Forensic science (often shortened to forensics) is the application of a broad spectrum of sciences to answer questions of interest to the legal system. This may be in relation to a crime or to a civil action. The use of the term "forensics" in place of "forensic science" could be considered incorrect; the term "forensic" is effectively a synonym for "legal" or "related to courts" (from Latin, it means "before the forum"). However, it is now so closely associated with the scientific field that many dictionaries include the meaning that equates the word "forensics" with "forensic science".
Connections to Biomatics may include:
- Drug effects on behavior
- Developmental effects
Decision theory in mathematics and statistics is concerned with identifying the values, uncertainties and other issues relevant in a given decision and the resulting optimal decision. Most of decision theory is concerned with identifying the best decision to take, assuming an ideal decision maker who is fully informed, able to compute with perfect accuracy, and fully rational. The practical application of this prescriptive approach (how people should make decisions) is called decision analysis, and aimed at finding tools, methodologies and software to help people make better decisions.
In the common law tradition, case law interprets laws, via precedents, based on how prior cases have been decided. Case law governs the impact court decisions have on future cases. Unlike most civil law systems, common law systems follow the doctrine of stare decisis in which lower courts usually make decisions consistent with previous decisions of higher courts.
One to One Correspondence
The notion of one-to-one correspondence is fundamental to counting. When we count out a set of cards, we say, 1, 2, 3, ... , 52, and as we say each number we lay down a card. Each number corresponds to a card. Technically, we can say that we have put the cards in the deck and the numbers from 1 to 52 in a one-to-one correspondence with each other.
In abstract algebra, a homomorphism is a (one to one) structure-preserving map between two algebraic structures (such as groups, rings, or vector spaces). The word homomorphism comes from the Greek language: homos meaning "same" and morphe meaning "shape". Note the similar root word "homoios," meaning "similar," which is found in another mathematical concept, namely homeomorphisms.
In abstract algebra, an isomorphism (Greek: ison "equal", and morphe "shape") is a (one to one and onto) bijective map f such that both f and its inverse f −1 are homomorphisms, i.e., structure-preserving mappings.
The study of morphisms and of the structures (called objects) over which they are defined, is central to category theory. Much of the terminology of morphisms, as well as the intuition underlying them, comes from concrete categories, where the objects are simply sets with some additional structure, and morphisms are functions preserving this structure. Nevertheless, morphisms are not necessarily functions, and objects over which morphisms are defined are not necessarily sets. Instead, a morphism is often thought of as an arrow linking an object called the domain to another object called the codomain. Hence morphisms do not so much map sets into sets, as embody a relationship between some posited domain and codomain.
The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms are functions; in topology, continuous functions; in universal algebra, homomorphisms; in group theory, group homomorphisms.
Statistical decision theory
Several statistical tools and methods are available to organize evidence, evaluate risks, and aid in decision making. The risks of Type I and type II errors can be quantified (estimated probability, cost, expected value, etc) and rational decision making is improved.
One example shows a structure for deciding guilt in a criminal trial:
|True Positive||False Positive
(i.e. guilt reported
| Verdict of
Bayes' theorem relates the conditional and marginal probabilities of events A and B, where B has a non-vanishing probability:
Each term in Bayes' theorem has a conventional name:
- P(A) is the prior probability or marginal probability of A. It is "prior" in the sense that it does not take into account any information about B.
- P(A|B) is the conditional probability of A, given B. It is also called the posterior probability because it is derived from or depends upon the specified value of B.
- P(B|A) is the conditional probability of B given A.
- P(B) is the prior or marginal probability of B, and acts as a normalizing constant.
Intuitively, Bayes' theorem in this form describes the way in which one's beliefs about observing 'A' are updated by having observed 'B'.
The Dempster-Shafer theory is a mathematical theory of evidence that was introduced in the late 1970s by Glenn Shafer. It is a way of representing epistemic plausibilities. It developed from a sequence of works of Arthur Dempster , who was Shafer's advisor. In this formalism the best representation of chance is a belief function rather than a Bayesian probability distribution. Probability values are assigned to sets of possibilities rather than single events: their appeal rests on the fact they naturally encode evidence in favor of propositions. Shafer's framework allows for belief about propositions to be represented as intervals, taking on two values, support and plausibility, with support ≤ plausibility. Support for a hypothesis indicates the probability mass given to sets of events that are enclosed by it. Or in other words, it gives the amount of belief that directly supports a given hypothesis. Plausibility is 1 minus the masses given to sets of events whose intersection with the hypothesis results in an empty set. Again, in other words, it gives an upper bound on the belief that the hypothesis could possibly happen, i.e. it "could possibly happen" up to that value, because there was not any evidence that would contradict that hypothesis. For example, suppose we have a support of 0.5 and a plausibility of 0.8 for a proposition, say "the cat in the box is dead." This means that we have evidence that allows us to state strongly that the proposition is true with probability 0.5. However, the evidence contrary to that hypothesis (i.e. "the cat is alive") only has probability 0.2. This means that it is possible that the cat is alive, up to 0.8, since the remaining probability mass of 0.3 is essentially "indeterminate," meaning that the cat could either be dead or alive. Essentially this interval represents the level of uncertainty based off of the evidence in your system. Beliefs are combined using Dempster's rule of combination . Note that the probability masses from propositions that contradict each other can also be used to obtain a measure of how much conflict there is in a system. This measure has been used before as a criteria for clustering multiple pieces of seemingly conflicting evidence around competing hypotheses. In addition, one of the advantages of the Dempster-Shafer framework is that priors and conditionals need not be specified, unlike Bayesian methods which often map unknown priors to random variables (i.e. assigning 0.5 to binary values).
Game theory is a branch of applied mathematics that is used in the social sciences, most notably in economics, as well as in biology, engineering, political science, international relations, computer science, and philosophy. Game theory attempts to mathematically capture behavior in strategic situations, in which an individual's success in making choices depends on the choices of others. It has been expanded to treat a wide class of interactions, which are classified according to several criteria. Today, "game theory is a sort of umbrella or 'unified field' theory for the rational side of social science, where 'social' is interpreted broadly, to include human as well as non-human players (computers, animals, plants)" (Aumann 1987).
Traditional applications of game theory attempt to find equilibria in these games. In an equilibrium, each player of the game has adopted a strategy that they are unlikely to change. Many equilibrium concepts have been developed (most famously the Nash equilibrium) in an attempt to capture this idea. These equilibrium concepts are motivated differently depending on the field of application, although they often overlap or coincide. This methodology is not without criticism, and debates continue over the appropriateness of particular equilibrium concepts, the appropriateness of equilibria altogether, and the usefulness of mathematical models more generally.
Although some developments occurred before it, the field of game theory came into being with the 1944 book Theory of Games and Economic Behavior by John von Neumann and Oskar Morgenstern. This theory was developed extensively in the 1950s by many scholars. Game theory was later explicitly applied to biology in the 1970s, although similar developments go back at least as far as the 1930s. Game theory has been widely recognized as an important tool in many fields. Eight game theorists have won Nobel prizes in economics, and John Maynard Smith was awarded the Crafoord Prize for his application of game theory to biology.
Uncertainty is a term used in subtly different ways in a number of fields, including philosophy, statistics, economics, finance, insurance, psychology, sociology, engineering, and information science. It applies to predictions of future events, to physical measurements already made, or to the unknown.
Harmonic analysis is the branch of mathematics which studies the representation of functions or signals as the superposition of basic waves. It investigates and generalizes the notions of Fourier series and Fourier transforms. The basic waves are called " harmonics", hence the name "harmonic analysis." In the past two centuries, it has become a vast subject with applications in areas as diverse as signal processing, quantum mechanics, and neuroscience. The classical Fourier transform on Rn is still an area of ongoing research, particularly concerning Fourier transformation on more general objects such as tempered distributions. For instance, if we impose some requirements on a distribution f, we can attempt to translate these requirements in terms of the Fourier transform of f. The Paley-Wiener theorem is an example of this. The Paley-Wiener theorem immediately implies that if f is a nonzero distribution of compact support (these include functions of compact support), then its Fourier transform is never compactly supported. This is a very elementary form of an uncertainty principle in a harmonic analysis setting. See also classic harmonic analysis. Fourier series can be conveniently studied in the context of Hilbert spaces, which provides a connection between harmonic analysis and functional analysis.
Bifurcation theory studies and classifies phenomena characterized by sudden shifts in behavior arising from small changes in circumstances, analysing how the qualitative nature of equation solutions depends on the parameters that appear in the equation. This may lead to sudden and dramatic changes, for example the unpredictable timing and magnitude of a landslide.
Catastrophe theory, which was originated with the work of the French mathematician René Thom in the 1960s, and became very popular due to the efforts of Christopher Zeeman in the 1970s, considers the special case where the long-run stable equilibrium can be identified with the minimum of a smooth, well-defined potential function (Lyapunov function).
Small changes in certain parameters of a nonlinear system can cause equilibria to appear or disappear, or to change from attracting to repelling and vice versa, leading to large and sudden changes of the behaviour of the system. However, examined in a larger parameter space, catastrophe theory reveals that such bifurcation points tend to occur as part of well-defined qualitative geometrical structures.
Fuzzy logic is derived from fuzzy set theory dealing with reasoning that is approximate rather than precisely deduced from classical predicate logic. It can be thought of as the application side of fuzzy set theory dealing with well thought out real world expert values for a complex problem (Klir 1997).
Degrees of truth are often confused with probabilities. However, they are distinct conceptually; fuzzy truth represents membership in vaguely defined sets, not likelihood of some event or condition. For example, if a 100-ml glass contains 30 ml of water, then, for two fuzzy sets, Empty and Full, one might define the glass as being 0.7 empty and 0.3 full.
Kinesics is the interpretation of body language such as facial expressions and gestures — or, more formally, non-verbal behavior related to movement, either of any part of the body or the body as a whole.
The term was first used (in 1952) by Ray Birdwhistell, a ballet dancer turned anthropologist who wished to study how people communicate through posture, gesture, stance, and movement. Part of Birdwhistell's work involved making film of people in social situations and analyzing them to show different levels of communication not clearly seen otherwise. The study was joined by several other anthropologists, including Margaret Mead and Gregory Bateson.
Drawing heavily on descriptive linguistics, Birdwhistell argued that all movements of the body have meaning (ie. are not accidental), and that these non-verbal forms of language (or paralanguage) have a grammar that can be analysed in similar terms to spoken language. Thus, a "kineme" is "similar to a phoneme because it consists of a group of movements which are not identical, but which may be used interchangeably without affecting social meaning" (Knapp 1972:94-95).
Birdwhistell estimated that "no more than 30 to 35 percent of the social meaning of a conversation or an interaction is carried by the words." He also concluded that there were no universals in these kinesic displays - a claim disputed by Paul Ekman's analysis of universals in facial expression.
In molecular biology, genetic memory resides in the genetic material of the cell and is expressed via the genetic code used to translate it into proteins. The genetic code enables cells to decode the information needed to construct the protein molecules that make up living cells and therefore record and store a one-dimensional blueprint for all the parts that make up an organism. This blueprint or genetic memory in the form of species-specific collections of genes (genotype or genome) is passed on from cell to cell and from generation to generation in the form of DNA molecules. DNA therefore functions as both a template for protein synthesis and as a biological clock. Genetic memory can be modified by epigenetic memory, a process by which changes in gene expression are passed on through mitosis or meiosis through factors other than DNA sequence.
“‘Evidence is unfairly prejudicial when there exists a danger that marginally probative evidence will be given undue or preemptive weight by the jury.’”
...However, because it is pertinent to defendant’s assertion that she received ineffective assistance of counsel, we note that voluntary manslaughter is a killing performed in the heat of passion caused by adequate provocation and without sufficient time for a reasonable person to control that passion. People v Tierney, 266 Mich App 687, 714; 703 NW2d 204 (2005). Defendant’s theory of her case was that she acted in self-defense, not out of passion caused by reasonable provocation. A voluntary manslaughter instruction would not be compatible with defendant’s theory of her case or her own testimony. The trial court did not err by failing to give a voluntary manslaughter instruction.
The Fight or Flight Response to Stress
Stress is what happens to the body when any "pleasant" or "unpleasant" demand is placed upon it.
The human body has an inborn, "pre-wired" response for dealing with dangerous situations - it is called the "fight or flight" response. Both fighting and fleeing require the same activities on the part of the body's organs. The purpose of this response is to prepare the individual for vigorous muscular activity in response to a perceived threat. By itself, this response is normal, healthy, and adaptive. It is when the "fight/flight" response occurs too frequently or is greatly prolonged that we begin to experience the negative effects of stress.
The human nervous system has a component that works automatically (the autonomic nervous system). The autonomic nervous system has two divisions: the 'sympathetic' and 'parasympathetic' divisions. When the sympathetic division is active we experience the fight/flight response. Interestingly, when the parasympathetic division is active we experience something quite opposite from "fight/flight" --- parasympathetic activity results in a response of rest & relaxation. These two systems work to help us maintain our physical balance.
Too much "fight/flight" activity without corresponding rest and relaxation is what distress is all about.
POTENTIAL AUTONOMIC NERVOUS SYSTEM RESPONSES
|ORGAN OR FUNCTION||"FIGHT/FLIGHT" (sympathetic)||"REST/RELAXATION" (parasympathetic)|
These results demonstrate that changes in histone phosphorylation in the hippocampus are regulated by ERK/MAPK following a behavioral fear conditioning paradigm.
Numbers shape the world. Twisting them for political, business or
personal advantage is widespread - and often undetected.
Mathematics won't tell us what to do, but we think that understanding the numbers can help us deal with our own uncertainty and allow us to look critically at stories in the media.