pascual Jordan in 1925. It was the first conceptually
Matrix_mechanics
(GSMF) formulation is a local meshfree method in the field of numerical analysis, completely integration free, working as a weighted-residual weak-form
Generalized-strain mesh-free formulation
Generalized-strain_mesh-free_formulation
are not Hilbertian. The WWA property (weak 'weak approximation', sic) for a variety V over a number field is weak approximation (cf. approximation in algebraic
Thin_set_(Serre)
Quantum field theory of electromagnetism
called cosmological constant problem or vacuum catastrophe. The first formulation of a quantum theory describing radiation and matter interaction is attributed
Quantum_electrodynamics
Quantum chromodynamics on a lattice
the strong force and the large coupling constant at low energies. This formulation of QCD in discrete rather than continuous spacetime naturally introduces
Lattice_QCD
Logical principle
in van Heijenoort, p. 421) footnote 9: "This is Leibniz's very simple formulation (see Nouveaux Essais, IV,2)" (ibid p 421) The principle was stated as
Law_of_excluded_middle
Evolutionary equation under renormalization group flow
theory Feynman diagram History Background Field theory Electromagnetism Weak force Strong force Quantum mechanics Special relativity General relativity
Callan–Symanzik_equation
Field-equations in general relativity
curvature of spacetime. These equations form the core of the mathematical formulation of general relativity. They readily imply the geodesic equation, which
Einstein_field_equations
Geometrical re-formulation of Newtonian gravity
Newton–Cartan theory (or geometrized Newtonian gravitation) is a geometrical re-formulation, as well as a generalization, of Newtonian gravity first introduced by
Newton–Cartan_theory
Italian mathematician
equations: he is known for being the first mathematician generalizing in a weak sense the notion of boundary value in a boundary value problem, and for doing
Gianfranco_Cimmino
Central concept in Kantian moral philosophy
evaluating motivations for action. It is best known in its original formulation: "Act only according to that maxim whereby you can at the same time will
Categorical_imperative
Mathematical optimization concept
– maximum in the primal becomes minimum in the dual and vice versa. The weak duality theorem states that the objective value of the dual LP at any feasible
Dual_linear_program
Study of subatomic particles and forces
Fitch brought new questions to matter-antimatter imbalance. After the formulation of the Standard Model during the 1970s, physicists clarified the origin
Particle_physics
Principle that scientific change is generational
Species, age was a minor factor. On a more specialized scale, it also was a weak factor in accepting cliometrics. A study of when different geologists accepted
Planck's_principle
Pictorial representation of the behavior of subatomic particles
tied to the functional integral formulation of quantum mechanics, also invented by Feynman—see path integral formulation. The naïve application of such
Feynman_diagram
Well-quasi-ordering of finite trees
TREE. The version given here is that proven by Nash-Williams; Kruskal's formulation is somewhat stronger. All trees we consider are finite. Given a tree
Kruskal's_tree_theorem
Physics concept expressed as E = mc²
the energy L by emitting light, its mass diminishes by L/c2. This formulation relates only a change Δm in mass to a change L in energy without requiring
Mass–energy_equivalence
Linear perturbations to solutions of nonlinear Einstein field equations
the gravitational field is weak. The usage of linearized gravity is integral to the study of gravitational waves and weak-field gravitational lensing
Linearized_gravity
Description of physical properties at the atomic and subatomic scale
mathematical formulation of quantum mechanics and survey its application to some useful and oft-studied examples. In the mathematically rigorous formulation of
Quantum_mechanics
Japanese physicist (1906-1979)
Press. ISBN 0-226-80794-0. Tomonaga, S. "On a Relativistically Invariant Formulation of the Quantum Theory of Wave Fields." Prog. Theor. Phys. 1, 27–42 (1946)
Shin'ichirō_Tomonaga
Sugar-based rocket propellant
aesthetics of the liftoff or flight. A traditional sugar propellant formulation is typically prepared in a 65:35 (13:7) oxidizer to fuel ratio. This
Rocket_candy
Physics experiment
can illustrate the path integral formulation of quantum mechanics provided by Feynman. The path integral formulation replaces the classical notion of
Double-slit_experiment
WEAK FORMULATION
WEAK FORMULATION
Boy/Male
Arabic, Muslim
Weak
Boy/Male
Biblical
Weak, slacked.
Surname or Lastname
English
English : variant spelling of Leake.
Boy/Male
Arabic, Muslim
Weak
Surname or Lastname
English
English : variant of Week.
Girl/Female
Indian
Peak
Boy/Male
Australian, Hindu, Indian
Peak
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.
Girl/Female
Australian, Christian, French, German, Hebrew
Hair; Lovelorn; Delicate; Weak
Girl/Female
Hindu, Indian, Traditional
Peak
Boy/Male
Arabic
One who has Weak Eyes
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’.
Biblical
weak; slacked
Boy/Male
Hindu, Indian
Week
Girl/Female
Tamil
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.
Boy/Male
Hindu
Peak
Boy/Male
Spanish
Weak.
Boy/Male
Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Sindhi, Tamil, Telugu
Peak
WEAK FORMULATION
WEAK FORMULATION
Girl/Female
Australian, Danish, German, Swedish
God is Merciful
Boy/Male
German
Bear-strength
Boy/Male
British, English
Life; Little
Boy/Male
Hindu, Indian, Traditional
Pure Love for the Lord
Male
Babylonian
, Agu the Maker of Brilliance.
Girl/Female
Muslim
From the name sabine An italian culture
Boy/Male
Buddhist, Indian
Pure Blessing
Girl/Female
Arabic, Gujarati, Hindu, Indian, Modern
Twilight
Boy/Male
Arabic, Indian, Muslim, Pashtun, Sindhi
Worshipper of Allah; One who does Much Prostrations; Obedient; Kneeling in Prayer
Boy/Male
Hindu, Indian, Traditional
Radiant
WEAK FORMULATION
WEAK FORMULATION
WEAK FORMULATION
WEAK FORMULATION
WEAK FORMULATION
v. i.
Not able to resist external force or onset; easily subdued or overcome; as, a weak barrier; as, a weak fortress.
v. i.
Not firmly united or adhesive; easily broken or separated into pieces; not compact; as, a weak ship.
v. i.
Not stiff; pliant; frail; soft; as, the weak stalk of a plant.
v. t.
To cause or make by friction or wasting; as, to wear a channel; to wear a hole.
a.
Having a weak mind, either naturally or by reason of disease; feebleminded; foolish; idiotic.
v. i.
Not able to sustain a great weight, pressure, or strain; as, a weak timber; a weak rope.
v. i.
Wanting in point or vigor of expression; as, a weak sentence; a weak style.
v. i.
Tending towards lower prices; as, a weak market.
n.
The upper aftermost corner of a fore-and-aft sail; -- used in many combinations; as, peak-halyards, peak-brails, etc.
v. i.
Not having power to convince; not supported by force of reason or truth; unsustained; as, a weak argument or case.
v. i.
Feeble of mind; wanting discernment; lacking vigor; spiritless; as, a weak king or magistrate.
a.
To make or become weak; to weaken.
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.
Not able to withstand temptation, urgency, persuasion, etc.; easily impressed, moved, or overcome; accessible; vulnerable; as, weak resolutions; weak virtue.
v. i.
Wanting in power to influence or bind; as, weak ties; a weak sense of honor of duty.
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.
a.
Having weak knees; hence, easily yielding; wanting resolution.
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.
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.
To rise or extend into a peak or point; to form, or appear as, a peak.